Probability Distributions, Information and Distributed Quantity
Francesco R. Ruggeri · Zenodo (CERN European Organization for Nuclear Research) · 2023
In many cases, probability distributions are obtained by considering the number of ways one may place particles in various cells. This leads to expressions involving combinatorics with Stirling’s high number approximation being used together with the idea of permutations of identical particles only counting once. Examples include a Maxwell-Boltzmann gas and a three dimensional spherical random walk problem (1). In general, the ln of a the number of arrangements (or probability of the arrangements) is maximized subject to a constraint. This constraint, however, is “special” in that it represents the distributed variable in the problem such that this variable is additive. In this note, we try to obtain the same results without combinatorics and Stirling’s approximation by considering the quantity which is distributed (for example energy or spatial volume) and the idea of a lack of information. We suggest that one should be able to directly consider the distributed variable because the constraint in the entropy maximization problem is not a general mathematical constraint, but the distributed variable. Thus the maximization approach is a little unusual. For example, in an MB gas, the constraint is Sum over i ei n(ei), but mathematically Sum over i ei*ei n(ei) could be used or any other ei function, but the exponent of the probability is governed by the constraint function. This suggests a more direct approach to obtaining the probability which we consider here.