Equivalence of Langevin and Fokker–Planck Equations

Abstract

In this chapter we show the equivalence between the Langevin approach and the Fokker–Planck equation, and derive the equation for the statistical moments of the process whose dynamics is described by the Langevin equation.

Problems

7.1

Stratonovich interpretation

Starting from the Langevin equation, Eq. (7.10), prove that probability density function p(x, t) satisfies the Fokker–Planck equation, Eq. (7.9), with the Stratonovich interpretation.

7.2

Kramers–Moyal coefficients

For the Langevin equation

$$ dx(t)= a(x,t)~ dt + b(x,t) ~ dW(t) $$

where W(t) is a Wiener process, use the Itô lemma to prove that

(a) $D^{(1)}(x,t) = \lim _{dt \rightarrow 0} \frac{1}{dt} \langle (x(t+dt) - x(t)) | x(t) =x \rangle = a(x,t)$

(b) $D^{(2)}(x,t) = \lim _{dt \rightarrow 0} \frac{1}{2 dt} \langle (x(t+dt) - x(t))^2 | x(t) =x \rangle = \frac{1}{2} b^2(x,t)$

and for some $\delta >0$

(c) $\lim _{dt \rightarrow 0} \frac{1}{dt} \langle (x(t+dt) - x(t))^{2+\delta } | x(t) =x \rangle = 0$.

which means that higher-order Kramers–Moyal coefficients are $D^{(n)}(x,t) =0$, for $n \ge 3$.

7.3

Transition probability distribution

For the Langevin equation

$$ dx(t)= a(t)~ dt + b(t) ~ dW(t) $$

use the corresponding Fokker–Planck equation to prove that the transition (conditional) probability distribution with initial condition $p(y,s|x,s)=\delta (x-y)$ is given by:

$$ p(y,t|x,s) = \frac{1}{\sqrt{2 \pi \int _s ^t b^2(u) du}} \exp \left[ - \frac{\left( y-x-\int _s ^t a(u) du\right) ^2}{ 2 \int _s ^t b^2(u) du} \right] . $$

7.4

Statistical moments of the first-passage times

Suppose that the process x is the solution of the white noise-driven Langevin equation

$$ \frac{dx}{dt} = a(x,t) + b(x,t) \varGamma (t). $$

Since the process x is stochastic, each realization (which starts at $t_0=0$ from $x_0 \in [A,B]$) of it reaches the barrier $x_A=A$ or $x_B=B$ for the first time at a different passage (exit) time T. Therefore, it has a first passage time probability density $f(T|x_0)$ and its statistical moments, assuming that they exist, are defined as $\tau _n(x_0) \equiv \langle T^n \rangle$. The density $f(T|x_0)$ depends on the starting position $x_0$ and on the chosen boundary conditions. For example, we can choose (i) reflecting at A and absorbing boundary conditions at B, or (ii) both boundaries to be absorbing. In case (i) we are interested in to study the statistics of the first-passage times to reach the boundary at B, while in the second case one is interested in looking at the first-passage time for reaching either boundary.

Define the survival probability $G(x_0, t)$ of a trajectory x(t) to be within the interval [A, B] at time t, if it started at $x_0\in [A, B]$ at time $t = 0$.

(a) Show that $G(x_0,t)$ is given in term of the conditional probability distribution function as

$$ G(x_0,t) = \int _A ^B p(x',t|x_0,0) dx'. $$

For the systems with time translation invariant, i.e., $a(x,t)=a(x)$ and $b(x,t)=b(x)$, we write

$$ p(x',t|x_0,0) = p(x',0|x_0,-t). $$

(b) By noting that $p(x',0|x_0,t)$ satisfies the backward Fokker–Planck equation (see problem 3.3)

$$ \frac{\partial }{\partial t} p(x',0|x_0,t) = - a(x_0) \frac{\partial }{\partial x_0} p(x',0|x_0,t) - \frac{1}{2} b^2(x_0) \frac{\partial ^2}{\partial x_0^2} p(x',0|x_0,t) $$

show that the survival probability $G(x_0, t)$ for case (i) satisfies the partial differential equation

$$ \frac{\partial }{\partial t} G(x_0,t) = a(x_0) \frac{\partial }{\partial x_0} G(x_0,t) + \frac{1}{2} b^2(x_0) \frac{\partial ^2}{\partial x_0^2} G(x_0,t) $$

with initial and boundary conditions, $G(x_0,0) = \left\{ \begin{array}{ll} 1 &{} \quad x_0 \in [A,B] \\ 0 &{} \quad x_0\notin [A,B] \\ \end{array} \right.$, absorbing boundary at B, $G(x_0=B,t) =0$ and reflecting boundary at A, $\partial _{x_0} G(x_0=A,t)=0$. Also $G(x_0,t \rightarrow \infty ) =0$.

