1549380323-Statistical Mechanics Theory and Molecular Simulation

604 Langevin and generalized Langevin equations

reasonable approximation toζ(t)onlyin the high-frequency limit. In fact, Berneet al.
showed formally thatζ(t) can be calculated in this limit from the correlation function

ζ(t)

μ

=

〈δfei

Lt ̄

δf〉

〈p^2 〉

, (15.7.35)

whereiL ̄is the Liouville operator for a system in which the oscillator coordinate
xis fixed atx=x 0. Eqn. (15.7.35) can be used, for example, to approximate the
friction on a harmonic diatomic molecule by replacing it with a rigid diatomicin
which the distance between the two atoms is constrained to the value of the equilibrium
bond length. Importantly, as the frequency of the oscillator decreases, this rigid-bond
approximation breaks down, and care is needed to determine whenφ(t) is a good
approximation toζ(t).

15.8 Problems

15.1. a. Use the definition ofR(t) for the harmonic-bath model in eqn. (15.2.11)

to show that〈q(0)R(t)〉= 0 and to derive eqn. (15.2.17).

b. WhenR(t) is taken to be a true random process, it can be expressed as

a Fourier series evaluated atMtime points 0,∆t,2∆t,...,(M−1)∆tas

R(k∆t) =

M∑− 1

j=0

[

ajcos

(

2 πjk

M

)

+bjsin

(

2 πjk

M

)]

,

where the coefficients are random numbers sampled from a Gaussiandis-

tribution

P(a 0 ,...,aM− 1 ,b 0 ,...,bM− 1 ) =

M∏− 1

k=0

1

2 πσ^2 k

e−(a

(^2) k+b (^2) k)/ 2 σk 2
and
σk^2 =
kT
M

M∑− 1

j=0

ζ(j∆t) cos

(

2 πjk

M

)

.

Using these definitions, prove that

〈R(0)R(k∆t)〉=kTζ(k∆t),

where the average is performed over all possible realizations ofR(k∆t).

Hint: Think carefully about what distinguishes one realization of the

random force from another and define the average accordingly.