9. Langevin approach. Ito vs. Stratanovich.

LAST TIME:

• We discussed transformations of the Fokker-Planck equation.
• One of these eliminated an heterogeneous diffusion term.
• We discussed in detail an example of variable diffusion constant.
• Heterogeneous diffusion produced net effective drift $\Rightarrow$ motor!

TODAY try a different approach.

Master equation approach considered time evolution of probability distribution.
Fokker-Planck equation was derived from master equation typically from system size expansion.

LANGEVIN EQUATION $\Rightarrow$ stochastic d.e.
When "solved" a stochastic differential equation yields a single realization of a possible outcome, rather than the full probability distribution. Let us consider the simplest case: Brownian motion with no external force. We write the equation of motion as



$v=$ velocity of Brownian particle.
$\gamma=$ friction term.$\;\;$ $L(t)=$ noise term.
Collisions with other molecules gives rise to average force (friction) $+$random force. To distinguish between the two types of forces we require:

$$\langle L(t)\rangle=0$$

The physical picture is that the Brownian particle is mesoscopic. We study its motion over a timescale where it undergoes a large number of collisions in an "infinitesimal" time interval, while each collision with molecules within the fluids in which the mesoscopic particle moves has a small effect. If $t-t'»\tau$, the mean time between molecular collisions, we require that forces applied at different times are uncorrelated.

$$\langle L(t')L(t)\rangle=0$$

Some definitions:
$M=$ mass of Brownian particle.
$p=M\nu=$ random part of impulse transmitted at collision.
$\langle \nu^2\rangle=$ variance of $\nu$.
$1/\tau$ collision rate
Let

$$\int_0^\Delta Ldt=x$$

so that $x$ is a stochastic variable.
By central limit theorem of statistics, $x$ characterized by Gaussian
probability distribution

$$P(x)=\frac{1}{\sqrt{2\pi \xi\Delta}}\exp(-\frac{x^2}{2\xi\Delta})$$

where

$$\xi=\frac{\langle \nu^2\rangle}{\tau}$$

Physically, Langevin equation corresponds to limit

$$\Delta»\tau\Rightarrow 0$$

Mathematically $L(t)$ realization of Wiener process


Langevin equation suitable for computer simulations! To see how it works let us discretize time

$$t_n=n\Delta$$

then

$$v_n-v_{n-1}=\Delta\gamma v_n+g_n\sqrt{\xi\Delta}$$

where $g_n$ picked from Gaussian distribution with variance 1 and

$$\langle g_ng_{n'}\rangle=0\; if\; n\neq n'$$

Random number generators typically generate uniformly distributed numbers in the range
$0<r<1$.
A simple method to generate a Gaussian distribution is to make use of results from Brownian motion!

$$g_n\approx\sqrt{\frac{12}{m}}\sum_{i=1}^m(r_i-\frac{1}{2})$$

Clearly $\langle g_n\rangle =0$ and we find

$$\langle g_n^2\rangle=12/m\sum_{i,j=1}^m\langle(r_i-\frac{1}{2}) (r_j-\frac{1}{2})$$

$$=\frac{12}{m}\sum_{i=1}^{m}\langle(r_i-\frac{1}{2})^2\rangle= 12\int_{-1/2}^{1/2}x^2dx=1$$

By the central limit theorem the result will be a Gaussian distributed stochastic variable with mean zero and variance 1 if the number of steps $m$ is large enough.

The Box Muller method is a more efficient approach to generating Gaussian distributed random numbers. Let $x_1$ and $x_2$ be two independent stochastic variables. We make a coordinate transformation to generate two new stochastic variable $y_1(x_1,x_2),\;y_2(x_1,x_2)$. If the probability distribution for $x_1,\;x_2$ is $P(x_1,x_2)$, the probability distribution for $y_1,,y_2$ will be

$$\Pi(y_1,y_2)dy_1\;dy_2=P(x_1,x_2)dx_1dx_2=P(x_1,x_2)\vert J\vert dy_1\;dy_2$$

where $J$ is the Jacobian determinant

$$J=\left\vert\begin{array}{cc} \frac{\partial x_1}{\partial y... ...}& \frac{\partial x_2}{\partial y_2}\\ \end{array}\right\vert$$

If we now choose

$$y_1=sqrt{-2\ln x_1}\cos(2\pi x_2)$$

$$y_2=sqrt{-2\ln x_1}\sin(2\pi x_2)$$

we find for the Jacobian

$$J=-\frac{1}{\sqrt{2\pi}}\exp(\frac{-y_1^2}{2})\frac{1}{\sqrt{2\pi}}\exp(\frac{-y_2^2}{2})$$

If now $x_1,x_2$ are uniformly distributed in the range $0<x<1$ independent variables, $y_1,y_2$ will be two Gaussian distributed independent variables with zero mean and variance 1. A simple program to generate the Box Muller algorithm is described in William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery [1992], Numerical Recipes in C, second edition, Cambridge University Press. This book is also available on the web at http://www.nr.com/. Numerical Recipes also gives a trick to avoid the time consuming evaluation of the trigonometric functions.

