Non-Gaussian particle number fluctuations in vicinity of the critical point for van der Waals equation of state

TL;DR

This paper analyzes non-Gaussian particle number fluctuations near the critical point of the van der Waals fluid, revealing universal divergence and sign changes in skewness and kurtosis, using analytical methods in a grand canonical framework.

Contribution

It provides analytical expressions for skewness and kurtosis near the critical point, demonstrating their universal behavior independent of specific van der Waals parameters.

Findings

Skewness is positive in the gas phase and negative in the liquid phase.

Kurtosis becomes significantly negative at the critical density and supercritical temperatures.

Skewness and kurtosis diverge at the critical point.

Abstract

The non-Gaussian measures of the particle number fluctuations -- skewness $S\sigma$ and kurtosis $\kappa \sigma^2$ -- are calculated in a vicinity of the critical point. This point corresponds to the end point of the first-order liquid-gas phase transition. The gaseous phase is characterized by the positive values of skewness while the liquid phase has negative skew. The kurtosis appears to be significantly negative at the critical density and supercritical temperatures. The skewness and kurtosis diverge at the critical point. The classical van der Waals equation of state in the grand canonical ensemble formulation is used in our studies. Neglecting effects of the quantum statistics we succeed to obtain the analytical expressions for the rich structures of the skewness and kurtosis in a wide region around the critical point. These results have universal form, i.e., they do not depend on particular values of the van der Waals parameters $a$ and $b$. The strongly intensive measures of particle number and energy fluctuations are also considered and show singular behavior in the vicinity of the critical point.