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Thermal Physics 2019 Lecture Note 101

7 II.

Particle Picture vs. Field Picture

N.G.van Kampen,

“Stochastic Processes in Physics and

Chemistry”

(North-Holland, 1981).

Langevin

md

dt~v(t) = −γ~v − ~∇U + ~Fr

d

dt~r(t) = ~v

(7–103)

trajectory

ensemble average

(~x, ~v) P (~x,~v; t)

Langevin Page 23

Fokker-Planck

7.1 One-Dimensional System

t∫ +∞

−∞

P (s; t)ds = 1

normalization

s P (s; t)

t → t+∆t

P (s; t+∆t) =

∫

T (s, s′; ∆t)P (s′; t)ds′ (7–104)

T (s, s′; ∆t) ∆t s′ t

T (s, s′; t,∆t)s transition probability

T (s, s′; ∆t)

lim∆t→0

∫

T (s, s′; ∆t)ds = 1 ∆t (7–105)

lim∆t→0

∫

(s− s′)T (s, s′; ∆t)ds = A(s′)∆t → (7–106)

lim∆t→0

∫

(s− s′)2T (s, s′; ∆t)ds = B(s′)∆t → (7–107)

lim∆t→0

∫

(s− s′)nT (s, s′; ∆t)ds = O((∆t)2)for n ≥ 3 T long tail (7–108)


∆t → 0

∂P (s; t)

∂t= −

∂

∂s[A(s)P (s; t)] +

1

2

∂2

∂s2[B(s)P (s; t)] (7–109)

P (s; t) stochastic

FP

Smoluchowski

second Kolmogorov

Adriaan Daniel Fokker (1887–1972)

Max Karl Ernst Ludwig Planck (1858–

1947)

1918

differential equation Fokker-Planck (FP)

• drift term

transport term, convection term

•

diffusion term fluctuation term

Fokker-Planck

∂P (s; t)

∂t= lim

∆t→0

1

∆t[P (s; t+∆t)− P (s; t)]

R(s) s

∂

∂t

∫

R(s)P (s; t)ds =∂

∂t〈R(s)〉

R

lim∆t→0

1

∆t

∫

R(s)[P (s; t+∆t)− P (s; t)]ds

= lim∆t→0

1

∆t

∫

R(s)

[∫

P (s′; t)T (s, s′; ∆t)ds′ − P (s; t)

]

ds

= lim∆t→0

1

∆t

[∫ ∫

R(s)P (s′; t)T (s, s′; ∆t)ds′ds−

∫

R(s′)P (s′; t)

]

s s′

= lim∆t→0

1

∆t

∫

P (s′; t)

[∫

R(s)T (s, s′; ∆t)ds−R(s′)

]

ds′

R(s) s Taylor

R(s) = R(s′) + (s− s′)R′(s′) +

1

2(s− s

′)2R′′(s′) +1

3!(s− s

′)3R′′′(s′) + · · ·

T (s, s′; t) (7–105)–(7–108)

lim∆t→0

1

∆tP (s′; t)

[∫

R(s)T (s, s′; ∆t)ds−R(s′)

]

ds′

=

∫

P (s′; t)[

R′(s′)A(s′) +

1

2R

′′(s′)B(s′)]

ds′

=

∫

R(s′)

[

−∂

∂s′[A(s′)P (s′; t)] +

1

2

∂2

∂s′2[B(s′)P (s′; t)]

]

ds′

lims′→±∞

R(s′) = 0

∫

R(s)∂

∂tP (s; t)ds =

∫

R(s)

[

−∂

∂s[A(s)P (s; t)] +

1

2

∂2

∂s2[B(s)P (s; t)]

]

ds

∫

R(s)

[

∂

∂tP (s; t) +

∂

∂s[A(s)P (s; t)]−

1

2

∂2

∂s2[B(s)P (s; t)]

]

ds = 0

R(s) [ ] Fokker–Planck


7.2 — Brown :

