Lecture 4: Diffusion and the Fokker-Planck equation
description
Transcript of Lecture 4: Diffusion and the Fokker-Planck equation
![Page 1: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/1.jpg)
Lecture 4: Diffusion and the Fokker-Planck equation
Outline:
• intuitive treatment• Diffusion as flow down a concentration gradient• Drift current and Fokker-Planck equation
![Page 2: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/2.jpg)
Lecture 4: Diffusion and the Fokker-Planck equation
Outline:
• intuitive treatment• Diffusion as flow down a concentration gradient• Drift current and Fokker-Planck equation
• examples:• No current: equilibrium, Einstein relation• Constant current, out of equilibrium:
![Page 3: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/3.jpg)
Lecture 4: Diffusion and the Fokker-Planck equation
Outline:
• intuitive treatment• Diffusion as flow down a concentration gradient• Drift current and Fokker-Planck equation
• examples:• No current: equilibrium, Einstein relation• Constant current, out of equilibrium:
• Goldman-Hodgkin-Katz equation• Kramers escape over an energy barrier
![Page 4: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/4.jpg)
Lecture 4: Diffusion and the Fokker-Planck equation
Outline:
• intuitive treatment• Diffusion as flow down a concentration gradient• Drift current and Fokker-Planck equation
• examples:• No current: equilibrium, Einstein relation• Constant current, out of equilibrium:
• Goldman-Hodgkin-Katz equation• Kramers escape over an energy barrier
• derivation from master equation
![Page 5: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/5.jpg)
Diffusion Fick’s law:
€
J = −D∂P
∂x
![Page 6: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/6.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
![Page 7: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/7.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
conservation:
€
∂P
∂t= −
∂J
∂x
![Page 8: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/8.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
conservation:
€
∂P
∂t= −
∂J
∂x
€
∂P
∂t= D
∂2P
∂x 2
=>
![Page 9: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/9.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
conservation:
€
∂P
∂t= −
∂J
∂x
€
∂P
∂t= D
∂2P
∂x 2
=> diffusion equation
![Page 10: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/10.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
conservation:
€
∂P
∂t= −
∂J
∂x
€
∂P
∂t= D
∂2P
∂x 2
=> diffusion equation
initial condition
€
P(x | 0) = δ(x)
![Page 11: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/11.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
conservation:
€
∂P
∂t= −
∂J
∂x
€
∂P
∂t= D
∂2P
∂x 2
=> diffusion equation
initial condition
€
P(x | 0) = δ(x)
solution:
€
P(x | t) =1
4πDtexp −
x 2
4Dt
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 12: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/12.jpg)
Diffusion Fick’s law: cf Ohm’s law
€
J = −D∂P
∂x
€
I = −g∂V
∂x
conservation:
€
∂P
∂t= −
∂J
∂x
€
∂P
∂t= D
∂2P
∂x 2
=> diffusion equation
initial condition
€
P(x | 0) = δ(x)
solution:
€
P(x | t) =1
4πDtexp −
x 2
4Dt
⎛
⎝ ⎜
⎞
⎠ ⎟
http://www.nbi.dk/~hertz/noisecourse/gaussspread.m
![Page 13: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/13.jpg)
Drift current and Fokker-Planck equationDrift (convective) current:
€
Jdrift (x, t) = u(x)P(x, t)
![Page 14: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/14.jpg)
Drift current and Fokker-Planck equation
Combining drift and diffusion: Fokker-Planck equation:
Drift (convective) current:
€
Jdrift (x, t) = u(x)P(x, t)
€
∂P
∂t= −
∂
∂xJdrift + Jdiff( )
![Page 15: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/15.jpg)
Drift current and Fokker-Planck equation
Combining drift and diffusion: Fokker-Planck equation:
Drift (convective) current:
€
Jdrift (x, t) = u(x)P(x, t)
€
∂P
∂t= −
∂
∂xJdrift + Jdiff( )
= −∂
∂xu(x)P − D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= −
∂
∂xu(x)P( ) + D
∂ 2P
∂x 2
![Page 16: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/16.jpg)
Drift current and Fokker-Planck equation
Combining drift and diffusion: Fokker-Planck equation:
Slightly more generally, D can depend on x:
Drift (convective) current:
€
Jdrift (x, t) = u(x)P(x, t)
€
∂P
∂t= −
∂
∂xJdrift + Jdiff( )
= −∂
∂xu(x)P − D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= −
∂
∂xu(x)P( ) + D
∂ 2P
∂x 2
€
Jdiff (x, t) = −∂
∂xD(x)P(x, t)( )
![Page 17: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/17.jpg)
Drift current and Fokker-Planck equation
Combining drift and diffusion: Fokker-Planck equation:
Slightly more generally, D can depend on x:
=>
Drift (convective) current:
€
Jdrift (x, t) = u(x)P(x, t)
€
∂P
∂t= −
∂
∂xJdrift + Jdiff( )
= −∂
∂xu(x)P − D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= −
∂
∂xu(x)P( ) + D
∂ 2P
∂x 2
€
Jdiff (x, t) = −∂
∂xD(x)P(x, t)( )
