1549380323-Statistical Mechanics Theory and Molecular Simulation

352 Free energy calculations

the reaction coordinates which, for finiteM, leads to a sum of force terms at different
times ∆t/M. This sum is in the form of a trapezoidal rule for a numerical integration
in time. Thus, when the limitM→ ∞is taken, these sums become continuous time
integrals.
Physically, eqn. (8.10.13) tells us that the force driving the slow reaction coordi-
nates is a time average over the motion of the 3N−nadiabatically decoupled fast
variables. If thenmasses assigned to the reaction coordinates are very large, the
remaining variables will follow the slow reaction coordinates approximately instanta-
neously and sample large regions of their phase space at roughly fixed values of the
reaction coordinates. In this limit, the time integrals in eqn. (8.10.13)can be replaced
by configuration-space integrals, assuming that the motion of thefast variables is
ergodic:

2

∆t

∫τ+∆t/ 2

τ

dt Fα[X,Yadb(Y(τ),Y ̇(τ),ΓY(τ),X;t)]

=

∫

dY Fα(X,Y)e−βV ̃(X,Y)

∫

dYe−βV ̃(X,Y)

=

∂

∂qα

1

β

lnZY(q 1 ,...,qn;β). (8.10.14)

Here

ZY(q 1 ,...,qn;β) =ZY(X;β) =

∫

dYe−β

V ̃(X,Y)

(8.10.15)

is the configurational partition function at fixed values of the reaction coordinatesX=
(q 1 ,...,qn). Eqn. (8.10.14) defines an effective potential, the potential of mean force,
on which the reaction coordinates move. Thus, we can define an effective Hamiltonian
for the reaction coordinates as

Heff(X,PX) =

∑n

α=1

p^2 α

2 m′α

−

1

β

lnZY(q 1 ,...,qn;β). (8.10.16)

Since we assume the dynamics to be adiabatically decoupled, thermostats applied to
this Hamiltonian yield the canonical distribution ofHeff(X,PX) at temperatureTq:

Padb(X) =Cn

[∫

dnpexp

{

−βq

∑n

α=1

p^2 α

2 m′α

}]

×exp

{

−βq

(

−

1

β

lnZY(q 1 ,...,qn)

)}

. (8.10.17)

From eqn. (8.10.17), we see that

Padb(X)∝[ZY(q 1 ,...,qn)]βq/β. (8.10.18)