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 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.