€
∂P
∂t= −
∂
∂xu(x)P( ) +
∂ 2
∂x 2D(x)P( )
![Page 18: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/18.jpg)
Drift current and Fokker-Planck equation
Combining drift and diffusion: Fokker-Planck equation:
Slightly more generally, D can depend on x:
=>
Drift (convective) current:
€
Jdrift (x, t) = u(x)P(x, t)
€
∂P
∂t= −
∂
∂xJdrift + Jdiff( )
= −∂
∂xu(x)P − D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥= −
∂
∂xu(x)P( ) + D
∂ 2P
∂x 2
€
Jdiff (x, t) = −∂
∂xD(x)P(x, t)( )
€
∂P
∂t= −
∂
∂xu(x)P( ) +
∂ 2
∂x 2D(x)P( )
First term alone describes probability cloud moving with velocity u(x)Second term alone describes diffusively spreading probability cloud
![Page 19: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/19.jpg)
Examples: constant drift velocityhttp://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
![Page 20: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/20.jpg)
Examples: constant drift velocity
€
u(x) = u0
P(x, t) =1
4πDtexp −
x − u0t( )2
4Dt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
![Page 21: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/21.jpg)
Examples: constant drift velocity
€
u(x) = u0
P(x, t) =1
4πDtexp −
x − u0t( )2
4Dt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
Stationary case:
![Page 22: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/22.jpg)
Examples: constant drift velocity
€
u(x) = u0
P(x, t) =1
4πDtexp −
x − u0t( )2
4Dt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
Stationary case:Gas of Brownian particles in gravitational field: u0 = μF = -μmg
![Page 23: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/23.jpg)
Examples: constant drift velocity
€
u(x) = u0
P(x, t) =1
4πDtexp −
x − u0t( )2
4Dt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
Stationary case:Gas of Brownian particles in gravitational field: u0 = μF = -μmg
μ =mobility
![Page 24: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/24.jpg)
Examples: constant drift velocity
€
u(x) = u0
P(x, t) =1
4πDtexp −
x − u0t( )2
4Dt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
Stationary case:Gas of Brownian particles in gravitational field: u0 = μF = -μmg
μ =mobilityBoundary conditions (bottom of container, stationarity):
€
P(x) = 0, x < 0;
J(x) = 0
![Page 25: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/25.jpg)
Examples: constant drift velocity
€
u(x) = u0
P(x, t) =1
4πDtexp −
x − u0t( )2
4Dt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
Stationary case:Gas of Brownian particles in gravitational field: u0 = μF = -μmg
μ =mobilityBoundary conditions (bottom of container, stationarity):
€
P(x) = 0, x < 0;
J(x) = 0 drift and diffusion currents cancel
![Page 26: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/26.jpg)
Einstein relation
FP equation:
€
μmgP(x) + DdP
dx= 0
![Page 27: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/27.jpg)
Einstein relation
FP equation:
Solution:
€
μmgP(x) + DdP
dx= 0
P(x) =μmg
D
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
μmgx
D
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 28: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/28.jpg)
Einstein relation
FP equation:
Solution:
But from equilibrium stat mech we know
€
μmgP(x) + DdP
dx= 0
P(x) =μmg
D
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
μmgx
D
⎛
⎝ ⎜
⎞
⎠ ⎟
P(x) =mg
T
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
mgx
T
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 29: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/29.jpg)
Einstein relation
FP equation:
Solution:
But from equilibrium stat mech we know
So D = μT
€
μmgP(x) + DdP
dx= 0
P(x) =μmg
D
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
μmgx
D
⎛
⎝ ⎜
⎞
⎠ ⎟
P(x) =mg
T
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
mgx
T
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 30: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/30.jpg)
Einstein relation
FP equation:
Solution:
But from equilibrium stat mech we know
So D = μT Einstein relation
€
μmgP(x) + DdP
dx= 0
P(x) =μmg
D
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
μmgx
D
⎛
⎝ ⎜
⎞
⎠ ⎟
P(x) =mg
T
⎛
⎝ ⎜
⎞
⎠ ⎟exp −
mgx
T
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 31: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/31.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions
![Page 32: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/32.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cell
![Page 33: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/33.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cellCan vary membrane potential experimentally by adding external field
![Page 34: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/34.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cellCan vary membrane potential experimentally by adding external fieldQuestion: At a given Vm, what current flows through the channel?
