Generalized Entropy Approach for Conserved Systems with Finite vs Infinite Entities: Insights into Non-Gaussian and Non-Chi-Square Distributions using Havrda-Charv\'at-Tsallis Entropy vs Gaussian Distributions via Boltzmann-Shannon Entropy

Abstract

We demonstrate that the most probable state of a conserved system with a limited number of entities or molecules is the state where non-Gaussian and non-chi-square distributions govern. We have conducted a thought experiment using a specific setup. We have verified the mathematical derivation of the most probable state accurately predicts the results obtained by computer simulations. The derived distributions approach the Gaussian and the chi-square distributions as the number of entities approaches infinity. The derived distributions of the most probable state will have an important role in the fields of medical research where the number of entities in the system of interest is limited. Especially, the non-chi-square distribution can be interpreted by an asset distribution achieved after a repetitive game where an arbitrary portion of one's assets is transferred to another arbitrary entity among a number of entities with equal abilities.