metody opisu dyfuzji wielu składników,

Metody opisu dyfuzji wielu składnikoacutew

unifikacja metody dyfuzji wzajemnej i

termodynamiki procesoacutew nieodwracalnych

Marek Danielewski

Interdisciplinary Centre for Materials Modeling

AGH Univ of Sci amp Technology Cracow Poland

Będlewo Czerwiec 2013

Diffusion equation (Fourier)

2

2

Fundamental or only numerology

t x

Diffusion equationsHeat

T t

Θ 2 T

x 2

Θ = α m2s-1

Diffusion of mass (1855)

t

Θ 2 x 2

Θ = D m2s-1

Diffusion equations

Hydrodynamics (noncompressible fluid)

υ t

Θ 2υ

x 2

Θ = ν m2s-1

Diffusion equations

Quantum mechanics

2

2t x φφ

2

i

m

2 2

2

i

m

i

2 1i

i

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m

Answerhellip P-K-C hypothesis

Economyhellip

hellip diffusion equation

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997

for a new method to determinehellip the value of derivatives

Robert C MertonHarvard University

Myron S ScholesLong Term Capital Management Greenwich CT USA

Economy amp diffusionhellipButhellip money is not conserved

Merton amp Sholes Nobel price bdquohelped inrdquohellip

hellipgrand failure in 2008

2

2t x

lt 10-35 - Planck scale- 10-35

nucleus - 10-16

atoms - 10-10

biology - 10-4

mechanics - 1 Earth - 107

cosmology ndash 1027

gt 1030

Fundamental

2

2 t x

Challengeseverywherehellip

Mechano-chemistry

Darken amp stressUniquenesshellip

Electro-chemistry

Nernst-Planck- Poisson + drift

AppliedReactive inter-diffusionhellipReal geometryhellip

Nernst-Planck-Poisson Problem Nernst-Planck-Poisson Problem

Simeacuteon DenisSimeacuteon Denis POISSON POISSON 1781 -18401781 -1840

grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F

Walther HermannWalther Hermann NERNST NERNST 1864-19411864-1941

MaxMax PLANCK PLANCK 1858-19471858-1947

div i iiz cE F

Nernst-Planck-Poisson Problem

1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck




Unsolved uniqueness quasi-stationary Problems multi-component ionic systemshellipUnsolved NPP + drifthellip

W Kucza (2009) convergehellip

Nernst-PlanckNernst-Planck

di i jj

d di i iJ c

B F

Flux is not limited to diffusionhellipFlux is not limited to diffusionhellip

d drifti i iJ J c

Bi-velocity Wagner (1933) Darken (1948) Danielewski amp Holly (Cracow gt1994)

( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt

Material reference frame (Darken 1948)Lagrange substantial material etchellipderivative

Internal reference frame (Darken 1948)Lagrange substantial materialhellip derivative

gradD

Dt t

or

material velocity = or

m

gradM

MD

Dt t

local centre of composition

M iii

c

c

gradm

mD

Dt t

local centre of mass

m iii

gradD

Dt t

local volume velocity

1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii

z FJ t x ( ) 0d drift

i iiJ t x c

We need different approachhellip

Darken

Bi-velocityhellip

Lattice sites not conserved

bdquobdquoZig-zag Roadrdquohellip Zig-zag Roadrdquohellip

to the targetto the target

~250 ag onl o y mEuler

Oumlttinger (2005) bdquosomething is missingrdquo

19th century Cauchy Navier Lameacutehellip

Stephenson (1988) drift amp m

up to 2007 only m

Brenner (2006) Fluid Mechanics Revisitedhellip

Cracow (1994) vd amp drift

Brenner in bdquoFluid Mechanics Revisitedrdquo (Physica A 2006)

1 Complemented volume fixed RF

2 Was polite to not notice

conflict between RFrsquos

hellip in our papers

150 years of diffusion equation

Defects bdquoeverywhere amp alwaysrdquohellip (1918 Frenkel)

Nonstoichiometry is a rulehellip (1933 Schottky amp Wagner)

Lattice sites are not conserved (1948 Kirkendall amp Darken)

Diffusion velocityhellip (~1900 Nernst amp Planck)

Darken problem has a unique solution (2008 Holly Danielewski amp Krzyżański)

Darken problem is self-consistent with LIT (ActaMat 2010 Danielewski amp Wierzba)

150 years of diffusion

Number of laws decreaseshellip

Complexity increaseshellip

Do we bdquostay withrdquo m ρυ q U only

B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)

Dynamics amp diffusion

1 mx f t x

Does xm depend on time ie xm(t) or xm= constI law F = 0 xm = const

B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem

Eg diffusion anddeformation stressreactions

Hopelesshellip fundamentals only

1 1 mr rf k x k x k f x x

1 11

r m

i rii

f k x k xx mk f x x

k x

Eulerrsquos theorem f (x1 xrhellip) is called homogeneous of the m-th degree in the

variables x1hellip xr if

several identities follow eg

1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1

The molar volume is the nonconserved property Buthellip is transported by components velocity field

Fundamentals II

The Liouville transport theorem

dd div d

di

i i

t t

i

ff fx x

t t

fi is a sufficiently smooth function (eg have first

derivative C1) and υi is defined on fi

Liouville

Conservation of component (fi = ci)

div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t

The Liouville theorem amp the Volume Continuity

dd div d

di ii

i i i i ii it t

cc x c x

t t

fi(tx) = bdquovolume densityrdquo = ci(tx) Ωi(tx)

ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic

The volume density conservation law orhellipequation of volume continuity at constantvolume

Overall drift velocity

drift D T tr drift d

Finally due to Liouvillethe bi-velocty method

dVolume continuity div

d i i iic

t

Conservation of mass div 0 for 1 2ii i i r

t

Momentum conservation DivD

rD

g adi

extiii

Vt

D 1 grad d

D 3ivEnergy conservation

i

mi

i

q

i i ii i

up

tJ

I

drift di i

Entropy production divDD

i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT

1 1grad grad grad grad 0

ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ

L-matrix is both as required by LITsymmetric and non - negative

Brenner 2009 Danielewski amp Wierzba 2010

bull Entropy production term is always positive

bull Bi-velecity method is consistent with LIT

Conservation laws in material reference frame

Bi-velocity vs LIT

Diffusion stress reactionsamp more

Planck-Kleinert Crystal

Reactions amp Interdiffusion Multiples

T= 150 180 200oC

t = 3h 30h 100h 7days 14dayshellip

Experiment

Cu-Sn-Ag Cu-Ag-Sn-Ni

Fabrication and vacuum annealing

020cm 300cm

350deg

Cu Ag Ni Sn

M Pawełkiewicz EMPA amp AGH

International PhD School Switzerland ndash Poland

Fabrication Sectioning


after heat treatment t=4h and T=180 C

Sn

Cu

Interdiffusion amp stress

mechano-chemistry

InterdiffusionInterdiffusion

C o5 2 w t N i t= 0F e

5 1 w t N i

D iffu s io n co u p le

Ni-Cu Ni-Fe

Model of InterdiffusionModel of Interdiffusion

tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec

tro-mechano-chemistry

d

i i

eli

chi

miB

Future

Bi-velocity methodhellip at the Planck scale

Oliver Heaviside (1850-1925)

bull Impedancebull Complex numbersbull Heaviside functionbull Maxwell reformulated

Mathematics is an experimental science and definitions do not come first but later on

I do not refuse my dinner simply because I do not understand the process of digestion


