phy 770 -- statistical mechanics 12:00 * - 1:45 p m tr olin 107

PHY 770 Spring 2014 -- Lecture 19 14/03/2014

PHY 770 -- Statistical Mechanics12:00* - 1:45 PM TR Olin 107

Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770

Lecture 19

Chap. 9 – Transport coefficients “Elementary” transport theory The Boltzmann equation

*Partial make-up lecture -- early start time

http://www.wfu.edu/~natalie/s14phy770


What is transport theory?Mathematical description of the averaged motion of particles or other variables through a host medium.

Examples of transport parameters• Thermal conductivity• Electrical conductivity• Diffusion coefficients• Viscosity


Simple transport theoryBase system – low density gas near thermal equilibrium;assume that the interaction energy is negligible compared to the kinetic energy of the particles.

23/22

For particles in volume , the probability of findinga particle with velocity between and :

( )2

mvkT

N Vv v d

k

v

mF v eT

as the average distance a particletravels between collisions. Collisions are assumed to occurrandomly. The average number of collisions per unit

Define the

lengthis

mean

given

free pat

by 1 T

/

h

.

00

/0

he probability of having no collisions in a

path length of is given byAverage distance between collisio

:ns

1

( )

( )col

r

r d

r P r e

P r rr


Estimation of mean free path for hard spheres

A

A

A

A

B

B

Collision radius dAB A

B


Estimation of mean free path for hard spheres

3 3) )

Define relative velocity:

Average relative speed between particles A and B:

( (

Relative mass:

Center of mass velocity: Tot

r A A A BAB

A Br A B AB

A B

A A B Bc

B

mA

B

B

v d v F v d v F v

mm

mmm

mm

m

v

v

v

v v

v vv

23/2 3/2

1

2

23 3

/2

1/

al mass:

exp2 2 2

8 for A

4 for A

AB A B

AB cm AB rAB ABr rAB

r ABAB

r AA

cm

A

r

m

vkT kT

kT

kTm

M m

M vMv d v d v vkT

v B

v B


dAA

2Collision volume: AAd

When there is only one type of particle (A):

2

2

21

2

AAA

A AA

d VN

n d


Self-diffusion

z

q

f

N

N

0

Density of detected particles: +..) .( (0) zT Tn

dznz n z

Assume that, in addition to geometric factors, the particle will reach the detector only if it does not have a collision.

r

/0

Recall the probability of not having a

collision in a distance : ( ) rrP er

0

2 /2

20 0

( )cossin4

r

T

vN N r dr ed d n z

r

q q f q

03

Tvz

N N n


Self-diffusion

z

q

f

N

N

r 03

-

T

D

nvN N

J

z

D

2

( , ) ( , )( , )Continuity condition: +

Self-diffusion equation:

( ) 0

( , ) ( ,

)

D r T

Tr D

Tr T

t D n tn t

tn t D n t

t

r rr J r

r r

J


The Boltzmann Equation(Additional reference: Statistical Mechanics, Kerson Huang)Assume a dilute gas of N particles of mass m in a box of volume V. In order to justify a classical treatment:

1/32

2V

mkT N

3 3

3 3

represents the number of particles in the6 dimensional

Define the distribution func

phase space a

tion ( , , ) :

( , , ) and at timebout .

( ,

, )

f t

f t dt

rd

f t

v

d rd v N

r vr v

r v

r v


The Boltzmann equation – continuedIn absence of collisions, the distribution of particles remains constant as

3 3 3 3

'

' '

( , , ) ( ', ', ')

( , , )

= ' '

, ,

mf t d f t d

f tm

t t t t t t

rd v r d v

f t t t t

Fr r r v v v v

r v r vFr v v r v

v

r

t

t+ t


The Boltzmann equation – continued

In the presence of collisions there is a diffence between thetwo distributions such that:

( , , )

( , ,

, ,

)

coll

r vcoll

f tm

ff t t t t tt

ft

tt

fm

Fr v v r v

Fv r v

The Boltzmann Equation:

( , , )r vcoll

f tm

ft t

Fv r v


The Boltzmann equation – continued

The Boltzmann Equation:

( , , )r vcoll

f tm

ft t

Fv r v

2

0

3/20 2

Linearized solution: ( , , ) 1 ( ,

where

, )

2

mvkT

f t f v h t

mfkT

v e

r v r v


Digression on two particle scattering theory (see Appendix E)

r1

r22 2

2

2

2 1

1 1

1

1

1

1

2

2

cmm m

m mM m m

m mm m

rr

r r

r R rz

x

y

2 2 21 2

1 2

2

Suppose that the interaction potential has the form( ):

( ) ( )2 2 2 2tot

V r

p p P pH V r V rm m M


Review of scattering analysis from classical mechanics class:


Scattering theory:


q

q

angleat detector into scattered is that beamincident of Area areaunit per particlesincident ofNumber

particle per target at particles detected ofNumber section cross alDifferenti

d

d

Figure from Marion & Thorton, Classical Dynamics

sin sin

qqqq

ddbb

dddbbd

dd

bdbdb


sin sin

qqqq

ddbb

dddbbd

dd

Differential cross section

?)( find can we How qb

separation largeat velocity mass reduced

: thatNote

v

bv


Note: The following is in the center of mass frame of reference.

In laboratory frame: In center-of-mass frame:

V1

m1

mtarget

v1

origin vCM

r

Also note: We are assuming that the interaction between particle and target V(r) conserves energy and angular momentum.