Example—Velocity Distribution of a Brownian Particle

Brown

s v Langevin

mdv

dt= −γv + Fr(t) (7–110)

∆v ≡ v(t+∆t)− v(t)

= −γ

mv(t)∆t+

∫ t+∆t

t

Fr(t′)

mdt′ (7–111)

A = lim∆t→0

〈∆v〉

∆t= −

γ

mv (7–112)

B = lim∆t→0

〈(∆v)2〉

∆t=

2kBTγ

m2(7–113)

B (1–9)

⟨

(∆v)2⟩

=

∫

t+∆t

t

∫

t+∆t

t

⟨

Fr(t′)

m

Fr(t′′)

m

⟩

dt′dt

′′

=1

m2

∫

t+∆t

t

∫

t+∆t

t

Cδ(t′ − t′′)dt′dt′′

=C

m2∆t

=2kBTγ

m2∆t

(2–62)

FP

∂

∂tP (v; t) =

γ

m

∂

∂v[vP (v; t)] +

kBTγ

m2

∂2

∂v2P (v; t) (7–114)

∂P∂t

= 0 P eq(v) (7–114)

γ

m

d

dv[vP eq(v)] +

kBTγ

m2

d2

dv2P eq(v) = 0 (7–115)

v

vP eq(v) +kBT

m

d

dvP eq(v) = const. (7–116)


limv→±∞

P eq(v) = 0

const. = 0 (7–116)

dP eq(v)

P eq= −

m

kBTvdv

P eq(v) ∝ exp

[

−mv2

2kBT

]

= exp

[

−Ekin

kBT

]

(7–117)

Boltzmann

7.3 : FP Equation in External Field

~F (~r) = −~∇U(~r)

~r ~v

md

dt~v = −γ~v − ~∇U(~r) + ~Fr(t)

d

dt~r = ~v

(7–118)

(~r,~v) P (~r,~v; t)

FP ∂2

∂~r2

⟨

(∆~r)2⟩

=⟨

~v2⟩

(∆t)2

∆t

∂

∂tP (~r,~v; t) = −

∂

∂~r[~vP (~r,~v; t)]

−∂

∂~v

[(

−γ

m~v −

1

m~∇U(~r)

)

P (~r,~v; t)

]

+kBTγ

m2

∂2

∂~v2P (~r,~v; t) (7–119)

Kramers

Hendrik Anthony Kramers

(1894–1952)

WKB

Kramers–Kronig(7–119) Boltzmann

P eq(~x,~v) ∝ exp

−

m

2v2 + U(~r)

kBT

(7–120)

(7–120) (7–119)

5.1

U(x)

kBT= 0.001x4 − 0.08x2 − 0.1x (7–121)

7–44 (x, v) = (−6, 0)

-2

-1

0

1

2

3

-10 -8 -6 -4 -2 0 2 4 6 8 10

Po

ten

tia

l E

ne

rgy

Position

()


#include <stdio.h>#include <math.h>

#define MAX_STEP 100000#define SAVE_STEP 100#define DT 0.01#define GAMMA 1.0

#define NX_CELL 40#define NV_CELL 20#define XMIN -10.0#define XMAX 10.0#define VMIN -5.0#define VMAX 5.0

double prob[NX_CELL][NV_CELL];double dprob[NX_CELL][NV_CELL];double dx,dv;

double potential(double x){

return 0.001*(x*x*x*x)-0.08*(x*x)-0.1*x;}double force(double x){

return -4*0.001*(x*x*x)+2*0.08*(x)+0.1;}void initial( ){

int ix, iv;

for (ix=0;ix<NX_CELL;ix++) {for (iv=0;iv<NV_CELL;iv++) {

prob[ix][iv]=0.0;} }

}void output(int nstep){

int ix, iv;char cdum[100];FILE *fout;

sprintf(cdum,"fp-%6.6d.dat",nstep);fout=fopen(cdum,"w");for (ix=0;ix<NX_CELL;ix++) {for (iv=0;iv<NV_CELL;iv++) {

fprintf(fout,"%6.2f %6.2f %9.5f",ix*dx+XMIN,iv*dv+VMIN,prob[ix][iv]);}

fprintf(fout,"\n");}fclose(fout);