![Page 35: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/35.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cellCan vary membrane potential experimentally by adding external fieldQuestion: At a given Vm, what current flows through the channel?
outside insidex
x=0 x=d
![Page 36: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/36.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cellCan vary membrane potential experimentally by adding external fieldQuestion: At a given Vm, what current flows through the channel?
outside insidex
V(x)Vm
Vout= 0
x=0 x=d
![Page 37: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/37.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cellCan vary membrane potential experimentally by adding external fieldQuestion: At a given Vm, what current flows through the channel?
outside insidex
V(x)Vm
Vout= 0
ρout
ρin
x=0 x=d
![Page 38: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/38.jpg)
Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cellCan vary membrane potential experimentally by adding external fieldQuestion: At a given Vm, what current flows through the channel?
outside insidex
V(x)Vm
Vout= 0
ρout
ρin
x=0 x=d?
![Page 39: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/39.jpg)
Reversal potential
If there is no current, equilibrium
=> ρin/ρout=exp(-βV)
![Page 40: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/40.jpg)
Reversal potential
If there is no current, equilibrium
=> ρin/ρout=exp(-βV)
This defines the reversal potential
at which J = 0.
€
Vr = T logρ out
ρ in
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 41: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/41.jpg)
Reversal potential
If there is no current, equilibrium
=> ρin/ρout=exp(-βV)
This defines the reversal potential
at which J = 0.
For Ca++, ρout>> ρin => Vr >> 0€
Vr = T logρ out
ρ in
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 42: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/42.jpg)
GHK model (2)
outside insidex
V(x)
Vout= 0
ρout
ρin
x=0 x=d?
Vm< 0: both diffusive current and drift current flow in
Vm
![Page 43: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/43.jpg)
GHK model (2)
outside insidex
V(x)Vout= 0
ρout
ρin
x=0 x=d?
Vm< 0: both diffusive current and drift current flow inVm= 0: diffusive current flows in, no drift current
![Page 44: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/44.jpg)
GHK model (2)
outside insidex
V(x)
Vm
Vout= 0
ρout
ρin
x=0 x=d?
Vm< 0: both diffusive current and drift current flow inVm= 0: diffusive current flows in, no drift currentVm> 0: diffusive current flows in, drift current flows out
![Page 45: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/45.jpg)
GHK model (2)
outside insidex
V(x)
Vm
Vout= 0
ρout
ρin
x=0 x=d?
Vm< 0: both diffusive current and drift current flow inVm= 0: diffusive current flows in, no drift currentVm> 0: diffusive current flows in, drift current flows outAt Vm= Vr they cancel
![Page 46: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/46.jpg)
GHK model (2)
outside insidex
V(x)
Vm
Vout= 0
ρout
ρin
x=0 x=d?
