1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 477

1

2

3

5 4

6

7

8

k=mωP^2

Fig. 12.10Classical isomorphism: The figure shows a cyclic polymer chain havingP= 8
described by the partition function in eqn. (12.6.2).

The classical isomorphism allows us, in principle, to introduce a molecular dynamics
scheme for eqn. (12.6.2), starting with the classical Hamiltonian

Hcl(x,p) =

∑P

k=1

[

p^2 k

2 m′

+

1

2

mω^2 P(xk+1−xk)^2 +

1

P

U(xk)

]∣∣

∣

∣

∣

xP+1=x 1

, (12.6.3)

which yields the following equations of motion:

x ̇k=

pk

m′

, p ̇k=−mω^2 p(2xk−xk+1−xk− 1 )−

1

P

∂U

∂xk

. (12.6.4)

If eqns. (12.6.4) are coupled to a thermostat, as discussed in Section 4.10, then the dy-
namics will sample the canonical distribution in eqn. (12.6.2). Althoughone of the first
path-integral molecular dynamics calculations by Parrinello and Rahman (1984) em-
ployed a scheme of this type, it was simultaneously recognized by Halland Berne (1984)
that path-integral molecular dynamics based on eqns. (12.6.4) cansuffer from very slow
convergence problems due to the wide range of time scales presentin the dynamics. If
one applies a normal mode transformation (see Section 1.7) to the harmonic coupling
term in eqn. (12.6.3), the normal-mode frequencies range densely from 0 to 4P/(β ̄h).
Thus, even for moderately largeP, this constitutes a broad frequency spectrum. The
time step that can be employed in a molecular dynamics algorithm is limitedby the
highest frequency, which means that the low-frequency modes, which are associated
with large-scale changes in the shape of the cyclic chain, will be inadequately sampled
unless very long runs are performed.
Because the normal-mode frequencies are closely spaced (approaching a contin-
uum asP → ∞), multiple time-scale integration algorithms such as RESPA (see
Section 3.11) are insufficient to solve the problem. However, if we candevise a suitable