New Insight into the Partition Theory of Integers Related to Problems of Thermodynamics and Mesoscopic Physics

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.