}//-------------------------------------------------int main( ){

int ix, iv, nstep;double dp;double x, v;double f, d;

dx=(XMAX-XMIN)/NX_CELL;dv=(VMAX-VMIN)/NV_CELL;

initial( );x=-6.0; v=0.0;ix=(x-XMIN)/dx;iv=(v-VMIN)/dv;prob[ix][iv]=1.0;

nstep=0;output(nstep);

for (nstep=1;nstep<=MAX_STEP;nstep++) {

for (ix=1;ix<NX_CELL-1;ix++) {x=dx*ix+XMIN;f=force(x);

for (iv=1;iv<NV_CELL-1;iv++) {v=dv*iv+VMIN;

d =-v*(prob[ix+1][iv]-prob[ix-1][iv])/(2*dx);

d+=GAMMA*prob[ix][iv];d+=(GAMMA*v-f)*(prob[ix][iv+1]-prob[ix][iv-1])/(2*dv);d+=GAMMA*(prob[ix][iv+1]-2*prob[ix][iv]+prob[ix][iv-1])/(dv*dv);dprob[ix][iv]=d;

} }for (ix=1;ix<NX_CELL-1;ix++) {for (iv=1;iv<NV_CELL-1;iv++) {

prob[ix][iv] += dprob[ix][iv]*DT;if (prob[ix][iv]<0.0) prob[ix][iv]=0.0;

} }printf("%d\n",nstep);if (nstep%SAVE_STEP==0) output(nstep);

}}

7–44: Fokker–Planck

explicit algorithm

DT

000000 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

000100 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

000300 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

001000 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

003000 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

010000 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

030000 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

100000 step

-4

-2

0

2

4

v

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

7–45:

d =-v*(prob[ix+1][iv]-prob[ix-1][iv])/(2*dx);

d+=GAMMA*prob[ix][iv];d+=(GAMMA*v-f)*(prob[ix][iv+1]-prob[ix][iv-1])/(2*dv);d+=GAMMA*(prob[ix][iv+1]-2*prob[ix][iv]+prob[ix][iv-1])/(dv*dv);dprob[ix][iv]=d;

prob[ix][iv] += dprob[ix][iv]*DT;


7.4 : Generalized Diffusion Equation

5.2

viscosity-dominant

d~r

dt=

1

γ

[

−~∇U(~r) + ~Fr(t′)]

(7–122)

lim〈∆~r〉

∆t= −

1

γ~∇U(~r) (7–123)

lim

⟨

∆~r2⟩

∆t=

2kBT

γ(7–124)

P (~r; t) Fokker-Planck

∂P (~r; t)

∂t= ~∇

[

1

γ

(

~∇U)

P

]

+kBT

γ

(

~∇)2

P (7–125)~∇ = grad =~i

∂

∂x+~j

∂

∂y~k

∂

∂z

( nabla)

∆ =(

~∇)2

=∂2

∂x2+

∂2

∂y2+

∂2

∂z2

Laplacianflux ~J = −1

γ

(

~∇U)

P −kBT

γ~∇P (7–126)

∂P (~r; t)

∂t= −~∇ ~J (7–127)

equation of continuity P (~r, t)∫

Pd~r = const. D ≡kBT

γ

2 D =kBT

γ(2–65)

generalized diffusion equation

∂P (~r; t)

∂t= ~∇

[

1

γ

(

~∇U)

P

]

+D∆P (7–128)

steady state∂P

∂t= 0

~∇

[

1

γ

(

~∇U)

P

]

+kBT

γ

(

~∇)2

P = 0

1

γ

(

~∇U)

P +kBT

γ~∇P = const.

