Partition asymptotics from one-dimensional quantum entropy and energy currents

Abstract

We give an alternative method to that of Hardy–Ramanujan–Rademacher to derive the leading exponential term in the asymptotic approximation to the partition function p(n,a), defined as the number of decompositions of a positive integer n into integer summands, with each summand appearing at most a times in a given decomposition. The derivation involves mapping to an equivalent physical problem concerning the quantum entropy and energy currents of particles flowing in a one-dimensional (1D) channel connecting thermal reservoirs, and which obey Gentile’s intermediate statistics with statistical parameter a. The method is also applied to partitions associated with Haldane’s fractional exclusion statistics.