Partition functions and symmetric polynomials

Abstract

We find a close correspondence between the partition functions of ideal quantum gases and certain symmetric polynomials. From this correspondence, it can be shown that a number of thermodynamic identities that have recently been considered in the literature are essentially of combinatorial origin and have been known for a long time as theorems on symmetric polynomials. For example, a recurrence relation for partition functions in the textbook by P. Landsberg is Newton's identity in disguised form. Conversely, a theorem on symmetric polynomials translates into a new and unexpected relation between fermion and boson partition functions, which can be used to express the former by means of the latter and vice versa. (C) 2002 American Association of Physics Teachers.