Computational Physics - Department of Physics

12.6 Langevin and Fokker-Planck Equations 409

If we want to get some useful information out of this, we have to average over all possible
realizations ofF(t), with the initial velocity as a condition. A useful quantityis then

〈v(t)·v(t)〉v 0 =v− 0 ξ^2 t+ 2

∫t

0

dτe−ξ(^2 t−τ)v 0 ·〈F(τ)〉v 0

+

∫t

0

dτ′

∫t

0

dτe−ξ(^2 t−τ−τ

′)

〈F(τ)·F(τ′)〉v 0.

In order to continue we have to make some assumptions about the conditional averages of
the stochastic forces. In view of the chaotic character of the stochastic forces the following
assumptions seem to be appropriate. We assume that

〈F(t)〉= 0 ,

and
〈F(t)·F(t′)〉v 0 =Cv 0 δ(t−t′).

We omit the subscriptv 0 when the quantity of interest turns out to be independent ofv 0. Using
the last three equations we get

〈v(t)·v(t)〉v 0 =v^20 e−^2 ξt+

Cv 0

2 ξ

( 1 −e−^2 ξt).

For largetthis should be equal to the well-known result 3 kT/m, from which it follows that

〈F(t)·F(t′)〉= 6

kT

mξ δ(t−t

′).

This result is called the fluctuation-dissipation theorem.
Integrating
v(t) =v 0 e−ξt+

∫t

0

dτe−ξ(t−τ)F(τ),

we get

r(t) =r 0 +v 0

1

ξ

( 1 −e−ξt)+

∫t

0

dτ

∫τ

0

τ′e−ξ(τ−τ

′)

F(τ′),

from which we calculate the mean square displacement

〈(r(t)−r 0 )^2 〉v 0 =

v^20

ξ

( 1 −e−ξt)^2 +

3 kT

mξ^2

( 2 ξt− 3 + 4 e−ξt−e−^2 ξt).

For very largetthis becomes

〈(r(t)−r 0 )^2 〉=

6 kT

mξt

from which we get the Einstein relation

D=

kT

mξ

where we have used〈(r(t)−r 0 )^2 〉= 6 Dt.
The standard approach in for example quantum mechanical diffusion Monte Carlo calcula-
tions, is to use the Langevin equation to propose new moves (for examples new velocities or
positions) since they will depend on the given probability distributions. These new proposed
states or values are then used to compute the transition probabilityT, where the latter is the
solution of for example the Fokker-Planck equation.