Microcanonical Ensemble

The following section is adapted from the very clear and concise feature article by Tuckerman and Martyna [20].

The equations of motion are time reversible (invariant to the transformation $t \rightarrow -t$ ) and the total energy is a constant of motion

$${\partial E \over \partial t} = {\partial {\cal H}({\bf R}^N,{\bf\dot R}^N) \over \partial t} = 0 \enspace .$$ (8)

These properties are important to establish a link between molecular dynamics and statistical mechanics. The latter connects the microscopic details of a system the physical observables such as equilibrium thermodynamic properties, transport coefficients, and spectra. Statistical mechanics is based on the Gibbs' ensemble concept. That is, many individual microscopic configurations of a very large system lead to the same macroscopic properties, implying that it is not necessary to know the precise detailed motion of every particle in a system in order to predict its properties. It is sufficient to simply average over a large number of identical systems, each in a different configuration; i.e. the macroscopic observables of a system are formulated in term of ensemble averages. Statistical ensembles are usually characterized by fixed values of thermodynamic variables such as energy, $E$ ; temperature, $T$ ; pressure, $P$ ; volume, $V$ ; particle number, $N$ ; or chemical potential $\mu$ . One fundamental ensemble is called the microcanonical ensemble and is characterized by constant particle number, $N$ ; constant volume, $V$ ; and constant total energy, $E$ , and is denoted the $NVE$ ensemble. Other examples include the canonical or $NVT$ ensemble, the isothermal-isobaric or $NPT$ ensemble, and the grand canonical or $\mu VT$ ensemble. The thermodynamic variables that characterize an ensemble can be regarded as experimental control parameters that specify the conditions under which an experiment is performed.

Now consider a system of $N$ particles occupying a container of volume $V$ and evolving under Hamilton's equation of motion. The Hamiltonian will be constant and equal to the total energy $E$ of the system. In addition, the number of particles and the volume are assumed to be fixed. Therefore, a dynamical trajectory (i.e. the positions and momenta of all particles over time) will generate a series of classical states having constant $N$ , $V$ , and $E$ , corresponding to a microcanonical ensemble. If the dynamics generates all possible states then an average over this trajectory will yield the same result as an average in a microcanonical ensemble. The energy conservation condition, ${\cal H}({\bf R}^N,{\bf\dot R}^N) = E$ which imposes a restriction on the classical microscopic states accessible to the system, defines a hypersurface in the phase space called a constant energy surface. A system evolving according to Hamilton's equation of motion will remain on this surface. The assumption that a system, given an infinite amount of time, will cover the entire constant energy hypersurface is known as the ergodic hypothesis. Thus, under the ergodic hypothesis, averages over a trajectory of a system obeying Hamilton's equation are equivalent to averages over the microcanonical ensemble. In addition to equilibrium quantities also dynamical properties are defined through ensemble averages. Time correlation functions are important because of their relation to transport coefficients and spectra via linear response theory [21,22].

The important points are: by integration of Hamilton's equation of motion for a number of particles in a fixed volume, we can create a trajectory; time averages and time correlation functions of the trajectory are directly related to ensemble averages of the microcanonical ensemble.