Computational Physics - Department of Physics

408 12 Random walks and the Metropolis algorithm

Mn=

1

τ

∫∞

−∞

ξnW(x+ξ,τ|x)dξ=

〈[∆x(τ)]n〉

τ

,

resulting in
∂W(x,s|x 0 )
∂s

=

∞

∑

n= 1

(−ξ)n

n!

∂n

∂xn

[W(x,s|x 0 )Mn].

Whenτ→ 0 we assume that〈[∆x(τ)]n〉→ 0 more rapidly thanτitself ifn> 2. Whenτis
much larger than the standard correlation time of system thenMnforn> 2 can normally be
neglected. This means that fluctuations become negligible at large time scales.
If we neglect such terms we can rewrite the ESKC equation as

∂W(x,s|x 0 )

∂s =−

∂M 1 W(x,s|x 0 )

∂x +

1

2

∂^2 M 2 W(x,s|x 0 )

∂x^2.

In a more compact form we have

∂W

∂s

=−∂M^1 W

∂x

+^1

2

∂^2 M 2 W

∂x^2

,

which is the Fokker-Planck equation. It is trivial to replace position with velocity (momentum).
The solution to this equation is a Gaussian distribution andcan be used to constrain pro-
posed transitions moves, that one can model the transition probabilitiesTfrom our discussion
of the Metropolis algorithm.

12.6.2Langevin Equation

Consider a particle suspended in a liquid. On its path through the liquid it will continuously
collide with the liquid molecules. Because on average the particle will collide more often on
the front side than on the back side, it will experience a systematic force proportional with its
velocity, and directed opposite to its velocity. Besides this systematic force the particle will
experience a stochastic forceF(t). The equations of motion then read

dr

dt

=v,

dv
dt=−ξv+F,
The last equation is the Langevin equation. The original Langevin equation was meant to
describe Brownian motion. It is a stochastic differential equation used to describe the time
evolution of collective (normally macroscopic) variablesthat change only slowly with respect
to the microscopic ones. The latter are responsible for the stochastic nature of the Langevin
equation. We can say that we model our ignorance about the microscopic physics in a stochas-
tic term. From the Langevin equation we can in turn derive forexample the fluctuation dissi-
pation theorem discussed below. To see, we need some information about the friction constant
from hydrodynamics. From hydrodynamics we know that the friction constantξis given by

ξ= 6 π ηa/m

whereηis the viscosity of the solvent andais the radius of the particle.
Solving the Langevin equation we get

v(t) =v 0 e−ξt+

∫t

0

dτe−ξ(t−τ)F(τ).