9. Langevin approach

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.
• Heteregeneous 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 stochastic d.e.
Simplest case: BROWNIAN MOTION.




v= velocity of Brownian particle.
$\gamma=$ friction term.$\;\;$ L(t)= noise term.
Collisions with other molecules gives rise to
average force (friction) +random force
Require:

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

if $t-t'>>\tau$ time between molcular collisions require

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

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

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

x 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!
Discretize time

$$t_n=n\Delta$$

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

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 range
0<r<1.
Simple method to generate 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

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

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

We next generalize to take the limit $\Delta=0$ 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.
For Gaussian process higher moments given by e.g.

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

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

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

Can formally solve Langevin equation with initial condition v=v₀ for t=0

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

Find

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

$$\langle v(t)^2\rangle=v_0^2e^{-2\gamma t}$$

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

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

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 of higher order in $\delta t$.
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

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

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?

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

Trouble is expression is ambiguous!
L is a process with discrete jumps.

Should C(v) be evaluated using value of v before or after jump?
Or, should an average value be chosen?
Choice matters! (We saw last time that heterogeneous diffusion gives rise to effective drift term.)

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

If noise internal:
E.g. birth and death processes
"Natural" to say that rate of processes should be calculated before process happens.
Ito interpretation:

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

Can show by calculating jump moments that result is Fokker-Planck equation on form



If noise external
Open system.
Must consider that no noise is truly white!
There is always a nonzero relaxation time.
"Natural" to evaluate C(v) for average value of v during process.
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



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? Need to go back to master equation and look at microscopic processes.

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