New insight into the partition theory of integers related to problems of thermodynamics and mesoscopic physics

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 → ∞ and N → ∞. 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.