M Danielewski ldquoThe Planck-Kleinert Crystalrdquo Z Naturforsch 62a 564-568 (2007)

httpwwwquantumunivieacatresearchc60Zeilinger soccer balls diffracthellip

Soccer Balls Diffract

httpwwwquantumunivieacatresearchc60

Professor Anton Zeilinger

Experiment amp theory for C60 and C70

C60F48 world record (108 atoms) in matter interferometry

J Clerk Maxwell Phil Trans R Soc Lond 155 (1865) 459-512

Already onhellip

bdquoPlanck-Kleinert Crystalrdquo

bdquoThe assumption therefore that gravitation arises fromthe action of the surrounding medium in the way pointedout leads to the conclusion that every part of this medium possesses when undisturbed an enormous intrinsic energy and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction As I am unable to understand in what way a medium can possess such properties I cannot go any further in this direction in searching for the cause of gravitationrdquo

Physics Today [1] rarr ldquoThe persistence of etherrdquo

Statistical mechanics [2] rarr dimensions become large

quantum properties emerge

Quantum space [3] rarr analogous to crystal

Kleinert [4] rarr Einstein gravity from a defect model

[1] F Wilczek Phys Today 52 11 (1999) [2] J L Lebowitz Rev of Modern Phys 71 S347-S357 (1999) [3] M Bojowald Nature 436 920-921 (2005)[4] H Kleinert Ann Phys 44 117 (1987)

Vacuumhellip No [5]rarr ldquoThere is no information without

representationrdquo

[5] W Żurek Nature 453 (2008) 23

Volume continuity div 0c

Conservation of mass div 0c

ct

DMomentum conservation Div grad

D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I

The Planck-Kleinert Crystal rarr World Crystal (three-dimensional quasi-continuum)

- Frenkel disorder

- defects form solid solution - defects diffuse

- bdquoclassicalrdquo conservation laws

- volume continuity amp material reference frame

- double valued deformation field

divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation

Navier-Lame+ diffusion

Energy conservation

Div 2 graddiv divgrad u uσ σ

P-KC single crystal bdquosuper idealrdquo enormous intrinsic energy bounded () etchellip

div 0c

m d

Included the entropy production as a result of defect formation and diffusionhellip

L D Landau E M Lifshits ldquoFluid Mechanicsrdquo 2nd ed (Butterworth-Heinemann Oxford 1987)

+ stationary travelingamp

their combinations

Processes

1 Transverse wave

2 Longitudinal wave

3 lattice deformation (Kleinert 2003)

4 Pi diffusion (mass)

5 Heat transferhellip

Physical Quantity Unit Symbol for unit Value in SI units SI unit Reference

Lattice parameter Planck length lP 161624(12)10-35 m NIST

Poisson ratio in ideal fcc crystal 025 -Cauchy ampPoisson

Mass of particle Planck mass mP 217645(16)10-8 kg NIST

Frequency of the internal process

Inverse of the Planck time

fP = 1tP 185486(98)1043 s-1 NIST

Lameacute constant Energy density 185323719410114 kgm-1s-2 This work

Number of particles in unit cell 4 This work

National Institute of Standards and Technology Reference on Constants Units and Uncertainty httpphysicsnistgov (2006)

The physical constants (ideal regular fcc lattice)

Rem

Imm

Re Imm m mi

The energy of volume deformation field

The energy of the torsion field

divt

DDiv

D

e

t

Mass conservation

Energyconservation

Volume continuity div 0d

Gravity

[defects] asymp const at T= const

in Planck-Kleinert crystal

div m dP P e

already Newtonhellip

Redivgrad 4mMG

2where P PG l c m

Simeacuteon-Denis Poisson

116674189 10G NIST data G = 66742(10) bull 10-11

4 The ldquodark energyrdquo rarr energy of the DIPPrsquos

1 The Diffusing Interstitial Planck Particles (DIPPrsquos) = WIMPrsquos

2 DIPPrsquos create the gravitational interaction between matter

3 The ldquodark matterrdquo rarr DIPPrsquos

Remark 1 Planck length = Schwarzschild radius

2 P WIMP gravitonm m m m Higg s boson

Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion

Only transverse wave

1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2

2 = 8 grad 8 grad divdiv

P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x

Transverse waves in P-KC

Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx

Equation of the transverse wave

299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x

Full set Maxwell eqs in vacuum

analogous simple transformations

divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm

Im Imgrad grad d m mM M M MJ B eB

The energy flux

1 The process that governs de Broglie waves is the fast internal process 2 We analyze the case when the driving force of the transport (the collective Planck mass movement ie the movement of a complex of

particles showing an energy E and mass M = E c-2) is controlled by the imaginary part of mechanical potential

Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2

exp 2 div exp 2 gradm m m m

mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im

exp 2 div exp 2 grad

exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M

The physical constants at the Planck scale and

four time scalesPhysical Quantity Value in SI units SI unit Reference

Volume of PC cell 4222 002 82810-105 m3 This work ampNIST

Planck density 2062 008 6621097 kg m-3 This work amp NIST

Young modulus 4633 092 98610114 kg m-1s-2 This work

Planck mass mobility 5391 213 98210-44 s This work

Defects self-diffusion coefficient 2422 685 81610-27 m2s-1 This work

Planck constant6626 069 31110-34

6626 069 3(11)10-34 kg m2 s-1 This workNIST

Gravitational constant66742(10)10-11

667417610-11 m3 kg-1 s-2 NISTThis work

Speed of longitudinal wave 519 255 240 m s-1 This work

Speed of transverse wave299 792 153299 792 458

m s-1 This workNIST

Physical reality at the Planck scale

bull Faster than light velocity of longitudinal wavehellip Kleinert fine-tuning to make all ldquosound speedsrdquo equal orhellip

consider

bull The different velocities are related to specific force field rarr a real quantities that mark different time-scales

bull The Diffusing Interstitial Planck Particles (DIPPrsquos) rarr gravity

bull The collective behavior of the Planck particles rarr the particle Schroumldinger equation follows

3L c

bull Transverse wave equiv electromagnetic wave

bull DIPPrsquos rarr Dark Matter rarr Dark Energy

bull Waves involving temperature equiv ldquothe second soundrdquo described by Landau and Lifschitz etchellip

Conclusions

Fluxes rarr Nernst-Planck formulae

Collective behavior rarr standing wave (rdquoparticlerdquo)

Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future

bull new experimental methods

bullnew processes to predict

bull all methods developed in math and physics will be usefull

Forthcoming

elec


d

i i

eli

chi

miB

Planck-Kleinert Crystal straightforward

vs

Remark

Complexity of diffusion processes in multicomponenthellip systems

E N D

Fundamentals I

Euler - the volume amp molar volume

1 1 r rkn kn T p k n n T p

hellip homogeneous of the 1st degree in the variables n1hellip nr

The volume and molar volume

Ni = cic is molar ratio


1 1 r rN N T p k n n T p

hellip homogeneous of the 1st degree in the variables N1hellip Nr

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

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Slide 30

Slide 31

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Slide 33

Slide 34

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Slide 36

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Slide 42

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Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

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Slide 58

Slide 59

Slide 60

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Slide 65

Slide 66

Slide 67

Slide 68

Slide 69

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Slide 71

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Slide 73

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Slide 76

Slide 77

Slide 78

Slide 79

Slide 80

Slide 82

Slide 83

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Slide 86

Slide 87

Slide 88

Slide 89

Slide 91

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Slide 94

Slide 95

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Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

Diffusion equation (Fourier)

2

2

Fundamental or only numerology

t x


T t

Θ 2 T

x 2

Θ = α m2s-1


t

Θ 2 x 2

Θ = D m2s-1

Diffusion equations


υ t

Θ 2υ

x 2

Θ = ν m2s-1

Diffusion equations

Quantum mechanics

2

2t x φφ

2

i

m

2 2

2

i

m

i

2 1i

i

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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T t

