Why Arrangements and their Maximization in Statistical Mechanics?

Addendum April 21, 2020 For further discussion, please also see: Why Arrangements and their Maximization in Statistical Mechanics? Part 2. Addendum April 20, 2020 In statitiscal texts, physical restrictions (fermions) and enhancements (bosons) are linked to factorial arrangement schemes. A fermion may physically occupy a state or not, so this is related to a physical arrangement. In the case, of bosons, however, it seems one has to be more careful. In the note below, we argue that B= (n+g-1)! /[n! (g-1)!] is the number of boson groupings which seems to be a physical "arrangement or configuration grouping of then n bosons. This grouping, however, is associated with a linear boson enhancement rate for scattering i.e. (n+1). For the sake of argument, if the enhancement were (n+1)*(n+1) (or any power), there would still be the same "physical arrangement or configuration" grouping given above, but that would not account for the (n+1) squared enhancement, while B*B would, but B*B (etc) does not seem to be a counting of physical boson groups, rather it is the square of such groupings. Thus, one has to be careful about taking physical arrangement or grouping arguements too seriously. This is not made clear in the note below. Statistical mechanics texts and also information theory emphasize the use of arrangements (usually using factorials). For example, the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) are often derived from factorial schemes (1) and in information theory, the number of arrangements is a key computation. For example, for a binary alphabet consisting of 0s and 1s, an entropy function equivalent to that of the Fermi-Dirac case is obtained. In a number of notes, we argued one may use reaction balancing to obtain equilibrium distributions, but also showed this could be mapped into arrangements because particles with different energies must be “next to each other” to collide. We also showed that for generalized cases like FD and BE, one may obtain an independent scattering object g(f(e)), where f(e) is the number of particles with energy e, such that g(f(e))=exp(-(e-u)/T). (2) In the case of Maxwell-Boltzmann statistics, it is often argued in texts that the number of configurations or positional arrangements is key and is linked to the entropy. We have argued in previous notes that it is the activity in a system which is key. For the MB case, f(e) is the number of particles, but also the probability for a particle with f(e) to collide so for this special case, one may argue either arrangements or activity. For FD and BE cases, we argue that activity is key, although physical arrangements are linked to this activity. For example, for the FD case, a state may have 1 or 0 particles (which is a configurational statement), but it is directly linked to whether scattering into this state may occur or not. For the BE case, one may have f(e) bosons with energy e, but for scattering considerations, it is the number of bosons in the same location in space (or in a degenerate box located at a specific position) which enhances the scattering probability into that same degenerate box. The purpose of this note is to try to examine why arrangements are used in statistical mechanics, how they link to activity and why they are maximized, especially given that one may obtain the equilibrium distribution f(e) using reaction balance which uses “averaging”, but does not directly use any arrangement or factorial schemes