Degeneracy, Average Occupancy and Microstates in Statistical Mechanics Part II

 In Part II we examine how information may be used to define physical distribution functions. In particular, we argue that a mathematical function may be equivalent to certain pieces of information. For example, for elastic two body scattering e1+e2 = e3+e4. Thus ei is one piece of information in the problem. A second piece is the distribution function n(ei/T) where T is a parameter. A third piece of information is that n(ei)n(ej) = n(ei+ej) or reaction balance:  n(e1)n(e2)=n(e3)n(e4). These three pieces of information lead to the result:  ln(n(ei)) = -ei/T ((AA))Thus we argue that the function exp(-ei/T) may be thought of in terms of ((AA)).

   Interestingly, we argue, ((AA)) is exactly of the form of a maximization of the function  -n(ei) ln(n(ei)) subject to the constraint ei n(ei). Why should maximization come into play if one has three pieces of information which already solve the problem? n(ei) is proportional to a probability for large N i.e. n(ei) = Np(ei). Given probabilities, one may consider repeated trials i.e. N of them for N very large. In the large N limit one expects ei to be associated with a probability p(ei) [ power Np(ei)]. Each trial of N events is a microstate and has the same probability, thus the microstates are permutations of each other. The probability should be 1/ { number of permutations }. This still does not define n(ei) if one writes: 1/ { number of permutations } = 1/ {  N!/ Product over i n(ei)! } with the constraint that Sum over i ei n(ei) = Eave. The constraint term shows that ei is one piece of information. One may take ln of 1/ { number of permutations }  in order to have a sum of terms instead of a product i.e.  -Sum  over i  ln(n(ei)!). The idea from the first set of 3 information pieces that ln(n(ei)) should be associated with ei is identical to maximizing ln(n(ei)!) with respect to n(ei). Thus the function exp(-ei/T) is associated with the maximization of  removed information  n(ei)! (because n(ei)! Is in the denominator of N!/ Product over i n(ei)!). 

    The second example we consider is the relaxation of both a fixed Etotal and N i.e. constraints Sum over i n(ei)= Nave and Sum over i ei n(ei) = Eave. If one uses exp(-(E-uN)/T), a function which arose from the three pieces of information (or in other ways, for example from the constraints Sum over i ei n(ei) and Sum over i n(ei)), and considers E and N in terms of single particle values i.e.  E= Sum over i  ei n and N= Sum over i n (in i) then for a fixed ei, one may sum over n. This means one considers something special about the level ei i.e. there is some information linked to a level ei which is not present in the simple Maxwell-Boltzmann distribution or in the simple reaction balance e1+e2=e3+e4 and n(e1)n(e2)=n(e3)n(e4). (This special information, however, must correspond to actual information which appears in nature.) 

   Given that exp(-ei/T) is linked to the constraint Sum over i ei n(ei) and also to maximization permutations, one may guess that the two constraints: Sum over i ei n(ei) and Sum over i n(ei) correspond to an exp(-(ei-u)/T) which in turn is also linked to some kind of permutations. Since one sums of n for a given ei, it seems one considers probabilities of placing more than one particle in each energy level, with a product factor exp(-(ei-u)/T) each time one adds another particle to ei (in the boson case). This is an extra piece of information and may be linked with the maximization of permutations within a state ei with gi degenerate boxes. Thus instead of simply removing information n(ei)! In the denominator of a permutation expression, one may be trying to maximize information in the numerator through an expression (a n(ei) + b)! where a and b are constants. Thus, sums of products of exp(-(ei-u)/T) factors may be ultimately linked to functions of the form: (a n(ei) + b)! / n(ei)!. This we argue is why the grand canonical distribution produces results identical to the maximization of permutations within energy levels ei with degeneracies of gi. The factor (ei-u)/T in exp(-(ei-u)/T) from the grand canonical partition function is identical to d/d n(ei) of the two constraints  Sum over i ei n(ei) and Sum over i n(ei) used in any maximization problem.