Vm< 0: both diffusive current and drift current flow inVm= 0: diffusive current flows in, no drift currentVm> 0: diffusive current flows in, drift current flows outAt Vm= Vr they cancel
€
Jdrift = μqEρ (x) = −μqdV
dxρ (x) = −
μqVm
dρ (x), Jdiff = −D
dρ
dx
![Page 47: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/47.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
![Page 48: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/48.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
![Page 49: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/49.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟Use Einstein relation:
![Page 50: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/50.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
Use Einstein relation:
![Page 51: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/51.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
ρ (x) = −J
μTκ+ ρ(0) +
J
μTκ
⎛
⎝ ⎜
⎞
⎠ ⎟exp −κx( )
Use Einstein relation:
Solution:
![Page 52: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/52.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
ρ (x) = −J
μTκ+ ρ(0) +
J
μTκ
⎛
⎝ ⎜
⎞
⎠ ⎟exp −κx( )
Use Einstein relation:
Solution:
We are given ρ(0) and ρ(d). Use this to solve for J:
![Page 53: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/53.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
ρ (x) = −J
μTκ+ ρ(0) +
J
μTκ
⎛
⎝ ⎜
⎞
⎠ ⎟exp −κx( )
€
J
μTκ1− exp −κd( )( ) = ρ out exp −κd( ) − ρ (d),
Use Einstein relation:
Solution:
We are given ρ(0) and ρ(d). Use this to solve for J:
![Page 54: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/54.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
ρ (x) = −J
μTκ+ ρ(0) +
J
μTκ
⎛
⎝ ⎜
⎞
⎠ ⎟exp −κx( )
€
J
μTκ1− exp −κd( )( ) = ρ out exp −κd( ) − ρ (d), ρ (d) = ρ in = ρ out exp −βqVr( )
Use Einstein relation:
Solution:
We are given ρ(0) and ρ(d). Use this to solve for J:
![Page 55: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/55.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
ρ (x) = −J
μTκ+ ρ(0) +
J
μTκ
⎛
⎝ ⎜
⎞
⎠ ⎟exp −κx( )
€
J
μTκ1− exp −κd( )( ) = ρ out exp −κd( ) − ρ (d), ρ (d) = ρ in = ρ out exp −βqVr( )
J =μTκ ρ out exp −κd( ) − ρ (d)( )
1− exp −κd( )
Use Einstein relation:
Solution:
We are given ρ(0) and ρ(d). Use this to solve for J:
![Page 56: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/56.jpg)
Steady-state FP equation
€
dJ
dx=
d
dx−D
dρ
dx−
μqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟= 0
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+
βqVm
dρ
⎛
⎝ ⎜
⎞
⎠ ⎟
−J
μT=
dρ
dx+ κρ, κ =
βqVm
d
ρ (x) = −J
μTκ+ ρ(0) +
J
μTκ
⎛
⎝ ⎜
⎞
⎠ ⎟exp −κx( )
€
J
μTκ1− exp −κd( )( ) = ρ out exp −κd( ) − ρ (d), ρ (d) = ρ in = ρ out exp −βqVr( )
J =μTκ ρ out exp −κd( ) − ρ (d)( )
1− exp −κd( )=
μqVmρ out exp −βqVm( ) − exp −βqVr( )( )
d 1− exp −βqVm( )( )
Use Einstein relation:
Solution:
We are given ρ(0) and ρ(d). Use this to solve for J:
![Page 57: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/57.jpg)
GHK current, another wayStart from
€
J = −Ddρ
dx−
μqVm
dρ = const.
