Lecture 4: Diffusion and the Fokker-Planck
equation
Outline:
• intuitive treatment
• Diffusion as flow down a concentration gradient
• Drift current and Fokker-Planck equation
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:
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
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
Diffusion
Fick’s law:

J  D
P
x
Diffusion
Fick’s law:

J  D
P
x
cf Ohm’s law

I  g
V
x
Diffusion
Fick’s law:
conservation:


J  D
P
x
P
J

t
x
cf Ohm’s law

I  g
V
x
Diffusion
Fick’s law:
conservation:

=>


J  D
P
x
P
J

t
x
P
2 P
D 2
t
x
cf Ohm’s law

I  g
V
x
Diffusion
Fick’s law:
conservation:

=>


J  D
P
x
P
J

t
x
P
2 P
D 2
t
x
cf Ohm’s law
I  g

diffusion equation
V
x
Diffusion
Fick’s law:
J  D
conservation:

=>

initial condition


P
x
P
J

t
x
P
2 P
D 2
t
x
P(x | 0)   (x)
cf Ohm’s law
I  g

diffusion equation
V
x
Diffusion
Fick’s law:
J  D
conservation:

=>

initial condition

solution:


P
x
P
J

t
x
P
2 P
D 2
t
x
cf Ohm’s law
I  g

diffusion equation
P(x | 0)   (x)
 x 2 
1
P(x | t) 
exp

4Dt
 4Dt 
V
x
Diffusion
Fick’s law:
J  D
conservation:

=>

initial condition

solution:

P
x
P
J

t
x
P
2 P
D 2
t
x
cf Ohm’s law
I  g
V
x

diffusion equation
P(x | 0)   (x)
 x 2 
1
P(x | t) 
exp

4Dt
 4Dt 
http://www.nbi.dk/~hertz/noisecourse/gaussspread.m

Drift current and Fokker-Planck equation
Jdrift (x,t)  u(x)P(x,t)
Drift (convective) current:

Drift current and Fokker-Planck equation
Drift (convective) current:
Jdrift (x,t)  u(x)P(x,t)
Combining drift and diffusion: Fokker-Planck equation:
P

  Jdrift  
Jdiff 
t
x

Drift current and Fokker-Planck equation
Drift (convective) current:
Jdrift (x,t)  u(x)P(x,t)
Combining drift and diffusion: Fokker-Planck equation:
P

  J drift  
J diff 
t
x
 
P  
 2P
  u(x)P  D   u(x)P   D 2
x 
x  x
x

Drift current and Fokker-Planck equation
Drift (convective) current:
Jdrift (x,t)  u(x)P(x,t)
Combining drift and diffusion: Fokker-Planck equation:
P

  J drift  
J diff 
t
x
 
P  
 2P
  u(x)P  D   u(x)P   D 2
x 
x  x
x
Slightly more generally, D can depend on x: Jdiff (x,t)    D(x)P(x,t)
x


Drift current and Fokker-Planck equation
Drift (convective) current:
Jdrift (x,t)  u(x)P(x,t)
Combining drift and diffusion: Fokker-Planck equation:
P

  J drift  
J diff 
t
x
 
P  
 2P
  u(x)P  D   u(x)P   D 2
x 
x  x
x
Slightly more generally, D can depend on x: Jdiff (x,t)    D(x)P(x,t)
x
P

2

=>
  u(x)P   2 D(x)P 
t
x
x


Drift current and Fokker-Planck equation
Drift (convective) current:
Jdrift (x,t)  u(x)P(x,t)
Combining drift and diffusion: Fokker-Planck equation:
P

  J drift  
J diff 
t
x
 
P  
 2P
  u(x)P  D   u(x)P   D 2
x 
x  x
x
Slightly more generally, D can depend on x: Jdiff (x,t)    D(x)P(x,t)
x
P

2

=>
  u(x)P   2 D(x)P 
t
x
x

First term alone describes probability cloud moving with velocity u(x)
Second term alone describes diffusively spreading probability cloud

Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
u(x)  u0
P(x,t) 

 x  u t 2 
1
0

exp
4Dt

 4Dt 

Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
u(x)  u0
P(x,t) 
Stationary case:

 x  u t 2 
1
0

exp
4Dt

 4Dt 

Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
u(x)  u0
P(x,t) 
 x  u t 2 
1
0

exp
4Dt

 4Dt 

Stationary case:
 in gravitational field: u0 = μF = -μmg
Gas of Brownian particles
Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
Solution (with no boundaries):
u(x)  u0
P(x,t) 
 x  u t 2 
1
0

exp
4Dt

 4Dt 

Stationary case:
 in gravitational field: u0 = μF = -μmg
Gas of Brownian particles
μ =mobility
Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
u(x)  u0
Solution (with no boundaries):
P(x,t) 
 x  u t 2 
1
0

exp
4Dt

 4Dt 

Stationary case:
 in gravitational field: u0 = μF = -μmg
Gas of Brownian particles
μ =mobility
Boundary conditions (bottom of container, stationarity):
P(x)  0, x  0;
J(x)  0

Examples: constant drift velocity
http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m
u(x)  u0
Solution (with no boundaries):
P(x,t) 
 x  u t 2 
1
0

exp
4Dt

 4Dt 

Stationary case:
 in gravitational field: u0 = μF = -μmg
Gas of Brownian particles
μ =mobility
Boundary conditions (bottom of container, stationarity):
P(x)  0, x  0;
J(x)  0

drift and diffusion currents cancel
Einstein relation
m gP(x)  D
FP equation:

dP
0
dx
Einstein relation
dP
0
dx
FP equation:
m gP(x)  D
Solution:
m g  m gx
P(x)  
exp

 D   D 

Einstein relation
dP
0
dx
FP equation:
m gP(x)  D
Solution:
m g  m gx
P(x)  
exp

 D   D 
But from equilibrium stat mech we know

m g  m gx
P(x)   exp

 T   T 
Einstein relation
dP
0
dx
FP equation:
m gP(x)  D
Solution:
m g  m gx
P(x)  
exp

 D   D 
But from equilibrium stat mech we know
So
D = μT

m g  m gx
P(x)   exp

 T   T 
Einstein relation
dP
0
dx
FP equation:
m gP(x)  D
Solution:
m g  m gx
P(x)  
exp

 D   D 
But from equilibrium stat mech we know
So
D = μT

Einstein relation
m g  m gx
P(x)   exp

 T   T 
Constant current: Goldman-Hodgkin-Katz
model of an (open) ion channel
Pumps maintain different inside and outside concentrations of ions
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
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
Can vary membrane potential experimentally by adding external field
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
Can vary membrane potential experimentally by adding external field
Question: At a given Vm, what current flows through the channel?
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
Can vary membrane potential experimentally by adding external field
Question: At a given Vm, what current flows through the channel?
x=0
outside
x=d
inside
x
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
Can vary membrane potential experimentally by adding external field
Question: At a given Vm, what current flows through the channel?
x=0
x=d
Vout= 0
inside
outside
x
V(x)
Vm
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
Can vary membrane potential experimentally by adding external field
Question: At a given Vm, what current flows through the channel?
ρout
x=0
x=d
ρin
x
inside
Vout= 0
outside
V(x)
Vm
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
Can vary membrane potential experimentally by adding external field
Question: At a given Vm, what current flows through the channel?
ρout
x=0
x=d
?
ρin
x
inside
Vout= 0
outside
V(x)
Vm
Reversal potential
If there is no current, equilibrium
=> ρin/ρout=exp(-βV)
Reversal potential
If there is no current, equilibrium
=> ρin/ρout=exp(-βV)
This defines the reversal potential
out 
Vr  T log 
 in 
at which J = 0.

