Mayer-series asymptotic catastrophe in classical statistical mechanics

Abstract

The problem of an asymptotic catastrophe related to calculating Mayer series coefficients is discussed. The mean squared error of the Monte Carlo estimate for the coefficient of an nth power of a variable tends to infinity catastrophically fast as n increases if we use the standard representation for coefficients of a Mayer series and forbid the rapid growth of the calculation volume. In contrast, if we represent these coefficients as tree sums, the error vanishes as n increases. We precisely define the notion of a powerseries asymptotic catastrophe that improves the description introduced by Ivanchik. For a nonnegative potential that rapidly decreases at infinity and has a hard core, the standard, representation of mayer coefficients results in the asymptotic catatrophe both in the sense of our approach and in the sense of Ivanchik. Virial coefficients are represented as polynomials in tree sums. These representations resolve the asymptotic catastrophe problem for the case of virial expansions.