![Page 58: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/58.jpg)
GHK current, another wayStart from
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 59: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/59.jpg)
GHK current, another wayStart from
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( )
![Page 60: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/60.jpg)
GHK current, another wayStart from
Note
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
![Page 61: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/61.jpg)
GHK current, another wayStart from
Note
Integrate from 0 to d:
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
![Page 62: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/62.jpg)
GHK current, another wayStart from
Note
Integrate from 0 to d:
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
€
J
κexp κd( ) −1( ) = −μT ρ in exp κd( ) − ρ out( )
![Page 63: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/63.jpg)
GHK current, another wayStart from
Note
Integrate from 0 to d:
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
€
J
κexp κd( ) −1( ) = −μT ρ in exp κd( ) − ρ out( )
J = −μTκρ in exp κd( ) − ρ out
exp κd( ) −1
![Page 64: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/64.jpg)
GHK current, another wayStart from
Note
Integrate from 0 to d:
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
€
J
κexp κd( ) −1( ) = −μT ρ in exp κd( ) − ρ out( )
J = −μTκρ in exp κd( ) − ρ out
exp κd( ) −1= −
μqVm ρ in exp βqVm( ) − ρ out( )
d exp βqVm( ) −1( )
![Page 65: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/65.jpg)
GHK current, another wayStart from
Note
Integrate from 0 to d:
(as before)
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
€
J
κexp κd( ) −1( ) = −μT ρ in exp κd( ) − ρ out( )
J = −μTκρ in exp κd( ) − ρ out
exp κd( ) −1= −
μqVm ρ in exp βqVm( ) − ρ out( )
d exp βqVm( ) −1( )
L =μqVmρ out exp −βqVm( ) − exp −βqVr( )( )
d 1− exp −βqVm( )( )
![Page 66: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/66.jpg)
GHK current, another wayStart from
Note
Integrate from 0 to d:
(as before)
Note: J = 0 at Vm= Vr
€
J = −Ddρ
dx−
μqVm
dρ = const.
= −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟
J exp κx( ) = −μTdρ
dx+ κρ
⎛
⎝ ⎜
⎞
⎠ ⎟exp κx( ) = −μT
d
dxρ exp κx( )( )
€
J
κexp κd( ) −1( ) = −μT ρ in exp κd( ) − ρ out( )
J = −μTκρ in exp κd( ) − ρ out
exp κd( ) −1= −
μqVm ρ in exp βqVm( ) − ρ out( )
d exp βqVm( ) −1( )
L =μqVmρ out exp −βqVm( ) − exp −βqVr( )( )
d 1− exp −βqVm( )( )
![Page 67: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/67.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
V
J
![Page 68: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/68.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
€
Vm → −∞ : qJ ≈ −μq2ρ out
Vm
d
V
J
![Page 69: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/69.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
€
Vm → −∞ : qJ ≈ −μq2ρ out
Vm
d= σE, E = −Vm /d,
V
J
![Page 70: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/70.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
€
Vm → −∞ : qJ ≈ −μq2ρ out
Vm
d= σE, E = −Vm /d, σ = μq2ρ out
V
J
![Page 71: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/71.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
€
Vm → −∞ : qJ ≈ −μq2ρ out
Vm
d= σE, E = −Vm /d, σ = μq2ρ out
Vm → +∞ : qJ ≈ −μq2ρ out exp(−βqVr)Vm
d= σE,
V
J
![Page 72: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/72.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
€
Vm → −∞ : qJ ≈ −μq2ρ out
Vm
d= σE, E = −Vm /d, σ = μq2ρ out
Vm → +∞ : qJ ≈ −μq2ρ out exp(−βqVr)Vm
d= σE, σ = μq2ρ out exp(−βqVr) = μq2ρ in
V
J
![Page 73: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/73.jpg)
GHK current is nonlinear
(using z, Vr for Ca++)
€
Vm → −∞ : qJ ≈ −μq2ρ out
Vm
d= σE, E = −Vm /d, σ = μq2ρ out
Vm → +∞ : qJ ≈ −μq2ρ out exp(−βqVr)Vm
d= σE, σ = μq2ρ out exp(−βqVr) = μq2ρ in
Vm ≈ Vr: qJ ≈μβq3Vr
d⋅
ρ outρ in
ρ out − ρ in
⋅(Vr −Vm )
V
J
![Page 74: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/74.jpg)
Kramers escape
Rate of escape from a potential well due to thermal fluctuations
www.nbi.dk/hertz/noisecourse/demos/Pseq.matwww.nbi.dk/hertz/noisecourse/demos/runseq.m
V1(x)P1(x)
P2(x)
V2(x)
