7 ¬ p a w ,Å II. { ³þ T Ô w ³þ Particle Picture vs. Field …...ä ú g » è [ Ê...
Transcript of 7 ¬ p a w ,Å II. { ³þ T Ô w ³þ Particle Picture vs. Field …...ä ú g » è [ Ê...
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)
Thermal Physics 2019 Lecture Note 102
∆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
Thermal Physics 2019 Lecture Note 103
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)
Thermal Physics 2019 Lecture Note 104
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
()
Thermal Physics 2019 Lecture Note 105
#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;
Thermal Physics 2019 Lecture Note 106
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
Thermal Physics 2019 Lecture Note 107
// 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){
int ix, iv;char cdum[100];FILE *fout;
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( );
nstep=0;output(nstep);
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;
Thermal Physics 2019 Lecture Note 108
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
Thermal Physics 2019 Lecture Note 109
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.
Thermal Physics 2019 Lecture Note 110
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)
Thermal Physics 2019 Lecture Note 111
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
Thermal Physics 2019 Lecture Note 112
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
Thermal Physics 2019 Lecture Note 113
// 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){
int ix, iv;char cdum[100];FILE *fout;
// 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( );
nstep=0;output(nstep);
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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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);