1549380323-Statistical Mechanics Theory and Molecular Simulation

340 Free energy calculations

p ̇i=Fi+λ

∂σ

∂ri

, (8.7.15)

whereλis the Lagrange multiplier needed to impose the constraint. In orderto carry
out the statistical mechanical analysis of the constrained dynamics, we shall make use
of Gauss’s principle of least constraint introduced in Section 1.10. There we showed
that the Lagrange multiplier is given by

λ=−

∑

jFj·∇jσ/mj+

∑

j,k∇j∇kσ··pjpk/(mjmk)

∑

j(∇jσ)

(^2) /m
j

. (8.7.16)

When eqn. (8.7.16) is substituted into eqn. (8.7.15), the result is a set of non-Hamiltonian
equations of motion that explicitly satisfy the two conservation lawsσ(r) = 0 and
σ ̇(r,p) = 0. In addition, the equations of motion conserve the HamiltonianH(r,p),
since the forces of constraint do no work on the system. The methods of classical
non-Hamiltonian statistical mechanics introduced in Section 4.9 allow us to derive
the phase space distribution sampled by eqns. (8.7.15). According to Section 4.9, we
need to determine the conservation laws and the phase space metric in order to con-
struct the “microcanonical” partition function in eqn. (4.9.21). Thepartition function
corresponding to eqns. (8.7.15) is given by

Z=

∫

dNrdNp

√

g(r,p)δ(H(r,p)−E)δ(σ(r))δ( ̇σ(r,p)). (8.7.17)

From eqn. (4.9.11), the metric factor is

√

g= exp(−w), where dw/dt=κ, andκis
the compressibility of the system,

κ=

∑N

i=1

[∇ri·r ̇i+∇pi·p ̇i]. (8.7.18)

Note that∇ri·r ̇i= 0 so that

κ=

∑

i

∇pi·p ̇i

=−

2

∑

i(∇iσ/mi)·∇i

∑

j∇jσ·pj/mj

∑

i(∇iσ)

(^2) /m
i
=−

2

∑

∑i∇iσ·∇iσ/m ̇ i

i(∇iσ)

(^2) /m
i

=−

d

dt

ln

[

∑

i

(∇iσ)^2 /mi

]

=

dw

dt

. (8.7.19)

The metric, therefore, becomes