![Page 75: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/75.jpg)
Kramers escape (2)
a b c
V(x)
![Page 76: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/76.jpg)
Kramers escape (2)
a b c
V(x)
J
![Page 77: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/77.jpg)
Kramers escape (2)
a b c
V(x)
Basic assumption: (V(b) – V(a))/T >> 1
J
![Page 78: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/78.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥Conservation (continuity):
![Page 79: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/79.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −∂
∂xμ
∂V
∂xP + D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
Conservation (continuity):
![Page 80: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/80.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −∂
∂xμ
∂V
∂xP + D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −D∂
∂x
∂(βV )
∂xP +
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
Conservation (continuity):
Use Einstein relation:
![Page 81: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/81.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −∂
∂xμ
∂V
∂xP + D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −D∂
∂x
∂(βV )
∂xP +
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
J = −Dexp −βV (x)( )∂
∂xexp βV (x)( )P[ ]
Conservation (continuity):
Use Einstein relation:
Current:
![Page 82: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/82.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −∂
∂xμ
∂V
∂xP + D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −D∂
∂x
∂(βV )
∂xP +
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
J = −Dexp −βV (x)( )∂
∂xexp βV (x)( )P[ ]
If equilibrium, J = 0,
Conservation (continuity):
Use Einstein relation:
Current:
![Page 83: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/83.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −∂
∂xμ
∂V
∂xP + D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −D∂
∂x
∂(βV )
∂xP +
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
J = −Dexp −βV (x)( )∂
∂xexp βV (x)( )P[ ]
If equilibrium, J = 0,
€
P(x)∝ exp −βV (x)( )
Conservation (continuity):
Use Einstein relation:
Current:
![Page 84: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/84.jpg)
Fokker-Planck equation
€
−∂P
∂t=
∂J
∂x=
∂
∂xu(x)P −
∂
∂xD(x)P( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −∂
∂xμ
∂V
∂xP + D
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
= −D∂
∂x
∂(βV )
∂xP +
∂P
∂x
⎡ ⎣ ⎢
⎤ ⎦ ⎥
J = −Dexp −βV (x)( )∂
∂xexp βV (x)( )P[ ]
If equilibrium, J = 0,
Here: almost equilibrium, so use this P(x)
€
P(x)∝ exp −βV (x)( )
Conservation (continuity):
Use Einstein relation:
Current:
![Page 85: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/85.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ] (J is constant)
![Page 86: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/86.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
(J is constant)
integrate:
![Page 87: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/87.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
≈ −exp βV (a)( )P(a)
(J is constant)
(P(c) very small)
integrate:
![Page 88: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/88.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
≈ −exp βV (a)( )P(a)
⇒ J =Dexp βV (a)( )P(a)
exp βV (x)( )dxa
c
∫
(J is constant)
(P(c) very small)
integrate:
![Page 89: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/89.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
≈ −exp βV (a)( )P(a)
⇒ J =Dexp βV (a)( )P(a)
exp βV (x)( )dxa
c
∫
If p is probability to be in the well, J = pr, where r = escape rate
(J is constant)
(P(c) very small)
integrate:
![Page 90: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/90.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
≈ −exp βV (a)( )P(a)
⇒ J =Dexp βV (a)( )P(a)
exp βV (x)( )dxa
c
∫
If p is probability to be in the well, J = pr, where r = escape rate
€
p = P(x)dxa−Δ
a +Δ
∫
(J is constant)
(P(c) very small)
integrate:
![Page 91: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/91.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
≈ −exp βV (a)( )P(a)
⇒ J =Dexp βV (a)( )P(a)
exp βV (x)( )dxa
c
∫
If p is probability to be in the well, J = pr, where r = escape rate