Reversal potential
If there is no current, equilibrium
=> ρin/ρout=exp(-βV)
This defines the reversal potential
out 
Vr  T log 
 in 
at which J = 0.
ForCa++, ρout>> ρin => Vr >> 0
GHK model (2)
Vm< 0: both diffusive current and drift current flow in
ρout
x=0
x=d
?
ρin
x
inside
Vout= 0
outside
V(x)
Vm
GHK model (2)
Vm< 0: both diffusive current and drift current flow in
Vm= 0: diffusive current flows in, no drift current
ρout
Vout= 0
outside
x=0
x=d
?
V(x)
ρin
x
inside
GHK model (2)
Vm< 0: both diffusive current and drift current flow in
Vm= 0: diffusive current flows in, no drift current
Vm> 0: diffusive current flows in, drift current flows out
ρout
x=0
x=d
?
V(x)
Vout= 0
outside
Vm
ρin
x
inside
GHK model (2)
Vm< 0: both diffusive current and drift current flow in
Vm= 0: diffusive current flows in, no drift current
Vm> 0: diffusive current flows in, drift current flows out
At Vm= Vr they cancel
ρout
x=0
x=d
?
V(x)
Vout= 0
outside
Vm
ρin
x
inside
GHK model (2)
Vm< 0: both diffusive current and drift current flow in
Vm= 0: diffusive current flows in, no drift current
Vm> 0: diffusive current flows in, drift current flows out
At Vm= Vr they cancel
Jdrift  qE(x)  q
ρout

dV
qVm
(x)  
(x),
dx
d
x=0
outside
d
dx
x=d
?
V(x)
Vout= 0
Jdiff  D
Vm
ρin
x
inside
Steady-state FP equation
dJ d  d qVm 
 D

 0

dx dx  dx
d

Steady-state FP equation
dJ d  d qVm 
 D

 0

dx dx  dx
d
d qVm
J  D

  const.
dx
d

Steady-state FP equation
Use Einstein relation:

dJ d  d qVm 
 D

 0

dx dx  dx
d
d qVm
J  D

  const.
dx
d
d qVm 
 T 

dx

d
Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const.
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const .
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

J
J 
 (x)  
 (0) 
Solution:
expx 
T 
T 

Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const .
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

J
J 
 (x)  
 (0) 
Solution:
expx 
T 
T 
We are given ρ(0) and ρ(d). Use this to solve for J:

Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const .
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

J
J 
 (x)  
 (0) 
Solution:
expx 
T 
T 
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd   

out expd    (d),
T
Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const .
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

J
J 
 (x)  
 (0) 
Solution:
expx 
T 
T 
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd   

out expd    (d),
T
(d)  in  out expqVr 
Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const .
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

J
J 
 (x)  
 (0) 
Solution:
expx 
T 
T 
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd   

out expd    (d),
T
J
T out expd  (d)
1 expd
(d)  in  out expqVr 
Steady-state FP equation
dJ d  d qVm 
 D

  0

dx dx  dx
d
d qVm
J  D

  const .
dx
d
d qVm 
Use Einstein relation:
 T 
 
dx

d
J
d
qVm


 ,

T dx
d

J
J 
 (x)  
 (0) 
Solution:
expx 
T 
T 
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd   

out expd    (d),
T
(d)  in  out expqVr 
T out expd  (d) qVm out expqVm   expqVr 
J

1 expd
d 1 expqVm 
GHK current, another way
Start from

J  D
d qVm

  const.
dx
d
GHK current, another way
Start from

d qVm

  const.
dx
d
d

 T   
dx

J  D
GHK current, another way
Start from

d qVm

  const.
dx
d
d

 T   
dx

d

J expx   T   expx 
dx

J  D
GHK current, another way
Start from
Note

d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
GHK current, another way
Start from
Note
d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
Integrate from 0 to d:

GHK current, another way
Start from
Note
d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
Integrate from 0 to d:


J

expd 1  T in expd  out 
GHK current, another way
Start from
Note
d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
Integrate from 0 to d:

J

expd 1  T in expd  out 
J  T

in expd   out
expd  1
GHK current, another way
Start from
Note
d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
Integrate from 0 to d:

J

expd 1  T in expd  out 
J  T

qVm in expqVm   out 
in expd   out

expd  1
dexpqVm  1
GHK current, another way
Start from
Note
d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
Integrate from 0 to d:

J

expd 1  T in expd  out 
J  T
qVm  in expqVm    out 
 in expd    out

expd  1
d expqVm  1
qVm  out expqVm   expqVr 
L 
d1 expqVm 

(as before)
GHK current, another way
Start from
Note
d qVm

  const.
dx
d
d

 T   
dx

d

d
J expx   T   expx   T  expx 
dx

dx
J  D
Integrate from 0 to d:

