96 Microcanonical ensemble
Ω(N,E) =
(
2 πE
h) 3 N
1
Γ(3N)
∏N
i=11
ω^3 i, (3.6.18)
or, using Stirling’s approximation Γ(3N)≈(3N)!≈(3N)^3 Ne−^3 N,
Ω(N,E) =
(
2 πE
3 Nh) 3 N
e^3 N∏N
i=11
ω^3 i. (3.6.19)
We can now calculate the temperature of the collection of oscillatorsvia
1
kT=
(
∂ln Ω(N,E)
∂E)
N=
3 N
E
, (3.6.20)
which leads to the familiar relationE= 3NkT. Note that this result is readily evi-
dent from the virial theorem eqn. (3.3.1), which also dictates that the average of the
potential and kinetic energies each be 3NkT/2.
The harmonic bath and ideal gas systems illustrate that the microcanonical en-
semble is not a particularly convenient ensemble in which to carry out equilibrium
calculations due to the integrations that must be performed over the Diracδ-function.
In the next three chapters, three different statistical ensembles will be considered that
employ different sets of thermodynamic control variables other thanN,V, andE. It
will be shown that all statistical ensembles become equivalent in the thermodynamic
limit, and therefore one has the freedom to choose the most convenient statistical en-
semble for a given problem (although some care is needed when applying this notion
to finite systems). The importance of the microcanonical ensemblelies not so much
in its utility for equilibrium calculations but rather in that it is the only ensemble in
which the dynamics of a system can be rigorously generated. In theremainder of this
chapter, therefore, we will begin our foray into the numerical simulation technique of
molecular dynamics, which we will show is capable both of sampling an equilibrium
distribution and producing true dynamical observables.
3.7 Introduction to molecular dynamics
Calculating the partition function and associated thermodynamic and equilibrium
properties for a general many-body potential that includes nonlinear interactions
becomes an insurmountable task if only analytical techniques are employed. Unless
a clever transformation can be devised, it is very unlikely that the integrals in eqns.
(3.2.16) and (3.2.22) can be performed analytically. In this case, theonly recourses are
to introduce simplifying approximations, replace a given system by a simpler model
system, or employ numerical methods. In the remainder of this chapter, our discus-
sion will focus on such a numerical approach, namely, the methodology of molecular
dynamics.
Molecular dynamics is a technique that allows a numerical “thought experiment”
to be carried out using a model that, to a limited extent, approximates a real physical
or chemical system. Such a “virtual laboratory” approach has the advantage that many
such “experiments” can be easily set up and carried out in succession by simply varying