Self-similar summation of virial expansions

TL;DR

This paper introduces a new, regular, and unique self-similar approximation method for summing virial expansions, overcoming limitations of Padé summation, and accurately predicting physical properties of fluids.

Contribution

A novel self-similar summation approach for virial expansions that is regular, uniquely defined, and capable of identifying physically motivated poles.

Findings

Self-similar summation matches or exceeds Padé and Monte Carlo accuracy.

Method reconstructs functions exactly in some cases.

Applicable to various systems like hard-disk and hard-sphere fluids.

Abstract

Virial expansions are the series in powers of density assumed to be small. However, the equations of state require to consider finite densities for which virial expansions, as a rule, diverge. In order to extrapolate a virial expansion to the values, where this expansion diverges, one uses summation methods. The most often used method is the Pad\'{e} summation, which has several deficiencies. First of all, Pad\'{e} approximants are not uniquely defined, suggesting a large table of admissible variants. Second, often there appear spurious unphysical poles. On the contrary, in those cases where the existence of a pole is physically motivated, Pad\'{e} approximants do not necessarily exhibit it. A new approach for the summation of virial expansions is suggested, based on self-similar approximation theory. The method is regular and uniquely defined. It allows for the determination of physically motivated poles. The accuracy of self-similar approximants is not worse than that of the best Pad\'{e} approximants with fitting parameters or of Monte Carlo simulations. The self-similar summation is based solely on virial expansions, involving no fitting parameters. In some cases, self-similar summation allows for reconstructing the sought functions exactly. The approach is illustrated by summing virial expansions for hard-disk fluids, hard-sphere fluids, and systems with power-law potentials.