Θ 2 T

x 2

Θ = α m2s-1


t

Θ 2 x 2

Θ = D m2s-1

Diffusion equations


υ t

Θ 2υ

x 2

Θ = ν m2s-1

Diffusion equations

Quantum mechanics

2

2t x φφ

2

i

m

2 2

2

i

m

i

2 1i

i

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

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Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


t

Θ 2 x 2

Θ = D m2s-1

Diffusion equations


υ t

Θ 2υ

x 2

Θ = ν m2s-1

Diffusion equations

Quantum mechanics

2

2t x φφ

2

i

m

2 2

2

i

m

i

2 1i

i

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

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Slide 39

Slide 40

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Slide 45

Slide 46

Slide 47

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Slide 49

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Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

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Slide 59

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Slide 63

Slide 64

Slide 65

Slide 66

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Slide 71

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Slide 89

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Slide 92

Slide 93

Slide 94

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Slide 96

Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


υ t

Θ 2υ

x 2

Θ = ν m2s-1

Diffusion equations

Quantum mechanics

2

2t x φφ

2

i

m

2 2

2

i

m

i

2 1i

i

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 124

Slide 125

Quantum mechanics

2

2t x φφ

2

i

m

2 2

2

i

m

i

2 1i

i

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

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Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

Quantum mechanics

2 2

22i

t m x

free particlehellip

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

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Slide 36

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Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

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Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

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Slide 59

Slide 60

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Slide 63

Slide 64

Slide 65

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Slide 71

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Slide 87

Slide 88

Slide 89

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Slide 92

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Slide 94

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Slide 96

Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

Question

Why

2

i

m


Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 125

Economyhellip









2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

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Slide 9

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Slide 120

Slide 121

Slide 124

Slide 125








2

2t x


nucleus - 10-16

atoms - 10-10

biology - 10-4



gt 1030

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

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Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

Fundamental

2

2 t x


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

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Slide 13

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Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


Mechano-chemistry


Electro-chemistry





grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 120

Slide 121

Slide 124

Slide 125



grad

div dii i

di i jj

ch elj i ij

cc

t

B F

F



div i iiz cE F


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

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Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


1( ) 2( ~ 1900)

3(~ 1960) 4

i Nernst Planck







di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

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Slide 121

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di i jj

d di i iJ c

B F


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 125


d drifti i iJ J c


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

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Slide 13

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Slide 31

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Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

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Slide 47

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Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

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Slide 59

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Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


( ) iic c t x const

Showhellip

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

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Slide 31

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Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 58

Slide 59

Slide 60

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

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Slide 69

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Slide 71

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Slide 87

Slide 88

Slide 89

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Slide 96

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Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

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Slide 108

Slide 109

Slide 110

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Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

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Slide 121

Slide 124

Slide 125

( ) iic c t x const

No stresshellip

Ωi = Ω = const

R1hellip

t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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t

d D DD

dt Dt Dt



gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

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gradD

Dt t

or


m

gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

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gradM

MD

Dt t


M iii

c

c

gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

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gradm

mD

Dt t


m iii

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 125

gradD

Dt t


1

r

i ii

i

c

c

None of them

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

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Slide 8

Slide 9

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Slide 116

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Slide 118

Slide 119

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Slide 121

Slide 124

Slide 125

( ) iic c t x const

If not

Then

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

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Slide 29

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Slide 31

Slide 32

Slide 33

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Slide 35

Slide 36

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Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

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Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 58

Slide 59

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Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

EOS Vegard law

( ) 0dii

J t x ( ) 0ii

J t x ( ) 0i ii


i iiJ t x c


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 117

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Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


Darken

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

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Slide 8

Slide 9

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Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

Bi-velocityhellip








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

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Slide 38

Slide 39

Slide 40

Slide 42

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Slide 44

Slide 45

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Slide 47

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Slide 49

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Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

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Slide 59

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Slide 101

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Slide 109

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Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125








up to 2007 only m



















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

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Slide 21

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Slide 29

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 58

Slide 59

Slide 60

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

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Slide 120

Slide 121

Slide 124

Slide 125

















B100

A100

A(x)

o

ΛΛ

t= 0+

xxmλ1(0)


