  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.
