Asynchronous finite differences in most probable distribution with finite numbers of particles

TL;DR

This paper develops a discrete calculus of variations formalism to accurately determine the most probable distribution in particle systems, including quantum statistics, without assuming infinite particle numbers.

Contribution

It introduces a novel approach using asynchronous finite differences and second order variations to identify true solutions for particle distributions.

Findings

Exact distributions for Boltzmann, Bose, and Fermi systems derived

Method works with finite particle numbers, including single-particle cases

Provides a new discrete calculus framework for statistical mechanics

Abstract

For a discrete function $f\left( x\right) $ on a discrete set, the finite difference can be either forward and backward. If $f\left( x\right) $ is a sum of two such functions $f\left( x\right) =f_{1}\left( x\right) +f_{2}\left( x\right) $, the first order difference of $\Delta f\left( x\right) $ can be grouped into four possible combinations, in which two are the usual synchronous ones $\Delta ^{f}f_{1}\left( x\right) +\Delta ^{f}f_{2}\left( x\right) $ and $\Delta ^{b}f_{1}\left( x\right) +\Delta ^{b}f_{2}\left( x\right) $, and other two are asynchronous ones $\Delta ^{f}f_{1}\left( x\right) +\Delta ^{b}f_{2}\left( x\right) $ and $\Delta ^{b}f_{1}\left( x\right) +\Delta ^{f}f_{2}\left( x\right) $, where $\Delta ^{f}$ and $\Delta ^{b}$ denotes the forward and backward difference respectively. Thus, the first order variation equation $\Delta f\left( x\right) =0$ for this function $f\left( x\right) $ gives at most four different solutions which contain both true and false one. \emph{A formalism of the discrete calculus of variations is developed to single out the true one by means of comparison of the second order variations, in which the largest value in magnitude indicates the true solution, yielding the exact form of the distributions for Boltzmann, Bose and Fermi system without requiring the numbers of particle to be infinitely large}. When there is only one particle in the system, all distributions reduce to be the Boltzmann one.