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

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.