advanced molecular dynamics velocity scaling andersen thermostat hamiltonian & lagrangian...

Department of Chemical Engineering

Advanced Molecular Dynamics

Velocity scaling

Andersen Thermostat

Hamiltonian & Lagrangian Appendix A

Nose-Hoover thermostat


Naïve approachVelocity scaling

Do we sample the canonical ensemble?

2

1

3 1 1

2 2

N

B ii

k T mvN

req

Ti i

Tv v


Maxwell-Boltzmann velocity distribution

3 2

2exp 22

P p p mm

23

1dp exp 2 dr exp r

!N N N

NVT iNQ p m U

h N

Partition function

2 2dpp P p p

3m

3 24 2d 4 exp 2

2p p p m

m


3 2

2exp 22

P p p mm

3 2

2 2 4 2dp d 4 exp 22

p P p p p p p mm

3m

2

4 4dp 15m

p P p p

Fluctuations in the momentum:

2

22 4 2

2 22 2

pp p

p p

2 2

2

15 3 2

33

m m

m

Fluctuations in the temperature

2 22

2 2B

B Bk T

B B

k T k T

k T k T

2

3N

2 2

1

1 1

3 3

N

B ii

k T p m N pN Nm

22 2

21

1

3

N

B ii

k T pmN

4 2 2

21 1 1

1

3

N N N

i i ji i j

j i

p p pmN

24 2

2

11

3N p N N p

mN

224 2 22

2 22

1Bk T

B

N p N N p N p

k T N p

24 2

22

1 2

3

p p

N Np


Andersen thermostat

Every particle has a fixed probability to collide with the Andersen demon

After collision the particle is give a new velocity

3 2

2exp 22

P v mvm

The probabilities to collide are uncorrelated (Poisson distribution)

; expP t v vt


i 2 i 2ip prL t L tL te e e

Velocity Verlet:

i 2: 0

2pL t t

e t tm

v v f

i : 2rL te t t t t t t r r v

i 2: 2

2pL t t

e t t t t tm

v v f


Andersen thermostat: static properties


Andersen thermostat: dynamic properties


Hamiltonian & Lagrangian

The equations of motion give the path that starts at t1 at position x(t1) and end at t2 at position x(t2) for which the action (S) is the minimum

2

1

dt

k p

t

S t U U

t

x

t2t1

S<S

S<S


Example: free particleConsider a particle in vacuum:

2

0

1

2

p

k

U

U mv

2 2

1 1

21d d

2

t t

k p

t t

S t U U t mv

av( )v t v t

2

1

2

av

1d

2

t

t

S m t v t

2 2

1 1

2

av av

1d d

2

t t

t t

S m tv t m t t

2 2

1 1

av d dt t

t t

v tv t t

2

1

av2 1

1d

t

t

t v tt t

2

1

av av2 1

1d

t

t

v v t tt t

2

1

d 0t

t

t t

2

1

2

av

1d

2

t

t

S m t t

Always > 0!!η(t)=0 for all t

v(t)=vav


Calculus of variation ( )x t x t t

S S S t True path for which S is minimum

η(t) should be such the δS is minimal

At the boundaries:η(t1)=0 and η(t2)=0

2

1

dt

k p

t

S t U U

221 1( )

2 2

dmv t m x t t

dt

212

2m x t x t t

dU x t

U x t U x t tdx

η(t) is small

Department of Chemical Engineering A description which is independent of the coordinates

2

1

212

2

t

t

dU x tS dt m x t x t t U x t t

dx

2

1

t

t

dU x tS dt mx t t

dx

This term should be zero for all η(t) so […] η(t)

2

1

t

t

dU x tS dt mx t t t

dx

If this term 0, S has a minimum

dU x tmx t

dx Newton

Integration by parts

d t h t

t h t t h tdt

d t h t

dt dt t h t dt t h tdt

1

2

t

tdt t h t t h t dt t h t

2

1

2

1

tt

tt

m dtx t t mx t t dtmx t t

Zero because of the boundaries η(t1)=0 and η(t2)=0


LagrangianCartesian coordinates (Newton) → Generalized coordinates (?)

,x x ,q q

, k pq q U q U q L

2 2

1 1

d , d ,t t

t t

S t x x t q q L L

, k px x U x U x LLagrangian

q t q t t

q t q t t

ActionThe true path plus deviation


, k pq q U q U q L q t q t t

q t q t t

, ,, ,

q q q qq q q q t t

q q

L L

L L

Desired format […] η(t)

2

1

, ,d

t

t

q q q qdS t t

dt q q

L L

2

1

d ,t

t

S t q q L

2

1

, ,d

t

t

q q q qS S t t t

q q

L L

, ,q q q qd

dt q q

L L

Conjugate momentum ,

q

q qp

q

L ,q

q qp

q

L

Equations of motion

Should be 0 for all paths

Partial integration

1

2 2

1 12

, , ,d d

tt t

t tt

q q q q q qdt t t t t

q q dt q

L L L

2 2

1 1

, ,d d

t t

t t

q q q qdt t t t

q dt q

L L

Lagrangian equations of motion


Newton?

