1549380323-Statistical Mechanics Theory and Molecular Simulation

576 Langevin and generalized Langevin equations

q ̇=

∂H

∂p

=

p

μ

p ̇=−

∂H

∂q

=−

dV

dq

−

∑

α

gαxα

x ̇α=

∂H

∂pα

=

pα

mα

p ̇α=−

∂H

∂xα

=−mαω^2 αxα−gαq, (15.2.1)

which can be written as the following set of coupled second-order differential equations:

μq ̈=−

dV

dq

−

∑

α

gαxα

mα ̈xα=−mαω^2 αxα−gαq. (15.2.2)

Eqns. (15.2.2) must be solved subject to a set of initial conditions

{q(0),q ̇(0),x 1 (0),...,xn(0),x ̇ 1 (0),...,x ̇n(0)}.

The second equation for the bath coordinates can be solvedin terms of the system
coordinateqby Laplace transformation, assuming that the system coordinateqacts
as a kind of driving term. The Laplace transform of a functionf(t), alluded to briefly
in Section 14.6, is one of several types of integral transforms defined to be

f ̃(s) =

∫∞

0

dte−stf(t). (15.2.3)

As we will now show, Laplace transforms are particularly useful forsolving linear
differential equations. A more detailed discussion of Laplace transforms is given in
Appendix D. From eqn. (15.2.3), it can be shown straightforwardly that the Laplace
transforms of df/dtand d^2 f/dt^2 are given, respectively, by

∫∞

0

dte−st

df

dt

=sf ̃(s)−f(0)

∫∞

0

dte−st

d^2 f

dt^2

=s^2 f ̃(s)−f′(0)−sf(0). (15.2.4)

Finally, the Laplace transform of a convolution of two functionsf(t) andg(t) can be
shown to be ∫∞

0

dte−st

∫t

0

dτf(τ)g(t−τ) =f ̃(s) ̃g(s). (15.2.5)

Eqns. (15.2.4) and (15.2.5), together with eqn. (D.2), are sufficientfor us to solve eqns.
(15.2.2).