J

expd 1  T in expd  out 
J  T
qVm  in expqVm    out 
 in expd    out

expd  1
d expqVm  1
qVm  out expqVm   expqVr 
L 
d1 expqVm 
Note: J = 0 at Vm= Vr

(as before)
GHK current is nonlinear
(using z, Vr for Ca++)
J
V

GHK current is nonlinear
(using z, Vr for Ca++)
J
V
Vm   : qJ  q 2  out
Vm
d

GHK current is nonlinear
(using z, Vr for Ca++)
J
V
Vm   : qJ  q 2  out
Vm
 E, E  Vm /d,
d

GHK current is nonlinear
(using z, Vr for Ca++)
J
V
Vm   : qJ  q 2  out
Vm
 E, E  Vm /d,   q 2  out
d

GHK current is nonlinear
(using z, Vr for Ca++)
J
V
Vm
 E, E  Vm /d,   q2 out
d
V
Vm   : qJ  q 2 out exp(qVr ) m  E,
d
Vm   : qJ  q 2 out

GHK current is nonlinear
(using z, Vr for Ca++)
J
V
Vm
 E, E  Vm /d,   q 2 out
d
V
Vm   : qJ  q 2 out exp(qVr ) m  E,
  q2 out exp(qVr )  q2 in
d
Vm   : qJ  q 2 out
GHK current is nonlinear
(using z, Vr for Ca++)
J
V
Vm
 E, E  Vm /d,   q 2 out
d
V
Vm   : qJ  q 2 out exp(qVr ) m  E,
  q2 out exp(qVr )  q2 in
d
q 3Vr out in
Vm  Vr:
qJ 

 (Vr  Vm )
d
out  in
Vm   : qJ  q 2 out
Kramers escape
Rate of escape from a potential well due to thermal fluctuations
P2(x)
P1(x)
V1(x)
V2(x)
www.nbi.dk/hertz/noisecourse/demos/Pseq.mat
www.nbi.dk/hertz/noisecourse/demos/runseq.m
Kramers escape (2)
V(x)
a
b
c
Kramers escape (2)
V(x)
J
a
b
c
Kramers escape (2)
V(x)
J
a
b
c
Basic assumption: (V(b) – V(a))/T >> 1
Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x

Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x
  V
P 
  
P  D 
x  x
x 

Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x
  V
P 
  
P  D 
x  x
x 
  (V )
P 

D
P

Use Einstein relation:
 x

x 
x 

Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x
  V
P 
  
P  D 

x x
x 
  (V )
P 

D
P

Use Einstein relation:
 x

x 
x 

Current:
J  DexpV (x) expV (x)P 
x

Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x
  V
P 
  
P  D 

x x
x 
  (V )
P 

D
P

Use Einstein relation:
 x

x 
x 

Current:
J  DexpV (x) expV (x)P 
x
If equilibrium,
J = 0,

Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x
  V
P 
  
P  D 

x x
x 
  (V )
P 

D
P

Use Einstein relation:
 x

x 
x 

Current:
J  DexpV (x) expV (x)P 
x
If equilibrium,
J = 0,


P(x) expV (x)
Fokker-Planck equation


Conservation (continuity):  P  J   u(x)P   D(x)P 

t x x 
x
  V
P 
  
P  D 

x x
x 
  (V )
P 

D
P

Use Einstein relation:
 x

x 
x 

Current:
J  DexpV (x) expV (x)P 
x
J = 0,
P(x) expV (x)

Here: almost equilibrium, so use this P(x)
If equilibrium,

Calculating the current


J

expV (x)  expV (x)P(x)
D
x
(J is constant)
Calculating the current
J

expV (x)  expV (x)P(x)
D
x

integrate: 

J
D

expV (x)dx  expV (x)P(x)a
c
a
c
(J is constant)
Calculating the current

integrate: 

J
D

c
a
J

expV (x)  expV (x)P(x)
D
x
(J is constant)
expV (x)dx  expV (x)P(x)a
c
 expV (a)P(a)
(P(c) very small)
Calculating the current
integrate: 
J
D

