3
The microcanonical ensemble and
introduction to molecular dynamics
3.1 Brief overview
In the previous chapter, it was shown that statistical mechanics provides the link be-
tween the classical microscopic world described by Newton’s laws of motion and the
macroscopic observables that are actually measured in experiments, including thermo-
dynamic, structural, and dynamical properties. One of the greatsuccesses of statistical
mechanics is its provision of a rational microscopic basis for thermodynamics, which
otherwise is only a phenomenological theory. We showed that the microscopic connec-
tion is provided via the notion of an ensemble—an imaginary collection ofsystems
described by the same Hamiltonian with each system in a unique microscopic state at
any given instant in time.
In this chapter, we will lay out the basic classical statistical mechanics of the sim-
plest and most fundamental of the equilibrium ensembles, that of anisolated system
ofNparticles in a container of volumeV and a total energyE corresponding to
a HamiltonianH(x). This ensemble is known as themicrocanonical ensemble. The
microcanonical ensemble provides a starting point from which all other equilibrium
ensembles are derived. Our discussion will begin with the classical partition function,
its connection to the entropy via Boltzmann’s relation, and the thermodynamic and
equilibrium properties that it generates. Several simple applicationswill serve to illus-
trate these concepts. However, it will rapidly become apparent that in order to treat
any realistic system, numerical solutions are needed, which will lead naturally to a
discussion of the numerical simulation technique known as molecular dynamics (MD).
MD is a widely used, immensely successful computational approach inwhich the clas-
sical equations of motion are solved numerically and the trajectories thus generated
are used to extract macroscopic observables. MD also permits direct “visualization”
of the detailed motions of individual atoms in a system, thereby providing a “win-
dow” into the microscopic world. Although such animations of MD trajectories should
never be taken too seriously, they can be useful as a guide towardunderstanding the
mechanisms underlying a given chemical process. At the end of the chapter, we will
consider a number of examples that illustrate the power and general applicability of
molecular dynamics to realistic systems.