PRINCIPLES OF STATISTICAL MECHANICS

Abstract

The procedure of taking time averages for a given system becomes highly complicated if the system consists of a large number of particles. Following the suggestion of Gibbs, it is more appropriate to adopt an alternative procedure of taking ensemble averages at a fixed time over a collection of dynamical systems each having the same construction as the actual system. The collection of systems is called an ensemble. All the copies of the given system, called elements of the ensemble, consist of N particles with f degrees of freedom and have the same Hamiltonian. Their representative points in phase space are assumed to be suitably random so that every microstate accessible to the actual system during its motion is represented at any given instant by at least one system. Each of these microstates must be consistent with the same macrostate associated with the ensemble. In any volume element Ωd of phase space there will be at time t a certain number of representative points If we consider a sufficiently large number of elements of the ensemble, it is convenient to define a phase density of the representative points in phase space in the form .dn ( tpqD kk ,, ) ( ) Ω= d dntpqD kk ,, (25.1) where is the volume element in phase space, given by Ωd (25.2) ff dpdpdpdqdqdqd …… 2121=Ω In the course of time, every element of the ensemble undergoes a continual change of microstates, so that the assembly of representative points moves with time through phase space like a fluid of density D. The number of phase points in the volume element dn Ωd will change with time at a rate given by Ω∂ ∂=∂ ∂ d t D t n (25.3) where gives the rate of change with respect to a fixed point in phase space, and this can be estimated by taking the difference between the number of points entering and leaving , as suggested in Figure 25.1.