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

Abstract

For a discrete function fx on a discrete set, the finite difference can be either forward and backward. If fx is a sum of two such functions fx=f1x+f2x, the first order difference of Δfx can be grouped into four possible combinations, in which two are the usual synchronous ones Δff1x+Δff2x and Δbf1x+Δbf2x, and other two are asynchronous ones Δff1x+Δbf2x and Δbf1x+Δff2x, where Δf and Δb denote the forward and backward difference respectively. Thus, the first order variation equation δfx=0 for this function fx 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.