1 mx f t x


B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

B100

A100

t=0+

xxmλ1(0)

a)

Λ Λ

A(x)o

A(x)tgt0b)

xλ1(tgt0) xm

c) tA = const

xλ1(t) xm

λ1(0)


Yes

1

mx const

tf t x

1 mx f t x

Central problem




1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

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Slide 9

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Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125



1 11

r m

i rii

f k x k xx mk f x x

k x




1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

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Slide 26

Slide 27

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Slide 29

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Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 58

Slide 59

Slide 60

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

Slide 67

Slide 68

Slide 69

Slide 70

Slide 71

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Slide 73

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Slide 76

Slide 77

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Slide 79

Slide 80

Slide 82

Slide 83

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Slide 86

Slide 87

Slide 88

Slide 89

Slide 91

Slide 92

Slide 93

Slide 94

Slide 95

Slide 96

Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

1 i

r iip T N N

NN

1 r ii ip T N N N

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

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Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

whole mixture 1

-th component

V

Vi i i

c

i c

Volume densities

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

From Euler theorem

dd dd d

d d di iit t

tc x c x

t t t

1


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

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Slide 13

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Slide 49

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Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

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Slide 59

Slide 60

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Slide 63

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Slide 71

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Slide 93

Slide 94

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Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


Fundamentals II


dd div d

di

i i

t t

i

ff fx x

t t



Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 58

Slide 59

Slide 60

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

Slide 67

Slide 68

Slide 69

Slide 70

Slide 71

Slide 72

Slide 73

Slide 74

Slide 75

Slide 76

Slide 77

Slide 78

Slide 79

Slide 80

Slide 82

Slide 83

Slide 84

Slide 86

Slide 87

Slide 88

Slide 89

Slide 91

Slide 92

Slide 93

Slide 94

Slide 95

Slide 96

Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

Liouville


div

dd d 0

d i

t

i i

t

icc x

tc x

t

div 0ii i

cc

t


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 58

Slide 59

Slide 60

Slide 62

Slide 63

Slide 64

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Slide 121

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Slide 125


dd div d

di ii

i i i i ii it t

cc x c x

t t


ddiv d

d i i iit

c xt

1 1

div 0i i ii

constc

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 121

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Slide 125

div 0i i iic






d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

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Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125





d i i iic

t


t


rD

g adi

extiii

Vt

D 1 grad d


i

mi

i

q

i i ii i

up

tJ

I

drift di i


i

ii

q

isp A

st

JT


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

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Slide 51

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Slide 101

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Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


i

ii

q

isp A

st

JT


ms d q di i i

i i i i i ii i i

pA u J

T T T T

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

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Slide 19

Slide 20

Slide 21

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Slide 38

Slide 39

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Slide 42

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Slide 44

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Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

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Slide 59

Slide 60

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

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Slide 71

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Slide 77

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Slide 83

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Slide 89

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Slide 92

Slide 93

Slide 94

Slide 95

Slide 96

Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125

11 12

21 22

grad ln

gradM

L Lq T

L LJ






Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


Bi-velocity vs LIT




T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


T= 150 180 200oC


Experiment



020cm 300cm

350deg

Cu Ag Ni Sn






Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

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Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125





Sn

Cu


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

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Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125


mechano-chemistry



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

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Slide 51

Slide 52

Slide 53

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Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125



5 1 w t N i


Ni-Cu Ni-Fe


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 125


tgt0

Diffusion couple

c(tx)

DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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DDiv grad

De p extV

t

B Wierzba 2008

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

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Slide 121

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Slide 125

elec


d

i i

eli

chi

miB

Future















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

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Slide 101

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Slide 103

Slide 104

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Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125















Already onhellip










representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

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Slide 89

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Slide 93

Slide 94

Slide 95

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Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125









representationrdquo

[5] W Żurek Nature 453 (2008) 23



ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

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ct


D m

mextV

t

DI law Grad div

D qJt

ddiv

d i i iic

t

div 0 for 1ii i

cc i r

t

DDiv grad

Di

extiii

Vt

D 1 grad div

D 3i

ii i i i qi i

p Jt

I


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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Slide 121

