3. Fokker Planck equation.
LAST TIME DISCUSSED
MALTHUS-VERHULST PROBLEM
Birthrate: $b(n)=\beta n$
Death rate: $d(n)=\alpha n +\frac{1}{\Omega}n(n-1)$
Made a system size expansion

$$n=\Omega\phi(t)+\sqrt{\Omega} x$$

Found that $\phi$ satisfied a macroscopic rate equation

$$\frac{d\phi}{dt}=(\beta-\alpha)\phi-\phi^2$$

If $\beta>\alpha,\; \phi_s=(\beta-\alpha)$ stable steady state
An initial fluctuation x₀ evolved so that

$$\langle x(t)\rangle=x_0e^{-(\beta-\alpha)t}$$

$$\sigma(t)^2=\langle x(t)^2\rangle -\langle x(t)\rangle^2=\beta(1-e^{-2(\beta-\alpha)t})$$

FOKKER PLANCK EQUATION FOR
FLUCTUATIONS ABOUT MACROSCOPIC STEADY STATE

$$\frac{\partial\Pi(x,t)}{\partial t}=(\beta-\alpha)\frac{\part... ...al x}x\Pi+\beta(\beta-\alpha)\frac{\partial^2}{\partial x^2}\Pi$$

Find time independent distribution $\Pi(x)$ for long times

$$\frac{d}{dx}[x\Pi+\beta\frac{d\Pi}{dx}]=0$$

Get

$$x\Pi+\beta\frac{d\Pi}{dx}=const$$

Requirement that $\Pi \geq 0,\; -\infty <x<\infty$ gives const=0
Normalization condition $\int dx\Pi=1$ gives

$$\Pi(x)=\frac{1}{\sqrt{2\pi\beta}}\exp \frac{-x^2}{2\beta}$$

Gaussian distribution!

TIME EVOLUTION OF FLUCTUATIONS

Can also solve time dependent equation
All necessary information in P(x,t|x₀,0) (Notation $\Rightarrow, \;\Pi(A\vert B)$ is the conditional probability for A given B) Can show (e.g. by Fourier methods)

$$P(x,t\vert x_0,0)=\frac {\exp {-\frac {(x-e^{-(\beta-\al... ...\alpha)t})}} } {\sqrt{2\pi\beta(1-e^{-2(\beta-\alpha)t})}}$$

Probability distribution Gaussian with
evolving variance and mean




Variance and mean same as before






So far have argued only from an example
More general form of FOKKER-PLANCK equation

$$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial}{\partial x}A(x)P(x,t)+\frac{1}{2}\frac{\partial^2}{\partial x^2}B(x)P(x,t)$$

Also called Smoluchowski equation, or
second Kolmogorov equation.
Structure of equation:



J=Probabiliy current
A(x)P(x,t)=drift term or transport term
$\frac{1}{2}\frac{\partial}{\partial x}B(x)P(x,t)=$ diffusion term

ANALOGY WITH
MACROSCOPIC CURRENTS
Suppose n(x)= concentration of conserved quantity e.g. particle number

$$\int n(x)dx=const$$

$\frac{\partial n}{\partial t}=-\nabla\cdot$j

j= particle current
Current from constitutive relation e.g.

j$=\mu$f $n-\nabla(Dn)$

$\mu$=mobility given by $\langle$v $\rangle =\mu$ f
f=force on particles (e.g. mg or eE.)
D= diffusion constant
In Fokker-Planck equation deal with probability density rather than particle density!

LINEAR PROCESSES

By definition
Fokker-Planck equation linear in P
Fokker-Planck equations commonly referred to as linear if A(x) linear and B(x)=const. Stationary processes governed by linear Fokker Planck equation

$$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial}{\partial x}[A_0+A_1x]P+\frac{B}{2}\frac{\partial^2P}{\partial x^2}$$

called Ornstein-Uhlenbeck processes. Stationarity requires A₁<0. Time dependent Fokker-Planck equation can be solved explicitly for linear processes (Malthus-Verhulst problem example). Many numerical methods available for nonlinear processes!

STEADY STATE SOLUTION
(when exist)
can be found for arbitrary $A(x), \;B(x)$
In steady state probability current is zero
for unbounded system

$$0=A(x)P_s(x)-\frac{1}{2}\frac{d}{d x}B(x)P_s(x)$$

with solution



PLANCK'S DERIVATION
Continuous version of master equation

$$\frac{\partial P(q,t)}{\partial t}=\int dq'[W(q\vert q')P(q',t)-W(q'\vert q)P(q,t)]$$

Define r=q-q' and w(q',r)=W(q|q')

$$\frac{\partial P(q,t)}{\partial t}=\int drw(q-r,r)P(q-r,t)-P(q,t)\int dr w(q,-r)$$

Expand $\int dr w(q-r,r)P(q-r,t)$ w.r.t. r in first argument

$$\int dr w(q,r)P(q,t)-\frac{\partial}{\partial q}\int drrw(q,r)P(q,t)$$

$$+ \frac{1}{2}\frac{\partial^2}{\partial q^2}\int dr r^2w(q,r)P(q,t)+\cdots$$

Next, define the jump moments

$$a_n=\int dr r^n w(q,r)$$

Get Fokker-Planck equation to 2nd order!



ADDED BONUS OF DERIVATION:

Physical interpretation of A(q), B(q) in terms of jump moments.

Why stop at second order? Include all terms in expansion! Kramers-Moyal expansion:



Pawula's theorem $\Rightarrow$ if break off at finite order beyond 2nd, P(q,t) no longer positive
definite! Expansion to all order equivalent to master
equation. System size expansion $\Rightarrow$ higher than 2nd order jump moments higher order contributions in $\Omega^{-1/2}$ (van Kampen).

SUMMARY

• Found steady state probability distribution in Malthus-Verhulst problem.
• Could also compute evolution of individual fluctuation.
• Considered a more general form of Fokker-Planck equation.
• Interpreted terms in equation in terms of probability current, drift and diffusion.
• Defined linear Fokker-Planck equation as
  describing Ornstein-Uhlenbeck process.
• Derived Fokker-Planck equation from jump moments.
• Showed that continuous master equation
  equivalent to Kramers-Moyal expansion.

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