1549380323-Statistical Mechanics Theory and Molecular Simulation

592 Langevin and generalized Langevin equations

15.5.1 Numerical integration of the Langevin equation

Because the simple Langevin equation is typically employed over the full GLE in
most simulations, we will focus on numerical integration of the Langevin equation in
this section. In developing an algorithm that is accurate to order ∆t^2 in the posi-
tions and velocities, we will follow the derivation introduced by Vanden-Eijnden and
Ciccotti (2006).
From a numerical standpoint, the most important thing to note about the Langevin
equation is that the random forceR(t) is not a continuous function oftbut rather
a stochasticprocessthat is nowhere differentiable. It can, however, be realized on as
fine a time scale as required. Let us begin by writing the Langevin equation as

μ ̈q(t) =F(q(t))−γμq ̇(t) +

√

2 kTγμη(t), (15.5.1)

whereγ=ζ 0 /μ, and where we have redefined the random forceR(t) =

√

2 kTγμη(t).
Since〈R(0)R(t)〉=kTζ 0 δ(t) = 2kTμγδ(t), it follows that〈η(0)η(t)〉=δ(t). Although
R(t) andη(t) are not differentiable, we can define integrals of these processes, and
therefore, it is useful to introduce a processw(t), known as aWiener process, such
thatη(t) = dw/dt. From the properties ofη(t),w(t) can be shown to satisfy several
important properties. Let ∆tbe a small time interval. Then the following relations
hold forw(t):

〈w(s)w(s′)〉= min(s,s′)

〈∫

t+∆t

t

ds(w(s)−w(t))

∫t+∆t

t

ds′(w(s′)−w(t))

〉

=

1

3

∆t^3

〈

(w(t+ ∆t)−w(t))

∫t+∆t

t

ds′(w(s′)−w(t))

〉

=

1

2

∆t^2. (15.5.2)

As a result of these properties, a representation of a Wiener process can be defined
thus: IfR(t) is a Gaussian random process of the type we described in Section 15.2.2,
then the properties in eqns. (15.5.2) will be satisfied if

w(t+ ∆t)−w(t) =

√

∆tξ

∫t+∆t

t

ds(w(s)−w(t)) = ∆t^3 /^2

(

1

2

ξ+

1

2

√

3

θ

)

. (15.5.3)

In the last two terms of the last line of eqn. (15.5.3),ξandθare Gaussian random
variables of zero mean, unit width, and zero cross-correlation:

〈ξ^2 〉=〈θ^2 〉= 1 〈ξθ〉= 0. (15.5.4)

Because of the stochastic nature of the Langevin equation, it is often represented not
as a continuous differential equation as in eqn. (15.5.1) but rather as a relationship