Factorial Counting in Ensembles and Statisical Mechanics

  In statistical mechanics, microcanonical factorial counting applied to systems in an ensemble is approximated by a canonical distribution with a weight of exp(-E/T) (where E is the energy of the fixed N particle system). Such an approach is even applied to a “hot” nucleus with E= Sum over i ei. In such a case, however, no mention is made of the number of particles in a system, implying such a statistical approach should apply to very small numbers of particles. It is only when the canonical distribution exp(-E/T) is approximated by the grand canonical exp(-(E-uN)/T) where N is the number of particles in a system that one finds that <N*N>-<N><N> approaches zero for large N. Then, restrictions to large numbers of particles are made. In previous notes, we have argued one may describe a statistical system in terms of reaction balance and have tried to map such a balance into a factorial scheme, thus linking it to traditional statistical mechanics. Thus, we argued there is no a priori reason for the counting of different arrangements to have meaning unless it maps into reaction balance. In this note, we wish to examine more closely equilibrium with a small number of particles to see if there is a simple map between factorial counting (arrangements of an ensemble) and reaction balance. We try to argue that there is not and that reaction balance alone applies to such cases.