€
p = P(x)dx = P(a)a−Δ
a +Δ
∫ exp β V (a) −V (x)( )[ ]dxa−Δ
a +Δ
∫
(J is constant)
(P(c) very small)
integrate:
![Page 92: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/92.jpg)
Calculating the current
€
−J
Dexp βV (x)( ) =
∂
∂xexp βV (x)( )P(x)[ ]
−J
Dexp βV (x)( )dx
a
c
∫ = exp βV (x)( )P(x)[ ]a
c
≈ −exp βV (a)( )P(a)
⇒ J =Dexp βV (a)( )P(a)
exp βV (x)( )dxa
c
∫
If p is probability to be in the well, J = pr, where r = escape rate
€
p = P(x)dx = P(a)a−Δ
a +Δ
∫ exp β V (a) −V (x)( )[ ]dxa−Δ
a +Δ
∫
≈ P(a) exp − 12 β ′ ′ V (a)y 2
[ ]dy−∞
∞
∫ = P(a)2π
β ′ ′ V (a)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
(J is constant)
(P(c) very small)
integrate:
![Page 93: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/93.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
![Page 94: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/94.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
![Page 95: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/95.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
= exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
![Page 96: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/96.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
= exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
€
r =J
p=
Dexp βV (a)( )P(a)
p exp βV (x)( )dxa
c
∫
![Page 97: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/97.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
= exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
€
r =J
p=
Dexp βV (a)( )P(a)
p exp βV (x)( )dxa
c
∫
=Dexp βV (a)( )P(a)
P(a)2π
β ′ ′ V (a)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
![Page 98: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/98.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
= exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
€
r =J
p=
Dexp βV (a)( )P(a)
p exp βV (x)( )dxa
c
∫
=Dexp βV (a)( )P(a)
P(a)2π
β ′ ′ V (a)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
=Dβ
2π
⎛
⎝ ⎜
⎞
⎠ ⎟ ′ ′ V (a) ′ ′ V (b)( )
12 exp −β V (b) −V (a)( )[ ]
![Page 99: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/99.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
= exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
€
r =J
p=
Dexp βV (a)( )P(a)
p exp βV (x)( )dxa
c
∫
=Dexp βV (a)( )P(a)
P(a)2π
β ′ ′ V (a)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
=Dβ
2π
⎛
⎝ ⎜
⎞
⎠ ⎟ ′ ′ V (a) ′ ′ V (b)( )
12 exp −β V (b) −V (a)( )[ ] =
μ
2π
⎛
⎝ ⎜
⎞
⎠ ⎟ ′ ′ V (a) ′ ′ V (b)( )
12 exp −βEb( )
![Page 100: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/100.jpg)
calculating escape rate
In integral integrand is peaked near x = b
€
exp βV (x)( )dxa
c
∫
€
exp βV (x)( )dxa
c
∫ ≈ exp βV (b)( ) exp − 12 β ′ ′ V (b) x − b( )
2
( )−∞
∞
∫ dx
= exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
€
r =J
p=
Dexp βV (a)( )P(a)
p exp βV (x)( )dxa
c
∫
=Dexp βV (a)( )P(a)
P(a)2π
β ′ ′ V (a)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
exp βV (b)( )2π
β ′ ′ V (b)
⎛
⎝ ⎜
⎞
⎠ ⎟
12
=Dβ
2π
⎛
⎝ ⎜
⎞
⎠ ⎟ ′ ′ V (a) ′ ′ V (b)( )
12 exp −β V (b) −V (a)( )[ ] =
μ
2π
⎛
⎝ ⎜
⎞
⎠ ⎟ ′ ′ V (a) ′ ′ V (b)( )
12 exp −βEb( )________
![Page 101: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/101.jpg)
More about drift current
Notice: If u(x) is not constant, the probability cloud can shrink or spread even if there is no diffusion
![Page 102: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/102.jpg)
More about drift current
Notice: If u(x) is not constant, the probability cloud can shrink or spread even if there is no diffusion(like density of cars on a road where the speed limit varies)
![Page 103: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/103.jpg)
More about drift current
Notice: If u(x) is not constant, the probability cloud can shrink or spread even if there is no diffusion(like density of cars on a road where the speed limit varies)
http://www.nbi.dk/~hertz/noisecourse/driftmovie.m
Demo: initial P: Gaussian centered at x = 2u(x) = .00015x
![Page 104: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/104.jpg)
Derivation from master equation
€
∂P(x, t)
∂t= d ′ x w(x ← ′ x )P( ′ x , t) − w( ′ x ← x)P(x, t)[ ]∫ w(x ← ′ x ) ≡ r( ′ x ;x − ′ x ) :