J

expV (x)  expV (x)P(x)
D
x

expV (x)dx  expV (x)P(x)a
c
a
c
 expV (a)P(a)
J

(J is constant)
DexpV (a)P(a)

c
a
expV (x)dx
(P(c) very small)
Calculating the current
integrate: 
J
D

J

expV (x)  expV (x)P(x)
D
x

expV (x)dx  expV (x)P(x)a
c
a
(J is constant)
c
 expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)

c
a
expV (x)dx
If p is probability to be in the well, J = pr, where r = escape rate

Calculating the current
integrate: 
J
D

J

expV (x)  expV (x)P(x)
D
x

expV (x)dx  expV (x)P(x)a
c
a
(J is constant)
c
 expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)

c
a
expV (x)dx
If p is probability to be in the well, J = pr, where r = escape rate


p

a 
a
P(x)dx
Calculating the current
integrate: 
J
D

J

expV (x)  expV (x)P(x)
D
x

expV (x)dx  expV (x)P(x)a
c
a
(J is constant)
c
 expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)

c
a
expV (x)dx
If p is probability to be in the well, J = pr, where r = escape rate


p

a 
a
P(x)dx  P(a)  a exp V (a)  V (x)dx
a 
Calculating the current
integrate: 
J
D

J

expV (x)  expV (x)P(x)
D
x

expV (x)dx  expV (x)P(x)a
c
a
(J is constant)
c
 expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)

c
a
expV (x)dx
If p is probability to be in the well, J = pr, where r = escape rate

p

a 
a
P(x)dx  P(a) 
a 
a
exp V (a)  V (x)dx
2



2

2
 P(a)  exp 12 V (a)y dy  P(a)


V (a) 
1
calculating escape rate
In integral


exp V (x)dx integrand is peaked near x = b
c
a
calculating escape rate
In integral

a
exp V (x)dx integrand is peaked near x = b


expV (x)dx  expV (b)  exp  12  V (b) x  b dx
c

a




c
2
calculating escape rate
In integral

exp V (x)dx integrand is peaked near x = b
c
a


 a expV (x)dx  expV (b)  exp  12  V (b) x  b dx

c


2
 2 2
 expV (b)





V
(b)


1
calculating escape rate

In integral
exp V (x)dx integrand is peaked near x = b
c
a


 a expV (x)dx  expV (b)  exp  12  V (b) x  b dx

c

r


2
 2 2
 expV (b)





V
(b)


1
J DexpV (a)P(a)

p p  cexpV (x)dx
a
calculating escape rate

In integral
exp V (x)dx integrand is peaked near x = b
c
a


 a expV (x)dx  expV (b)  exp  12  V (b) x  b dx

c

r

2
 2 2
 expV (b)





V
(b)


1
J DexpV (a)P(a)

p p  cexpV (x)dx
a

DexpV (a)P(a)
 2 2
 2 
P(a)

 expV (b)







V
(a)

V
(b)




1
2
1
calculating escape rate

In integral
exp V (x)dx integrand is peaked near x = b
c
a


 a expV (x)dx  expV (b)  exp  12  V (b) x  b dx

c

r

2
 2 2
 expV (b)





V
(b)


1
J DexpV (a)P(a)

p p  cexpV (x)dx
a

DexpV (a)P(a)
 2 2
 2 
P(a)

 expV (b)







V
(a)

V
(b)




1
D 
  V (a) V (b) 2 exp V (b)  V (a)
2 
1
2
1
calculating escape rate

In integral
exp V (x)dx integrand is peaked near x = b
c
a


 a expV (x)dx  expV (b)  exp  12  V (b) x  b dx

c

r

2
 2 2
 expV (b)





V
(b)


1
J DexpV (a)P(a)

p p  cexpV (x)dx
a

DexpV (a)P(a)
 2 2
 2 
P(a)

 expV (b)







V
(a)

V
(b)




1
1
D 
  
2
  V (a) V (b)  exp V (b)  V (a)   V (a) V (b) 2 expE b 
2 
2 
1
2
1
calculating escape rate

In integral
exp V (x)dx integrand is peaked near x = b
c
a


 a expV (x)dx  expV (b)  exp  12  V (b) x  b dx

c

r

2
 2 2
 expV (b)





V
(b)


1
J DexpV (a)P(a)

p p  cexpV (x)dx
a

DexpV (a)P(a)
 2 2
 2 
P(a)

 expV (b)







