Kramers–Moyal Expansion and Fokker–Planck Equation

Abstract

In this chapter, we present the details of Kramers–Moyal (KM) expansion and prove the Pawula theorem. The Fokker–Planck equation is then introduced and its short-term propagator is presented. Finally, we derive the master equation from the Chapman–Kolmogorov equation.

Problems

3.1

Equation of statistical moments on the Kramers–Moyal equation

From the more general Kramers–Moyal expansion (3.7) for the probability density p(x, t), derive the following differential equations for the nth-order statistical moments (assuming that they exist) of x,

$$\begin{aligned} \frac{\partial }{\partial t} \left\langle x^n \right\rangle = \sum \limits _{k=1}^{n} \, \frac{n!}{(n-k)!} \, \left\langle \, x^{n-k} \, D^{(k)}(x,t) \, \right\rangle \; . \end{aligned}$$

3.2

Path integral solution of the Fokker–Planck equation

The short-term propagator (or transition probability) (3.18) is needed for path integral solution of the Fokker–Planck equation. Dividing time difference $t-t_0$ into N small intervals of length $\tau = (t-t_0)/N$, defining $t_n=t_0 + n \tau$, and by repeatedly applying the Chapman–Kolmogorov equation (3.19) show that in the limit $N \rightarrow \infty$ (or $\tau \rightarrow 0$)

(a) the probability distribution function p(x, t) can be written in terms of initial probability density $p(x_0,t_0)$ as:

$$ p(x,t)= \lim _{N \rightarrow \infty } \int dx_{N-1} \cdots \int dx_{0}~ p(x,t|x_{N-1},t_{N-1}) \cdots p(x_1,t_1|x_{0},t_{0}) ~ p(x_0,t_0) . $$

(b) Using Eq. (3.18), derive the path integral solution of the Fokker–Planck equation as,

$$\begin{aligned} p(x,t)= & {} \lim _{N \rightarrow \infty } \int \underbrace{\cdots }_{\text {N~ times}} \int \left\{ \Pi _{i=0} ^{N-1} {\left( 4 \pi D^{(2)} (x_i,t_i) \tau \right) }^{-1/2} dx_i \right\} \\\\ \nonumber&\times \exp \left( - \sum _{i=0} ^{N-1} \frac{\left[ x_{i+1} - x_i - D^{(1)}(x_i,t_i) \tau \right] ^2}{4 D^{(2)} (x_i,t_i) \tau } \right) ~p(x_0,t_0) \end{aligned}$$

where

$$ \lim _{N \rightarrow \infty } \sum _{i=0} ^{N-1} \frac{\left[ x_{i+1} - x_i - D^{(1)}(x_i,t_i) \tau \right] ^2}{4 D^{(2)} (x_i,t_i) \tau } =\int _{t_0} ^t \frac{ \left[ {\dot{x}}(t') - D^{(1)}(x(t'),t') \right] ^2 }{ 4 D^{(2)}(x(t'),t') } ~ dt' $$

which means that for small diffusion coefficient $D^{(2)}(x(t),t)$, only the paths near the deterministic solution ${\dot{x}}(t) = D^{(1)}(x(t),t)$, contribute to p(x, t).

(c) Use the result of part (b) and argue that the distribution function p(x, t) must remain positive, if one starts with a positive distribution $p(x_0,t_0)$.

3.3

Backward Kramers–Moyal equation

Starting from the following Chapman–Kolmogorov equation

$$ p(x,t|x',t') = \int p(x,t|x'',t'+\tau ) ~ p(x'',t'+\tau |x',t') dx'' $$

with $t \ge t'+\tau \ge t'$,

(a) Show that $p(x,t|x',t')$ obey the following backward Kramers–Moyal equation

$$ \frac{\partial p(x,t|x',t')}{\partial t'}= - \sum _{n=1}^{\infty } D^{(n)}(x',t') \Big (\frac{\partial }{\partial x'}\Big )^n ~ p(x,t|x',t'). $$

(b) Show that the operator

$$ \mathcal L_{KM}^{\dagger } = \sum _{n=1}^{\infty } D^{(n)}(x',t') \Big (\frac{\partial }{\partial x'}\Big )^n $$

is the adjoint operator of¹

$$ \mathcal L_{KM} = \sum _{n=1}^{\infty } \Big (-\frac{\partial }{\partial x'}\Big )^n D^{(n)}(x',t'). $$

3.4

The master equation: Random walk and diffusion equation

Let p(i, N) denote the probability that a random walker is at site i after N steps. Assume that walkers has an equal probability to walk one step left and right.

(a) Use the master equation and show that

$$ p(i,N) = \frac{1}{2} p(i+1,N-1) + \frac{1}{2} p(i-1,N-1) . $$

(b) To obtain the continuum limit of this equation, define $t=N\tau$ and $x=i a$, by assuming that $D=a^2/ 2 \tau$ is finite in the limit $\tau \rightarrow 0$ and $a \rightarrow 0$, show that p(x, t) satisfies the diffusion equation,

$$ \frac{\partial p(x,t)}{\partial t} = D ~\frac{\partial ^2 p(x,t)}{\partial x ^2} . $$

where D is the diffusion constant.

(c) Show that the solution of diffusion equation is given by:

$$ p(x,t) = \frac{1}{\sqrt{4\pi D t}} \exp \left\{ - \frac{x^2}{4Dt} \right\} . $$

(d) Show that the conditional probability distribution of the diffusion equation with initial condition $p(x',t | x,t) = \delta (x'-x)$ is given by:

$$ p(x',t+\tau | x,t) = \frac{1}{\sqrt{4\pi D \tau }} \exp \left\{ - \frac{(x-x')^2}{ 4D\tau } \right\} . $$

(e) Show that second statistical moment of x is given by

$$ \langle x^2 (t) \rangle = 2 D~ t . $$

3.5

The master equation: Poisson process

Poisson process is defined by the jump rate, $w_{n,n'} = \lambda ~ \delta _{n,n'+1}$ for $n=0,1,2,\ldots$ and positive constant $\lambda >0$.

(a) Show that the Master equation (3.28) reduces to

$$ \dot{p}_n = \lambda p_{n-1} - \lambda p_n \forall ~ n . $$

(b) Show that the solution for ${p}_n$ is,

$$ p_n = \frac{(\lambda t) ^ n}{n!} \exp \{-\lambda t\}. $$

(c) Compute the first and second statistical moments $\langle n \rangle$ and $\langle n^2 \rangle$.