The Microcanonical Ensemble

In this chapter we apply Monte Carlo methods to the simulation of the equilibrium properties of systems with many degrees of freedom. This application will allow us to explore the methodology of statistical mechanics and to introduce important concepts.

We first discuss as closed system of fixed number of particles $N$, volume $V$, and total energy $E$. We also assume that the system is isolated, i.e. that the influence of external parameters such as gravitational and magnetic fields can be ignored. We know that in general a closed macroscopic system tends to a time-independent equilibrium stat of maximum randomness or entropy. The macrostate of the system is the specified by the values of $N$, $V$ and $E$. At a microscopic level there are in general a large number of different ways or configurations in which a microstate $(N,V,E)$ can be realized. A particular configuration or ``microstate'' is ``accessible'' if its properties are consistent with the specified macrostate.

Since in principle we don not have a reason to prefer a microstate over another, it is reasonable to postulate that at any given time, the system is equally likely to be on any one of its accessible microstates. This can be formulated through a probability $P_s$ of finding the system in a microstate $s$

$$P_s = 1/\Omega, \mathrm{if} s \mathrm{is accessible}.$$ (110)

The sum of $P_s$ over all the $\Omega$ states is equal to unity.

To estimate the average of a physical quantity it is convenient to formulate statistical averages at a fixed instant of time. Imagine a collection or ``ensemble'' of systems which are mental replicas characterized by the same macrostate. The number of systems equals the number of possible microstates. An ensemble of such systems specified by $E$, $N$, $V$ which is described by the probability distribution (111) is called a ``microcanonical ensemble''.

Supouse that a physical quantity $A$ has the value $A_s$ when the system is in the state $s$. Then, the ensemble average of $A$ is given by

$$\langle A \rangle = \sum_s{A_sP_s}.$$ (111)