Abstract:
It is shown in the paper that the number $p_N(M)$ of partitions of
a positive integer $M$ into $N$ positive integer summands coincides
with the Bose and Fermi distributions with logarithmic accuracy if
one identifies $M$ with energy and $N$ with the number of
particles. We use the Gentile statistics (a.k.a. parastatistics) to
derive self-consistent algebraic equations that enable one to
construct the curves representing the least upper bound and the
greatest lower bound of the repeated limits as $M\to \infty$ and
$N\to \infty$. The resulting curves allow one to generalize the notion
of BKT (Berezinskii–Kosterlitz–Thouless) topological phase
transition and explaining a number of phenomena in thermodynamics
and mesoscopic physics.
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