1549380323-Statistical Mechanics Theory and Molecular Simulation

594 Langevin and generalized Langevin equations

is simple yet elegant. We begin by integrating eqns. (15.5.5) fromttot+ ∆tto yield
a pair of integral equations

q(t+ ∆t)−q(t) =

∫t+∆t

t

ds v(s)

v(t+ ∆t)−v(t) =

∫t+∆t

t

ds f(q(s))−γ

∫t+∆t

t

ds v(s)

+σ[w(t+ ∆t)−w(t)]. (15.5.9)

Note that the second line in eqn. (15.5.9) also holds for anys∈[t,t+ ∆t]:

v(s) =v(t) +

∫s

t

du f(q(u))−γ

∫s

t

du v(u) +σ[w(s)−w(t)]. (15.5.10)

Sinces∈[t,t+ ∆t], for small ∆t, eqn. (15.5.10) can be approximated as

v(s)≈v(t) + (s−t)f(q(t))−(s−t)γv(t) +σ[w(s)−w(t)]. (15.5.11)

Integrating eqn. (15.5.11) fromttot+ ∆tyields

∫t+∆t

t

ds v(s) = ∆tv(t) +

1

2

∆t^2 [f(q(t))−γv(t)]

+σ

∫t+∆t

t

ds[w(s)−w(t)]. (15.5.12)

Similarly, we can evaluate time integrals of the force appearing in eqn.(15.5.9). By
integrating the identity df/dt= (∂f/∂q) ̇q= (∂f/∂q)vfromttos, we obtain

f(q(s)) =f(q(t)) +

∫s

t

du v(u)f′(q(u))

≈f(q(t)) + (s−t)v(t)f′(q(t)). (15.5.13)

Hence, integrating eqn. (15.5.13) fromttot+ ∆tyields

∫t+∆t

t

ds f(q(s)) = ∆tf(q(t)) +

1

2

∆t^2 v(t)f′(q(t)). (15.5.14)

Finally, substituting eqns. (15.5.14) and (15.5.12) into eqn. (15.5.9) and using the
properties of the Wiener process in eqns. (15.5.2) and (15.5.3) yieldsthe following
evolution scheme for the Langevin equation: