1549380323-Statistical Mechanics Theory and Molecular Simulation

Molecular dynamics 593

between stochastic processes (Kuboet al., 1985). The latter is expressed in differential
form as

dq(t) =v(t)dt

dv(t) =f(q(t))dt−γv(t)dt+σdw(t), (15.5.5)

whereσ=

√

2 kTγ/μandf(q) =F(q)/μ.
Before proceeding, let us note that eqns. (15.5.5) can be easily generalized to a
system ofncoordinates as

dqi(t) =vi(t)dt

dvi(t) =f(q 1 (t),...,qn(t))dt−γivi(t)dt+σidwi(t), (15.5.6)

whereσi=

√

2 kTγi/μi. The properties in eqns. (15.5.2) for thenWiener processes
w 1 (t),...,wn(t) in eqns. (15.5.6) become

〈wi(s)wj(s′)〉= min(s,s′)δij

〈∫

t+∆t

t

ds(wi(s)−wi(t))

∫t+∆t

t

ds′(wj(s′)−wj(t))

〉

=

1

3

∆t^3 δij

〈

(wi(t+ ∆t)−wi(t))

∫t+∆t

t

ds′(wj(s′)−wj(t))

〉

=

1

2

∆t^2 δij. (15.5.7)

The Wiener processes, themselves, are defined analogously to eqn. (15.5.3)

wi(t+ ∆t)−wi(t) =

√

∆tξi

∫t+∆t

t

ds(wi(s)−wi(t)) = ∆t^3 /^2

(

1

2

ξi+

1

2

√

3

θi

)

(15.5.8)

withnindependent Gaussian random variablesξ 1 ,...,ξnandθ 1 ,...,θnfor which〈ξiξj〉=
〈θiθj〉=δijand〈ξiθj〉= 0. Since eqns. (15.5.6), (15.5.7) and (15.5.8) are the only gen-
eralizations needed to describe a system ofnvariables, in order to keep the notation
simple, we will proceed with the single-particle system in eqns. (15.5.5), noting that
the generalization of the algorithm for thencoupled Langevin equations (15.5.6) is
straightforward.
An operator-based method for deriving numerical solvers of time-dependent sys-
tems was proposed by Suzuki (1993); however, it is only applicable to systems with con-
tinuous time-dependent driving terms such as those discussed in Chapter 13. Because
of the stochastic nature of the Langevin equation, however, application of the opera-
tor formalism used in Chapters 3–5 is rather subtle and cannot be applied straightfor-
wardly (Melchionna, 2007). The alternative derivation of Vanden-Eijnden and Ciccotti