Conjugate momentum

,x

x xp mx

x

L

, px

U xx xp F

x x

L

, k pq q U q U q L

21,

2 px x mx U x L

, ,q q q qd

dt q q

L L

Valid in any coordinate system: Cartesian


PendulumEquations of motion in terms of l and θ

sin

cos

x l

y l

2 2 2 21 1

2 2KU m x y ml

1 cosPU l

, , , k pl l U U L

Conjugate momentum

2,

p ml

L

,sinpU

p l

L

sin

ml


Lagrangian dynamics

We have:

2nd order differential equation

Two 1st order differential equations

,q q q

,q p q p Change dependence:

, ,q q q p

With these variables we can do statistical thermodynamics


Legrendre transformation

1 2 3, , , , nf f x x x x

1

dn

i ii

f u x

ii

fu

x

1

n

i ii r

g f u x

1 1, , , , ,r r ng g x x u u

We have a function that depends on and we would like

1 2 3, , , , nx x x x 1 1, , , , ,r r nx x u u

Example: thermodynamics

d d - dE T S p V

( , )E E S V

We prefer to control T: S→T

Legendre transformation

d d dF E TS

,F F T V

d d dVF S T p

Helmholtz free energy


Hamiltonian ,q qL

,q qp

q

L

, ,q p qp q q H L

d , d d ,q p qp q q H L

, ,d d d d

q q q qq p p q q q

q q

L L

,q qp

q

L

d dp q q p

d , d dq p q pq p

H HH

qp

H

pq

H

Hamilton’s equations of motion

, ,q q q p


Newton?

Conjugate momentum

,x xp mx

x

L

21,

2 px x mx U x L

, ,x p xp x x H LHamiltonian

2

21

2 p

pmx U x

m

21

2 p

pU x

m

px

p m

H

pU xp

x x

H


Nosé thermostat

2 2 2Nose

1

1 1ln

2 2

NN

ii

gms r U r Qs s

L

Extended system 3N+1 variables

Associated mass

Lagrangian

Hamiltonian

Nose1

,N

i i si

r p sp x x

H L

Conjugate momentum

2i i

i

p ms rr

L

sp Qss

L

2

Nose 21

ln2 2

NNi s

i

p p gU r s

ms Q

H

Nose ', ln2

sp gp r s

Q H

'p p s


2

31d d ' d d ', ln

! 2N N N s

s

p Lp p r ss p r s E

N Q

H

Nosé and thermodynamics

2

Nose Nose ', ln2

sp Lp r s

Q H H

Nose Nose

1d d d d

!N N

sQ p p r s EN

H

3 1 ',21

d d ' d d!

spN E p rL QN N

s

sp p r s s e

N L

H

3 1',

21d d ' d

!

sN pE p r

L QN Nsp p r e

N L

H

3 1',

d ' d!

Np rN N L

Cp r e

N

H

',d ' d!

p rN NCp r e

N H

3 1L N

Gaussian integral

Constant plays no role in thermodynamics

Recall

MD

MC

1d d ,

!N N N N

NVEQ p r E r pN

H

1d d exp ,

!N N N N

NVTQ p r r pN

H

Delta functions

d ' dsh s h s h s h s 0ds s s

0 0h s

0d ' dsh s h s s s s

0

'

s sh s

h s

2

', ln2

sp Lh s p r s E

Q H

1'

gh s

s

2

0 exp ',2

sps E p r

L Q

H

2

', ln2

sp Lp r s E

Q

H

2

',2

spE p r

L Qss e

L

H

0

'

s sh s

h s


Equations of Motion 2 2 2

Nose1

1 1ln

2 2

NN

ii

gms r U r Qs s

L

Lagrangian

Hamiltonian

Conjugate momenta

sp Qss

L2

i ii

p ms rr

L

2 2

Nose 21

ln2 2

NNi s

i

p p gU r s

ms Q

H

Equations of motion:

Nose2

d

di i

i

r p

t p ms

H

Nosed

d

N

i

i i

U rp

t r r

H

Nosed

ds

s

ps

t p Q

H

2Nose

2

d 1

ds ip p g

t s s ms

H


Nosé Hoover

advanced molecular dynamics velocity scaling andersen thermostat hamiltonian & lagrangian...

Documents

cartesian slide

deviation slide

av slide

temperature slide

small slide

partial integration

boundaries t

nosehoover thermostat