1549380323-Statistical Mechanics Theory and Molecular Simulation

98 Microcanonical ensemble

This fact suggests an intimate connection between the microcanonical ensemble and
classical Hamiltonian mechanics. In the latter, we have seen that the equations of
motion conserve the total energy, dH/dt= 0⇒H(x) = const. Imagine that we have
a system evolving according to Hamilton’s equations:

q ̇α=

∂H

∂pα

, p ̇α=−

∂H

∂qα

. (3.7.1)

Since the equations of motion conserve the HamiltonianH(x), a trajectory computed
via Eqns. (3.7.1) will generate microscopic configurations belonging to a microcanon-
ical ensemble with energyE. Suppose, further that given an infinite amount of time,
the system with energyEis able to visit all configurations on the constant energy
hypersurface. A system with this property is said to beergodicand can be used to
generate a microcanonical ensemble. In general, dynamical systems provide a powerful
approach for generating an ensemble and its associated averages, and they form the
basis of the molecular dynamics methodology, which has evolved into one of the most
widely used techniques for solving statistical mechanical problems.
Given an ergodic trajectory generated by a HamiltonianH(x), microcanonical
phase space averages can be replaced by time averages over the trajectory according
to

〈a〉=

∫

dxa(x)δ(H(x)−E)

∫

dxδ(H(x)−E)

= lim

T→∞

1

T

∫T

0

dt a(xt)≡ ̄a. (3.7.2)

In a molecular dynamics calculation, eqns. (3.7.1) are solved numerically subject to
a given set of initial conditions. Doing so requires the use of a particularnumerical
integratororsolverfor the equations of motion, a topic we shall take up in the next
section. An integrator generates phase space vectors at discrete times that are multiples
of a fundamental time discretization parameter, ∆t, known as thetime step. Starting
with the initial condition x 0 , phase space vectors xn∆twheren= 0,...,Mare generated
by applying the integrator or solver iteratively. The ensemble average of a property
a(x) is then related to the discretized time average by

A=〈a〉=

1

M

∑M

n=1

a(xn∆t)≡ ̄a. (3.7.3)

The molecular dynamics method has the particular advantage of yielding equilibrium
averages and dynamical information simultaneously. This is an aspect of molecular
dynamics that is not shared by other equilibrium methods such as Monte Carlo (see
Chapter 7). Although we will develop the foundations of molecular dynamics in detail
over the next few chapters, we will restrict our usage of the technique, for now, to the
calculation of equilibrium averages only. We will not see how to use the dynamical
information available from molecular dynamics calculations until Chapter 13.
In the preceding discussion, many readers will have greeted the assumption of
ergodicity, which seems to underly the molecular dynamics approach, with a dose
of skepticism. Indeed, this assumption is a rather strong one thatclearly will not
hold for a system whose potential energyU(r) possesses high barriers—regions where
U(r)> E, leading to separatrices in the phase space. In general, it is not possible to