Transcript ppt
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
4Dt
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
4Dt
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
4Dt
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
4Dt
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
4Dt
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
4Dt
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
4Dt
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
4Dt
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.
ForCa++, 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:
expx
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:
expx
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:
expx
T
T
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd
out expd (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:
expx
T
T
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd
out expd (d),
T
(d) in out expqVr
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:
expx
T
T
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd
out expd (d),
T
J
T out expd (d)
1 expd
(d) in out expqVr
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:
expx
T
T
We are given ρ(0) and ρ(d). Use this to solve for J:
J
1 expd
out expd (d),
T
(d) in out expqVr
T out expd (d) qVm out expqVm expqVr
J
1 expd
d 1 expqVm
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 expx T expx
dx
J D
GHK current, another way
Start from
Note
d qVm
const.
dx
d
d
T
dx
d
d
J expx T expx T expx
dx
dx
J D
GHK current, another way
Start from
Note
d qVm
const.
dx
d
d
T
dx
d
d
J expx T expx T expx
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 expx T expx T expx
dx
dx
J D
Integrate from 0 to d:
J
expd 1 T in expd out
GHK current, another way
Start from
Note
d qVm
const.
dx
d
d
T
dx
d
d
J expx T expx T expx
dx
dx
J D
Integrate from 0 to d:
J
expd 1 T in expd out
J T
in expd out
expd 1
GHK current, another way
Start from
Note
d qVm
const.
dx
d
d
T
dx
d
d
J expx T expx T expx
dx
dx
J D
Integrate from 0 to d:
J
expd 1 T in expd out
J T
qVm in expqVm out
in expd out
expd 1
dexpqVm 1
GHK current, another way
Start from
Note
d qVm
const.
dx
d
d
T
dx
d
d
J expx T expx T expx
dx
dx
J D
Integrate from 0 to d:
J
expd 1 T in expd out
J T
qVm in expqVm out
in expd out
expd 1
d expqVm 1
qVm out expqVm expqVr
L
d1 expqVm
(as before)
GHK current, another way
Start from
Note
d qVm
const.
dx
d
d
T
dx
d
d
J expx T expx T expx
dx
dx
J D
Integrate from 0 to d:
J
expd 1 T in expd out
J T
qVm in expqVm out
in expd out
expd 1
d expqVm 1
qVm out expqVm expqVr
L
d1 expqVm
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 DexpV (x) expV (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 DexpV (x) expV (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 DexpV (x) expV (x)P
x
If equilibrium,
J = 0,
P(x) expV (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 DexpV (x) expV (x)P
x
J = 0,
P(x) expV (x)
Here: almost equilibrium, so use this P(x)
If equilibrium,
Calculating the current
J
expV (x) expV (x)P(x)
D
x
(J is constant)
Calculating the current
J
expV (x) expV (x)P(x)
D
x
integrate:
J
D
expV (x)dx expV (x)P(x)a
c
a
c
(J is constant)
Calculating the current
integrate:
J
D
c
a
J
expV (x) expV (x)P(x)
D
x
(J is constant)
expV (x)dx expV (x)P(x)a
c
expV (a)P(a)
(P(c) very small)
Calculating the current
integrate:
J
D
J
expV (x) expV (x)P(x)
D
x
expV (x)dx expV (x)P(x)a
c
a
c
expV (a)P(a)
J
(J is constant)
DexpV (a)P(a)
c
a
expV (x)dx
(P(c) very small)
Calculating the current
integrate:
J
D
J
expV (x) expV (x)P(x)
D
x
expV (x)dx expV (x)P(x)a
c
a
(J is constant)
c
expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)
c
a
expV (x)dx
If p is probability to be in the well, J = pr, where r = escape rate
Calculating the current
integrate:
J
D
J
expV (x) expV (x)P(x)
D
x
expV (x)dx expV (x)P(x)a
c
a
(J is constant)
c
expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)
c
a
expV (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
expV (x) expV (x)P(x)
D
x
expV (x)dx expV (x)P(x)a
c
a
(J is constant)
c
expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)
c
a
expV (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
expV (x) expV (x)P(x)
D
x
expV (x)dx expV (x)P(x)a
c
a
(J is constant)
c
expV (a)P(a)
J
(P(c) very small)
DexpV (a)P(a)
c
a
expV (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
expV (x)dx expV (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 expV (x)dx expV (b) exp 12 V (b) x b dx
c
2
2 2
expV (b)
V
(b)
1
calculating escape rate
In integral
exp V (x)dx integrand is peaked near x = b
c
a
a expV (x)dx expV (b) exp 12 V (b) x b dx
c
r
2
2 2
expV (b)
V
(b)
1
J DexpV (a)P(a)
p p cexpV (x)dx
a
calculating escape rate
In integral
exp V (x)dx integrand is peaked near x = b
c
a
a expV (x)dx expV (b) exp 12 V (b) x b dx
c
r
2
2 2
expV (b)
V
(b)
1
J DexpV (a)P(a)
p p cexpV (x)dx
a
DexpV (a)P(a)
2 2
2
P(a)
expV (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 expV (x)dx expV (b) exp 12 V (b) x b dx
c
r
2
2 2
expV (b)
V
(b)
1
J DexpV (a)P(a)
p p cexpV (x)dx
a
DexpV (a)P(a)
2 2
2
P(a)
expV (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 expV (x)dx expV (b) exp 12 V (b) x b dx
c
r
2
2 2
expV (b)
V
(b)
1
J DexpV (a)P(a)
p p cexpV (x)dx
a
DexpV (a)P(a)
2 2
2
P(a)
expV (b)
V
(a)
V
(b)
1
1
D
2
V (a) V (b) exp V (b) V (a) V (a) V (b) 2 expE b
2
2
1
2
1
calculating escape rate
In integral
exp V (x)dx integrand is peaked near x = b
c
a
a expV (x)dx expV (b) exp 12 V (b) x b dx
c
r
2
2 2
expV (b)
V
(b)
1
J DexpV (a)P(a)
p p cexpV (x)dx
a
DexpV (a)P(a)
2 2
2
P(a)
expV (b)
V
(a)
V
(b)
1
1
D
2
V (a) V (b) exp V (b) V (a) V (a) V (b) 2 expE 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
dxw(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)
dxr( 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)
dsr(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)
dsr(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)
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
Derivation from master equation (2)
expand:
P(x,t)
t
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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
dsr(x s;s)P(x s,t) r(x;s)P(x,t)
2
1 2
dsr(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)