V
(a)

V
(b)




1
1
D 
  
2
  V (a) V (b)  exp V (b)  V (a)   V (a) V (b) 2 expE b 
________
2 
2 
1
2
1
More about drift current
Notice: If u(x) is not constant, the probability cloud can shrink or
spread even if there is no diffusion
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)
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)
Demo: initial P: Gaussian centered at x = 2
u(x) = .00015x
http://www.nbi.dk/~hertz/noisecourse/driftmovie.m
Derivation from master equation
P(x,t)

t
 dx w(x  x )P( x ,t)  w( x  x)P(x,t)
w(x  x )  r( x ;x  x ) :
Derivation from master equation
P(x,t)

t
 dx 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)
Derivation from master equation
P(x,t)

t
 dx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)

 dxr( x ;x  x)P( x ,t)  r(x; x x)P(x,t)
x  x  s :
Derivation from master equation
P(x,t)

t
 dx 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)
 dx r( x ;x  x )P( x ,t)  r(x; x  x)P(x,t)
  dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)

x  x  s :
Derivation from master equation
P(x,t)

t
 dx 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)
 dx r( x ;x  x )P( x ,t)  r(x; x  x)P(x,t)
  dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)

x  x  s :
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.
Derivation from master equation
P(x,t)

t
 dx 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)
 dx r( x ;x  x )P( x ,t)  r(x; x  x)P(x,t)
  dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
x  x  s :

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.
x
s
Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)

L

r(x;s)P(x,t)



2

x

x


Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
2 

x

x




x
2
 sr(x,s)ds P(x,t)  x 2




1
2


s2 r(x,s)ds P(x,t)  L
Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
2 

x

x



2

sr(x,s)ds P(x,t)  2  12 s2 r(x,s)ds P(x,t)  L

x
x

1 2
  r1(x)P(x,t) 
r2 (x)P(x,t)  L
x
2 x 2






Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
2 

x

x



2

sr(x,s)ds P(x,t)  2  12 s2 r(x,s)ds P(x,t)  L

x
x

1 2
Kramers-Moyal expansion
  r1(x)P(x,t) 
r (x)P(x,t)  L
2  2
x
2 x






Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
2 

x

x



2

sr(x,s)ds P(x,t)  2  12 s2 r(x,s)ds P(x,t)  L

x
x

1 2
Kramers-Moyal expansion
  r1(x)P(x,t) 
r (x)P(x,t)  L
2  2
x
2 x
Fokker-Planck eqn if drop






terms of order >2
Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
x
x 2



2

sr(x,s)ds P(x,t)  2  12 s2 r(x,s)ds P(x,t)  L

x
x

1 2
Kramers-Moyal expansion
  r1(x)P(x,t) 
r (x)P(x,t)  L
2  2
x
2 x
Fokker-Planck eqn if drop






terms of order >2
rn (x) 
 s r(x,s)ds
n
Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
x
x 2



2

sr(x,s)ds P(x,t)  2  12 s2 r(x,s)ds P(x,t)  L

x
x

1 2
Kramers-Moyal expansion
  r1(x)P(x,t) 
r (x)P(x,t)  L
2  2
x
2 x
Fokker-Planck eqn if drop






terms of order >2
rn (x) 
 s r(x,s)ds
n
rn(x)Δt = nth moment of distribution of step size in time Δt
Derivation from master equation (2)
expand:
P(x,t)

t
 dsr(x  s;s)P(x  s,t)  r(x;s)P(x,t)
2



1 2 
  dsr(x;s)P(x,t)  s r(x,s)P(x,t)  2 s
r(x,s)P(x,t)  L  r(x;s)P(x,t)
x
x 2



2

sr(x,s)ds P(x,t)  2  12 s2 r(x,s)ds P(x,t)  L

x
x

1 2
Kramers-Moyal expansion
  r1(x)P(x,t) 
r (x)P(x,t)  L
2  2
x
2 x
Fokker-Planck eqn if drop






terms of order >2
rn (x) 
 s r(x,s)ds
n
rn(x)Δt = nth moment of distribution of step size in time Δt
r1 (x)  u(x), r2 (x)  2D(x)