![Page 105: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/105.jpg)
Derivation from master equation
€
∂P(x, t)
∂t= d ′ x w(x ← ′ x )P( ′ x , t) − w( ′ x ← x)P(x, t)[ ]∫ w(x ← ′ x ) ≡ r( ′ x ;x − ′ x ) :
(1st argument of r: starting point; 2nd argument: step size)
![Page 106: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/106.jpg)
Derivation from master equation
€
∂P(x, t)
∂t= d ′ x w(x ← ′ x )P( ′ x , t) − w( ′ x ← x)P(x, t)[ ]∫ w(x ← ′ x ) ≡ r( ′ x ;x − ′ x ) :
= d ′ x r( ′ x ;x − ′ x )P( ′ x , t) − r(x; ′ x − x)P(x, t)[ ]∫ ′ x = x − s :
(1st argument of r: starting point; 2nd argument: step size)
![Page 107: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/107.jpg)
Derivation from master equation
€
∂P(x, t)
∂t= d ′ x w(x ← ′ x )P( ′ x , t) − w( ′ x ← x)P(x, t)[ ]∫ w(x ← ′ x ) ≡ r( ′ x ;x − ′ x ) :
= d ′ x r( ′ x ;x − ′ x )P( ′ x , t) − r(x; ′ x − x)P(x, t)[ ]∫ ′ x = x − s :
= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
(1st argument of r: starting point; 2nd argument: step size)
![Page 108: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/108.jpg)
Derivation from master equation
€
∂P(x, t)
∂t= d ′ x w(x ← ′ x )P( ′ x , t) − w( ′ x ← x)P(x, t)[ ]∫ w(x ← ′ x ) ≡ r( ′ x ;x − ′ x ) :
= d ′ x r( ′ x ;x − ′ x )P( ′ x , t) − r(x; ′ x − x)P(x, t)[ ]∫ ′ x = x − s :
= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
Small steps assumption: r(x;s) falls rapidly to zero with increasing |s| on the scale on which it varies with x or the scale on which P varies with x.
(1st argument of r: starting point; 2nd argument: step size)
![Page 109: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/109.jpg)
Derivation from master equation
€
∂P(x, t)
∂t= d ′ x w(x ← ′ x )P( ′ x , t) − w( ′ x ← x)P(x, t)[ ]∫ w(x ← ′ x ) ≡ r( ′ x ;x − ′ x ) :
= d ′ x r( ′ x ;x − ′ x )P( ′ x , t) − r(x; ′ x − x)P(x, t)[ ]∫ ′ x = x − s :
= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
Small steps assumption: r(x;s) falls rapidly to zero with increasing |s| on the scale on which it varies with x or the scale on which P varies with x.
(1st argument of r: starting point; 2nd argument: step size)
x
s
![Page 110: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/110.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
expand:
![Page 111: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/111.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
expand:
![Page 112: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/112.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
expand:
![Page 113: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/113.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
expand:
![Page 114: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/114.jpg)
Derivation from master equation (2)
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∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
expand:
Kramers-Moyal expansion
![Page 115: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/115.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
expand:
Kramers-Moyal expansionFokker-Planck eqn if drop terms of order >2
![Page 116: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/116.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
rn (x) = snr(x,s)ds∫
expand:
Kramers-Moyal expansionFokker-Planck eqn if drop terms of order >2
![Page 117: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/117.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t)
⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
rn (x) = snr(x,s)ds∫
expand:
rn(x)Δt = nth moment of distribution of step size in time Δt
Kramers-Moyal expansionFokker-Planck eqn if drop terms of order >2
![Page 118: Lecture 4: Diffusion and the Fokker-Planck equation](https://reader035.fdocuments.in/reader035/viewer/2022081420/56813b63550346895da46089/html5/thumbnails/118.jpg)
Derivation from master equation (2)
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= ds r(x;s)P(x, t) − s∂
∂xr(x,s)P(x, t)( ) + 1
2 s2 ∂ 2
∂x 2 r(x,s)P(x, t)( ) +L − r(x;−s)P(x, t) ⎧ ⎨ ⎩
⎫ ⎬ ⎭
∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
rn (x) = snr(x,s)ds∫
r1(x) = u(x), r2(x) = 2D(x)
expand:
rn(x)Δt = nth moment of distribution of step size in time Δt
Kramers-Moyal expansionFokker-Planck eqn if drop terms of order >2