Self-similar summation of virial expansions

V. I. Yukalov; E. P. Yukalova

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arXiv · 2604.00294

Self-similar summation of virial expansions

V. I. Yukalov·E. P. Yukalova

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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.

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V. I. Yukalov, E. P. Yukalova. 2026-03-31. Self-similar summation of virial expansions. https://arxiv.org/abs/2604.00294

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