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Slide 125


- Frenkel disorder





divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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Slide 121

Slide 124

Slide 125

divt

DDiv

D m

m

t

DDiv

D

e

t

Volume continuity

Mass conservation


Energy conservation



div 0c

m d




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125




their combinations

Processes

1 Transverse wave

2 Longitudinal wave










fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 4

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Slide 121

Slide 124

Slide 125







fP = 1tP 185486(98)1043 s-1 NIST





Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 120

Slide 121

Slide 124

Slide 125

Rem

Imm

Re Imm m mi



divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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divt

DDiv

D

e

t

Mass conservation

Energyconservation


Gravity



div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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div m dP P e


Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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Redivgrad 4mMG

2where P PG l c m


116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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116674189 10G NIST data G = 66742(10) bull 10-11







Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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Electromagnetism

Zero diffusion

So far

042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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042223

Dgraddiv divgrad

DL L

P PP Pm f m ft

σ σNavier-Lame

amp no diffusion


1) 0d

2) ρ = ρ0 const

3) div 0

3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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3 3

2


P P

P P

L Ll lm mt

xx + x

2

2

2 2 = rot rotLc ct

x

x + x


Equivalent form

2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









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2

3 1

2

2 = rotr

8

ot1

km299 792 s

P PLc l

t

m

c

xx


299 792 5 (NIST)

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 121

Slide 124

Slide 125

= rotrotm Lt

x

B

t

0

B

t

rotE

mB

22

rot rotLP

P

ff

x

20

1 L

Pf

2rotPfE x



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125



divt

DDiv

D

e

t

Mass conservation

EnergyConservation

andhellip

Quantum mechanics

Re Imm m mi

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 125

Imm


The energy flux



Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 3

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Slide 117

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Slide 120

Slide 121

Slide 124

Slide 125

Im

div

gradd mM M

ee

t

J eB

Imdiv grad mM

eB e

t

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

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Slide 125

Im

0Re Im 0

div grad

exp 22

mM

m m

M P P

eB e

t

e i

B m M B

Re Im Re ImIm2 2


mP Pi im B

t c M c

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

Slide 4

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Slide 125

Re Im Re ImIm2 2

2Re Im


exp

m m m mmP P

m m

i im B

t c M c

i c

2

2Imdiv grad mP Pm B

t M

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

Slide 3

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Slide 117

Slide 118

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Slide 121

Slide 124

Slide 125

2

div grad 2P Pm B c

Mi E

t

22 P Ph m B c

div4

grad i Et

h

M














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125














m s-1 This workNIST



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Slide 51

Slide 52

Slide 53

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Slide 71

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Slide 78

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Slide 82

Slide 83

Slide 84

Slide 86

Slide 87

Slide 88

Slide 89

Slide 91

Slide 92

Slide 93

Slide 94

Slide 95

Slide 96

Slide 97

Slide 98

Slide 99

Slide 100

Slide 101

Slide 102

Slide 103

Slide 104

Slide 106

Slide 108

Slide 109

Slide 110

Slide 111

Slide 112

Slide 113

Slide 114

Slide 115

Slide 116

Slide 117

Slide 118

Slide 119

Slide 120

Slide 121

Slide 124

Slide 125



consider




3L c




Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


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Fundamentals I









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Conclusions



Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

Slide 2

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Conclusions

Multi-phase and

multi-component

Today in R1

Tomorrowhellip R3

Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Future




Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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Forthcoming

elec


d

i i

eli

chi

miB


vs

Remark


E N D

Fundamentals I









Slide 1

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vs

Remark


E N D

Fundamentals I









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E N D

Fundamentals I









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Fundamentals I









Slide 1

Slide 2

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metody opisu dyfuzji wielu składników,

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