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

For a discrete function f (x) on a discrete set, the finite difference can be either forward and backward. If f (x) is a sum of two such functions f (x) = f(1) (x)+ f(2) (x), the first order difference of Delta f (x) can be grouped into four possible combinations, in which two are the usual synchronous ones Delta(f) f(1) (x)+ Delta(f) f(2) (x) and Delta(b)f(1) (x)+ Delta(b)f(2) (x), and other two are asynchronous ones Delta(f) f(1) (x)+ Delta(b)f(2) (x) and Delta(b)f(1) (x)+ Delta(f) f(2) (x), where.f and.b denote the forward and backward difference respectively. Thus, the first order variation equation delta f (x) = 0 for this function f (x) gives at most four different solutions which contain both true and false one. 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. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

This article introduces the finite difference method for discrete functions and the formalism of discrete calculus of variations to determine the true solution of the function. By comparing second-order variations, the exact distributions of Boltzmann, Bose, and Fermi systems can be obtained without requiring an infinitely large number of particles.