P = 0

Boltzmann :

P eq(~r) ∝ exp

[

−U(~r)

kBT

]

(7–129)

7–47

TEMPERATURE

7–47

∆t

DT

implicit algorithm


// 1-D Diffusion in Potential Field#include <stdio.h>#include <math.h>

#define MAX_STEP 2000#define SAVE_STEP 10double DT = 0.001;double GAMMA = 1.0;double TEMPERATURE = 1.0;

#define NX_CELL 100double XMIN = -5.0;double XMAX = 5.0;

double prob[NX_CELL];double dprob[NX_CELL];double dx;

double potential(double x){

return 0.5*x*x;}double force(double x){

return -x;}

void initial( ){

int ix;

dx=(XMAX-XMIN)/NX_CELL;

for (ix=0;ix<NX_CELL;ix++) {prob[ix]=0.0;

}prob[5]=1.0/dx; //

}

void output(int nstep){


printf("%d\n",nstep);sprintf(cdum,"diffusion.%6.6d",nstep);fout=fopen(cdum,"w");

for (ix=0;ix<NX_CELL;ix++) {fprintf(fout,"%6.2f %9.5f\n",ix*dx+XMIN,prob[ix]);

}

fclose(fout);}//--------------------------------------------------------int main( ){

int ix, nstep;double dp;double x,xm,xp;double f, d;

initial( );


for (nstep=1;nstep<=MAX_STEP;nstep++) {

for (ix=1;ix<NX_CELL-1;ix++) { //x = dx*ix+XMIN;xm = x-dx;xp = x+dx;dprob[ix] = -(force(xp)*prob[ix+1]-force(xm)*prob[ix-1])/(2*dx)/GAMMA;dprob[ix]+= (prob[ix+1]-2*prob[ix]+prob[ix-1])/(dx*dx)*(TEMPERATURE/GAMMA);

}for (ix=1;ix<NX_CELL-1;ix++) { //

prob[ix] += dprob[ix]*DT;if (prob[ix]<0.0) prob[ix] = 0.0;

}

if (nstep%SAVE_STEP==0) output(nstep); //}return 0;

}

7–46:

U(x) = 12x

2

0

2

4

6

8

10

-5 0

P

x

T=1.0

0 step10 step20 step50 step

100 step200 step500 step

1000 step2000 step

U(x)

0

2

4

6

8

10

-5 0

P

x

T=0.25

0 step10 step20 step50 step

100 step200 step500 step

1000 step2000 step

U(x)

7–47: U(x) = 12x

2

dprob[ix] = -(force(xp)*prob[ix+1]-force(xm)*prob[ix-1])/(2*dx)/GAMMA;dprob[ix]+= (prob[ix+1]-2*prob[ix]+prob[ix-1])/(dx*dx)*(TEMPERATURE/GAMMA);

prob[ix] += dprob[ix]*DT;


7.5 H :

Entropy in Non-equilibrium Systems

( )

(7–128)

P (~r, t)

7.5.1 : Boltzmann’s Entropy

S =d−Q

T(7–130)

degeneracy W (E)

BoltzmannWien Boltzmann

S = kB logW (7–131)

W

p = 1W

p

−kB log pp −kB log p

informatics e 2

on off

−kB logep = −kB

log2 p

log2 e

= −kB

log2 elog2 p

kB

log2 ebit

1996 ISO

“ bit” “ shannon

” JIS

random variable X event x

probability p(x)

x 0 ≤ p(x) ≤ 1 (7–132)

∑

x∈X

p(x) = 1 (7–133)

normalization x

x = −kB log p(x) (7–134)

X


S = −∑

x

p(x)kB log p(x) = −kB∑

x

p(x) log p(x) (7–135)

Claude Elwood Shannon (1916–2001)

1948

“AMathematical

Theory of Communication”

Wiener, von

Neuman, Turing

1985

Wikipedia

(2012)

