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

TL;DR

This paper explores how the most probable states of conserved systems with finite entities are governed by non-Gaussian and non-chi-square distributions, using generalized entropy measures, and how these distributions converge to classical forms as the number of entities increases.

Contribution

It introduces a generalized entropy framework to describe the most probable states of finite conserved systems and demonstrates the convergence to classical distributions with increasing system size.

Findings

Derived distributions accurately predict simulation results.

Distributions approach Gaussian and chi-square forms as entities increase.

Non-Gaussian distributions are relevant for systems with limited entities.

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.