TABLE OF CONTENTS For the canonical ensemble, the total energy of the ensemble is a constant and the interactions among its member systems are weak. This means that the energy for most of the member systems cannot be very far from the average energy U and there must be some distribution { n i} for which the number of complexions is very large relative to that for other complexions. That is, there must be a distribution that maximizes the number of complexions, subject, of course, to the conditions that the number of systems in the ensemble and the total ensemble energy are both constant. The n i that maximize W{ n i} subject to constant X and constant UX can be found by the method of undetermined multipliers (see appendix 2). It is more convenient to apply this method to ln W{ n i} than to W{ n i}. Accordingly, the equations to be solved are
where (2.10.1) follows from (2.9.1).
Note that we have adopted the continuum notation of the calculus of variations even though the n i can change only in discrete steps. The number of systems can be taken to be so large that this is an excellent approximation. This is often the case in the statistical mechanics of macroscopic systems, and we will use either the discrete or the continuum notation as convenient.
Equations (2.10.1) (2.10.3) are readily simplified by using Stirling's...