16

S = −kB∑ 1

6log

1

6= kB log 6

S′ = −kB

[

1

6log

1

6+

2

6log

2

6+

1

6log

1

6+

1

6log

1

6+

1

6log

1

6+

0

6log

0

6

]

= −kB

[

4

6log

1

6+

2

6log

2

6

]

= −kB

[

log1

6+

2

6log 2

]

S − S′ =

2

6kB log 2 > 0

N (On, Off) On, Off12

S = −NkB∑ 1

2log

1

2= NkB log 2

2 kB

log2 e

N bit N On, Off

N bit redundancy

noise

p 1− p

S = −kB [p log p+ (1− p) log(1− p)]

p = 12

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1

Entropy

p

-x*log(x)-(1-x)*log(1-x)

[ p(x) p(x)

],

a ≤ x ≤ b p(x)

p(x) = const.


p(x) functional kB = 1

S[p(x)] =

∫

b

a

p(x) log p(x)dx

∫

b

a

p(x)dx = 1

S[p]

Lagrange

δ

δp(x)

[∫

b

a

p(x) log p(x)dx− λ

[∫

b

a

p(x)dx− 1

]]

= 0

differential of a functional

C.F. :

1997 1.5

δF [φ]

δφ(ξ)= lim

h→0

1

h(F [φ(x) + hδ(x− ξ)]− F [φ(x)])

log p(x) + 1− λ = 0

p(x) p(x) = const.

7.5.2 H : H Function and its Evolution

P (~r; t) Fokker-Planck

(7–128)∂

∂tP (~r; t) = −~∇

[

~A(~r)P (~r; t)]

+∆ [D(~r)P (~r; t)]

= ~∇[

− ~A(~r)P (~r; t) + ~∇ [D(~r)P (~r; t)]]

(7–136)

P eq(~r) 7.5.3 f(x) = x log x

N.G.van Kampen,

“Stochastic Processes in Physics and

Chemistry”

(North-Holland, 1981)

0 ≤ x < ∞ f(x)

f ′′(x) > 0 (7–137)

f(x) x = PP eq functional H[P ]

H[P ] =

∫

P eq(~r) · f

(

P (~r; t)

P eq(~r)

)

d~r (7–138)

H P (~r; t)

Boltzmann originally used the letter

E for quantityH; most of the literature

after Boltzmann uses the letter H.

(Wikipedia)


dH

dt=

∫

P eq · f ′ ·1

P eq

∂P

∂td~r

=

∫

f ′ · ~∇[

− ~A(~r)P (~r; t) + ~∇ [D(~r)P (~r; t)]]

d~r (7–136)

= −

∫

[

− ~A(~r)P (~r; t) + ~∇ [D(~r)P (~r; t)]]

~∇f ′d~r (7–139)

P → 0 ~∇f ′

~∇f ′ = f ′′ · ~∇

(

P (~r; t)

P eq(~r)

)

= f ′′ ·~∇P · P eq − P · ~∇P eq

(P eq)2 (7–140)

P eq Fokker-Planck (7–136) =

− ~A(~r)P eq(~r) + ~∇ [D(~r)P eq(~r)] = 0 (7–141)

~∇P eq =1

D(~r)

[

~A(~r)− ~∇D(~r)]

P eq (7–142)

(7–140)

~∇f ′ = f ′′ ·

~∇P · P eq − P ·[

1D(~r)

[

~A(~r)− ~∇D(~r)]

P eq]

(P eq)2

= f ′′ ·

[

− ~A(~r)P +D(~r)~∇P + P ~∇D(~r)]

P eq

D (P eq)2

= f ′′ ·− ~A(~r)P + ~∇ [D(~r)P (~r; t)]

D(~r)P eq(7–143)

(7–139)

dH

dt= −

∫

f ′′ ·1

D(~r)P eq·[

− ~A(~r)P + ~∇ [D(~r)P (~r; t)]]2

d~r (7–144)

f ′′ > 0, P eq ≥ 0 D(~r) > 0

D

D < 0dH

dt≤ 0 (7–145)

