Degeneracy, Average Occupancy and Microstates in Statistical Mechanics

 An interesting result arises in information theory, namely that if one considers a microstate with large N trials, each representing a state i with probability p(i), then even though this is only one of many possible microstates (each being a permutation of each other) it carries information about all of the permutations based on the average occupancy number. In particular, the probability of a microstate may be written as:  Product over i  p(ei) [power Np(ei) ] for large N. This holds for any microstate and so all have the same probability. Given that all of the microstates are permutations in the large N case,  this probability is equivalent to 1/ {number of permutations} where Number of permutations = N! / { Product over i  n(ei) }.

   We point out the above example, because it seems there are at least two main strategies in statistical mechanics. One is to begin with a set of average occupancy numbers n(ei/T) whose specific form is unknown. The idea then seems to be to seek degeneracy in the system, possibly among the arrangement of the n(ei)’s or degeneracy within single levels ei and to maximize the number of degenerate arrangements in order to account for as little information as possible. This approach may be used to calculate the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. In the case of the latter two, degeneracy in the single energy levels ei is key. One then maximizes the ln of the degenerate arrangements subject to constraints such as Sum over i ei n(ei) and Sum over i n(ei) = N. The degeneracies themselves are associated with momentum degeneracies yielding the same energy.

    A completely different approach, it seems, is to consider exp(-(E-uN)/T) as representing the probability of a state with energy E and N particles and to use either the grand canonical ensemble i.e. sum the various exp(-(E-uN)/T)’s. These may then be broken into single particle sums using E(N) = sum over i  ei n(ei) for n(ei) variable or sum over i ei for N fixed. Even though momentum degeneracy still exists, it is not part of the n sum of exp(-(E-uN)/T) factors which gives a single energy level probability distribution with no maximization necessary. Thus we point out that exp(-(E-uN)/T) for microstates seems to incorporate information about degeneracies without them needing to be explicitly added. On the other hand, if one starts with average n(ei) (without knowing their form),  one must explicitly consider degeneracies in order to maximize so that there is no unnecessary information in the system. Both approaches should yield the same result. We consider the Bose-Einstein example of a photon gas as an example.