It is common to formally generalize to take the limit $\Delta\rightarrow 0$ and write the Langevin equation on the form

$$\frac{dv}{dt}=-\gamma v+L(t)$$

where

$$\langle L(t)\rangle=0$$

and

$$\langle L(t)L(t')\rangle =\Gamma\delta(t-t')$$

with $\delta(x)$ the Dirac $\delta$-function.
Noise with these properties called white. It should always be understood that when the Langevin equation is written on the above form it should be understood as the result of such a limiting process.

For Gaussian process higher moments given by e.g.

$$\langle L(t_1)L(t_2)L(t_3)L(t_4)\rangle$$

$$= \Gamma^2[\delta(t_1-t_2)\delta(t_3-t_4)+\delta(t_1-t_3)\delta(t_2-t_4)$$

$$+\delta(t_1-t_4)\delta(t_2-t_3)]$$

We can formally find the mean and variance of a Langevin process with initial condition $v=v_0$ for $t=0$. To see this let us write the solution of the equation as

$$v(t)=e^{-\gamma t}(v_0+\int_0^te^{\gamma t'}L(t')dt')$$

Find

$$\langle v(t)\rangle =v_0e^{-\gamma t}$$

$$\langle v(t)^2\rangle=v_0^2e^{-2\gamma t}+ e^{-2\gamma t}\int_0^t\int_0^tdt'dt''e^{\gamma(t'+t'')} \langle L(t')L(t'')\rangle$$

$$=v_0^2e^{-2\gamma t}+\frac{\Gamma}{2\gamma}(1-e^-2\gamma t)$$

We next wish to demonstrate an equivalence between the Langevin and Fokker Planck approaches. Let us restrict $t$ to small value $\delta t$. The speed will then change by small amount $\delta v$.
Get jump moments

$$a_1=\frac{\langle\delta v\rangle}{\delta t}\Rightarrow-\gamma v$$

$$a_2=\frac{\langle(\delta v)^2\rangle}{\delta t}=\Gamma$$

Higher order jump moments will be of higher order in $\delta t$.
We find that Langevin equation equivalent to Fokker-Planck (Rayleigh) equation

$$\frac{\partial P(v,t)}{\partial t}=\gamma\frac{\partial vP}{\partial v}+\frac{\Gamma}{2} \frac{\partial ^2P}{\partial v^2}$$

Easy to generalize to case of nonlinear "force" with additive noise e.g.

$$\frac{d v}{d t}=A(v)+L(t)$$

assume as before

$$\langle L\rangle=0;\;\langle L(t)L(t')\rangle=\Gamma\delta(t-t')$$

We can calculate jump moments and find Langevin equation equivalent to Fokker-Planck equation

$$\frac{\partial P(v,t)}{\partial t}=-\frac{\partial A(v)P}{\partial v} +\frac{\Gamma}{2}\frac{\partial ^2P}{\partial v^2}$$

What about nonlinear noise?

$$\frac{d v}{dt}=A(v)+C(v)L(t)$$

The trouble with this is that the expression is ambiguous! (van Kampen describes the above equation as a meaningless string of symbols).
$L$ is a process with discrete jumps. Should $C(v)$ be evaluated using value of $v$ before or after each jump?
Or, should an average value be chosen? Our choice here actually matters! This is related to the fact found last time that heterogeneous diffusion gives rise to an effective drift term. Mathematically the problem is that in general

$$\langle C(v(t))L(t)\rangle\neq 0$$

In e.g. birth and death processes the noise is internal, which is to say that the noise is intrinsic to the system. It is then "natural" to say that rate of processes should be calculated before process happens. This situation is described as the Ito interpretation:

$$v(t+\delta t)=v(t)+A(v)\delta t+C(v(t))\int_t^{t+\delta t} L(t')dt'$$

It can be shown by calculating jump moments that result is Fokker-Planck equation on form



If the noise external (e.g. due to the effects of the environment in an open system) we must consider that no noise is truly white! There is always a nonzero relaxation time. It is then "natural" to evaluate $C(v)$ for average value of $v$ during process. This situation is known as the Stratanovich interpretation

$$v(t+\delta t)=v(t)+A(v)\delta t+C(\frac{v(t+\delta t) +v(t)}{2})\int_t^{t+\delta t} L(t')dt'$$

Get Fokker-Planck equation on form



Clearly the results according to the two interpretations will be different.

SUMMARY

• Langevin approach gives rise to stochastic differential equation.
• For additive white Gaussian noise Langevin equation equivalent to Fokker-Planck equation.
• When noise nonlinear need to impose interpretation.
• Ito and Stratanovich interpretation most common.
• No real mathematical difficulty. Can transform equation with Ito interpretation into one with Stratanovich interpretation or vice versa.
• Difficulty with physics. Which formula gives the correct result? To resolve this problem need to go back to master equation and look at microscopic processes.

Further reading: The material in this section is described in more detail in the book: N.G. van Kampen, Stochastic processes in physics and chemistry, North Holland (1981). See also N.G. van Kampen [1981], Ito versus Stratanovich, J. of Statistical Physics 24 175-187.