P = P eq

L. E.

Boltzmann 1872

Boltzmann

J. W. Gibbs

H H–theorem

f(x) (7–138) H

P → P eq H

H[P → P eq] =

∫

P eq(~r)f(1)d~r = f(1) (7–146)

P eq


7.5.3 : Non-Equilibrium Entropy

(7–137)

(7–135) x log x

f ′(x) = log x+ 1

f ′′(x) =1

x> 0

-1

0

1

2

3

4

5

0 1 2 3 4

x

x*log(x)

f(x) = x log x (7–147)

H[P ] =

∫

P eq ·P

P eqlog

P

P eqd~r =

∫

P (~r; t) logP (~r; t)

P eq(~r)d~r (7–148)

H H

S∗[P (t)] ≡ −kB

∫

P (~r; t) logP (~r; t)

Peq(~r)d~r (7–149)

H S∗

(1) S∗

(2) ( P → Peq) log 1 = 0 S∗ = 0

S∗

7–46

7–48


// 1-D Diffusion in Potential Field: Entropy Change#include <stdio.h>#include <math.h>

#define MAX_STEP 5000#define SAVE_STEP 10double DT = 0.001;double GAMMA = 1.0;double TEMPERATURE = 1.00;

#define NX_CELL 200double XMIN = -5.0;double XMAX = 5.0;

double prob[NX_CELL];double dprob[NX_CELL];double prob_eq[NX_CELL];double dx;//-------------------------------------------------------double potential(double x){

return 0.5*x*x;}double force(double x){

return -x;}

void initial( ){

int ix;double sum, factor;

dx=(XMAX-XMIN)/NX_CELL;

for (ix=0;ix<NX_CELL;ix++) {prob[ix]=0.0;

}prob[5]=1.0/dx; //

sum=0.0;for (ix=0;ix<NX_CELL;ix++) { //

prob_eq[ix]=exp(-potential(dx*ix+XMIN)/TEMPERATURE);sum+=prob_eq[ix];

}factor=1/(dx*sum);

for (ix=0;ix<NX_CELL;ix++) {prob_eq[ix]*=factor;

}}

void output(int nstep){


// printf("%d\n",nstep);sprintf(cdum,"T10\\diffusion.%6.6d",nstep);fout=fopen(cdum,"w");

for (ix=0;ix<NX_CELL;ix++) {fprintf(fout,"%6.2f %9.5f\n",ix*dx+XMIN,prob[ix]);

}

fclose(fout);}

//--------------------------------------------------------int main( ){

int ix, nstep;double dp;double x,xm,xp;double f, d;double entropy;

initial( );


for (nstep=1;nstep<=MAX_STEP;nstep++) {for (ix=1;ix<NX_CELL-1;ix++) { //

x = dx*ix+XMIN;xm = x-dx;xp = x+dx;dprob[ix] = -(force(xp)*prob[ix+1]-force(xm)*prob[ix-1])/(2*dx)/GAMMA;dprob[ix]+= (prob[ix+1]-2*prob[ix]+prob[ix-1])/(dx*dx)*(TEMPERATURE/GAMMA);

}entropy=0.0;

for (ix=1;ix<NX_CELL-1;ix++) { //prob[ix] += dprob[ix]*DT;if (prob[ix]<=0.0) {

prob[ix] = 0.0;}else { //

entropy+=-prob[ix]*log(prob[ix]/prob_eq[ix]);}

}entropy*=dx;printf("%8.3f %10.5f\n",DT*nstep, entropy);if (nstep%SAVE_STEP==0) output(nstep); //

}return 0;

}

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En

tro

py

Time

T=1.00T=0.50T=0.25

7–48:

entropy+=-prob[ix]*log(prob[ix]/prob_eq[ix]);

prob_eq[ix]=exp(-potential(dx*ix+XMIN)/TEMPERATURE);

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