(c) Argue that

$$ f(T|x_0) = - \partial _T G(x_0,T). $$

(d) Show that the statistical moments of the first-passage times, T, i.e. $\tau _n(x_0)=\int _0^\infty T^n f(T|x_0) dt$, satisfies the following Darling and Siegert’s recursive relation

$$ - n \tau _{n-1}(x) = a(x) \frac{\partial }{\partial x} \tau _n(x) + \frac{1}{2} b^2(x) \frac{\partial ^2}{\partial x^2} \tau _n(x) $$

where $x\equiv x_0$ with $\tau _0(x)=1$ and $\tau _n(A)=\tau _n(B)=0$, for $n=1,2,\ldots$. This yields a hierarchy of equations that can be solved recursively: once we have solved for the nth moment, we can go to moment $n + 1$ to solve it.

7.5

Mean first-passage time

Consider a stochastic process x that evolves within the interval [A, B] according to the white noise-driven Langevin equation

$$ \frac{dx(t)}{dt} = - \frac{d U(x)}{dx} + \varGamma (t) $$

where $-\frac{dU(x)}{dx}$ is a force due to the potential U(x). Use the Darling and Siegert’s recursive relation to compute the mean first-passage time $\tau _1(x_0)$ for $U(x)=-x$ and $U(x)= \frac{1}{2}x^2$ with the boundary conditions

(a) A reflecting and B absorbing,

(b) both A and B absorbing.

7.6

Kramers’ escape time

Consider the stochastic process x that evolves according to the white noise-driven Langevin equation

$$ \frac{dx(t)}{dt} = - \frac{d U(x)}{dx} + \epsilon \varGamma (t), ~~ x(0) =x_0 $$

where $-\frac{d U(x)}{dx}$ is a force due to the potential U(x). The limit $\epsilon \rightarrow 0$ is known as the small-noise limit. Assume that U(x) has a single non-degenerate minimum at $x=a$ and maximum at $x=b$, and that $U(x)\rightarrow \infty$ as $x\rightarrow -\infty$.

(a) Show that mean exit time satisfies the following equation with boundary conditions,

$$ \left[ \frac{1}{2} \epsilon ^2 \frac{d^2 }{dx^2} - U'(x) \frac{d }{dx}\right] \tau (x) = -1, ~~\tau (b)=0, ~~ \tau _x(-\infty ) =0. $$

(b) Using the boundary conditions show that

$$ \tau (x) = - \frac{2}{\epsilon ^2} \int _b ^x e^{2U(z)/\epsilon ^2} \int _{-\infty } ^z e^{-2U(y)/\epsilon ^2} dy dz. $$

(c) Suppose that $x_0\approx a$ and show, in the small-noise limit, that the mean exit time from the interval $(-\infty ,b]$ is given by

$$ \tau (x_0) \approx \pi \exp \left\{ \frac{2(U(b) - U(a))}{\epsilon ^2}\right\} ~ \sqrt{\frac{1}{ U''(a) |U''(b)|}} \quad . $$

Hint: There is no possibility of exit from $x\rightarrow -\infty$. Thus, we impose reflecting boundary condition at $x =-\infty$. Therefore, the boundary conditions in the Darling and Siegert’s relation (with $n=1$) will be $\tau (b)=0$ and $\tau _x(-\infty )=0$.

7.7

Novikov–Furutsu–Donsker relation

For a Gaussian-distributed noise $\eta (t)$ with the covariance

$$ C(t,t') = \langle \eta (t) \eta (t') \rangle - \langle \eta (t) \rangle \langle \eta (t') \rangle $$

show that correlation between a functional $g[\eta ]$ and noise $\eta (t)$ can be written in terms of $C(t,t')$ as

$$ \langle \eta (t) g[\eta ] \rangle = \langle \eta (t) \rangle \langle g[\eta ] \rangle + \int _0 ^t C(t,t') ~ \left\langle \frac{\delta g[\eta ])}{\delta \eta (t')} \right\rangle ~ dt' $$

where $\frac{\delta g[\eta ]}{\delta \eta (t')}$ is the functional derivative of $g[\eta ]$ with respect to the Gaussian random function $\eta (t')$.