1vr μ

mmmm

μ

target1

target1


)(22

1

)(21

:frame mass ofcenter in theenergy ofon Conservati

2

22

2

rVrdt

dr

rVdtdE

r


)(2 2

2

rVr

E

rmin

In center of mass reference frame:


2

2

)()(:sy variable trajectoroftion Transforma

:momentumangular ofon Conservati

rddr

dtd

ddr

dtdr

rtr

dtdr

ff

f

f

f

)(22

1

)(22

1

2

22

2

2

22

rVrrd

dr

rVrdt

drE

f


)(22

1

)(22

1

2

22

2

2

22

rVrrd

dr

rVrdt

drE

f

)(2

2

/

)(2

2

for Solving

2

2

2

2

2

2

42

rVr

E

rdrd

rVr

Erddr

(r))r(

f

f

ff


)(2

2

/

2

2

2

rVr

E

rdrd

f

bE

vE

bv

221

:separation largeat tion simplificaFurther

2


),(

)(1

/

)(2

2

/

:clearsdust When the

2

2

2

2

2

2

EbErV

rb

rbdrd

rVr

E

rdrd

f

f

f


minmin /1

022

2

2

2

2

2

2

//11)(1

/),(

)(1

/

rr

mEuVub

dubdr

ErV

rb

rbEb

ErV

rb

rbdrd

f

f


r min

rf

f


sin sin 2

2qqqq

ddbb

ddbb

dd

Evaluation of scattering expression:

fq )(2 m

22//11),(

min/1

022

qf

r

mEuVub

dubEb


Relationship between scattering angle q and impact parameter b for interaction potential V(r):

ErV

rb

rdrbr )(1

/12

2

2

2

min

q

sin qq

ddbb

dd

0)(1

:where

min2

min

2

ErV

rb


Hard sphere scattering

In this case:

cos2

b D

4sin

2Dddbb

dd


The results above were derived in the center of mass reference frame; relationship between normal laboratory reference and center of mass:

Laboratory reference frame: Before After

u1 u2=0 v1

v2

yz

m1 m2

Center of mass reference frame: Before After

U1 U2

V1

V2

q

m1 m2

q


Relationship between center of mass and laboratory frames of reference

2211212211

212211

212211

mass ofcenter of Definition

vvVuuRrr

RrrR

mmmmmmmmmm

mmmm

CM

CM

CM

CM

21

22111

21

1

:caseour In

mmmm

mmm

CM

vvuV

V1

VCM

v1yqCMVUu 11 CMVVv 11

U1

u1

VCM


Relationship between center of mass and laboratory frames of reference -- continued

CMCM

CMCMCM

mmm

mm

mmm

mmm

m

VuUVUu

VuUVUuuV

121

1222

1

21

21

21111

21

1

2

:restat initially is Since

CM

CM

VVvVVv

22

11


V1

VCM

v1yq

Relationship between center of mass and laboratory frames of reference

211

11

11

11

/cossin

/cossintan

coscossinsin

mmVV

VVvVv

CM

CM

CM

qq

qqy

qyqy

VVv

For elastic scattering


Digression – elastic scattering

2

21212

22212

1121

2212

12222

12112

1

CM

CM

VmmVmVm

VmmUmUm

Also note:

0 0

21

21

22112211

CMCMmm

mmmm

VUVU

VVUU

/mm/UV/VV

mm

CMCM 2111

21

: thatSo

: thatnote Also

and

21

CM2211

UU

VVUVU

PHY 770 Spring 2014 -- Lecture 19 36

v1V1

VCM

yq

4/03/2014

Relationship between center of mass and laboratory frames of reference – continued (elastic scattering)

211

11

11

11

/cossin

/cossintan

coscossinsin

mmVV

VVvVv

CM

CM

CM

qq

qqy

qyqy

VVv

22121

21

/cos/21

/coscos :Alsommmm

mm

q

qy


Differential cross sections in different reference frames

yq

yq

yq

qy

coscos

sinsin

dd

dd

dd

dd

dd

dd

LAB

CM

LAB

CM

CM

CM

LAB

LAB

2/322121

21

22121

21

/cos/21

1cos/coscos

/cos/21

/coscos

:Using

mmmm

mmdd

mmmm

mm

q

qqy

q

qy


Differential cross sections in different reference frames – continued:

yqqy

coscos

dd

dd

dd

CM

CM

LAB

LAB

1cos/

/cos/21

21

2/322121

q

qqymm

mmmmd

dd

d

CM

CM

LAB

LAB

21 /cossintan :where

mm

qqy


Example: suppose m1 = m2

1cos/

/cos/21

21

2/322121

q

qqymm

mmmmd

dd

d

CM

CM

LAB

LAB

21 /cossintan :where

mm

qqy

20 that note

2

1cossintan :case In this

y

qyqqy

yyy cos42

CM

CM

LAB

LAB

dd

dd


Back to Boltzmann equation: If we can assume that the collisions are due to binary interactions, such that particles 1 and 2 interact:

1 2 1 2

312 2 1 2 12

2 1 2 1 1 1 2

1

2

, , ' ( ) :

( , , ) ( )

' w

'

where

ith cross section

' ( , ' , ) ( ,

'

' ' , ) ( , , ) ( , , )col

dv d f f f

f f

f t

f

ft

f f t f t f t f t

v v

r v v

r v r v r v r v

v v

v

phy 770 -- statistical mechanics 12:00 * - 1:45 p m tr olin 107

Documents

lecture 1927rminrff4032014phy

partial makeup lecture

scattering theory

distribution of particles

interaction energy

type of particle

laboratory frame

estimation of mean free