Particle Number Fluctuations for van der Waals Equation of State

TL;DR

This paper reformulates the van der Waals equation of state in the grand canonical ensemble to study particle number fluctuations across the phase diagram, revealing finite fluctuations in the mixed phase and divergence at the critical point.

Contribution

It transforms the VDW equation into the GCE, enabling analysis of particle number fluctuations, which are absent in the canonical ensemble, and explores their behavior near the critical point.

Findings

Fluctuations are finite within the mixed phase.

Fluctuations diverge at the critical point.

GCE formulation aids in statistical description of hadronic systems.

Abstract

The van der Waals (VDW) equation of state describes a thermal equilibrium in system of particles, where both repulsive and attractive interactions between them are included. This equation predicts an existence of the 1st order liquid-gas phase transition and the critical point. The standard form of the VDW equation is given by the pressure function in the canonical ensemble (CE) with a fixed number of particles. In the present paper the VDW equation is transformed to the grand canonical ensemble (GCE). We argue that this procedure can be useful for new physical applications. Particularly, the fluctuations of number of particles, which are absent in the CE, can be studied in the GCE. For the VDW equation of state in the GCE the particle number fluctuations are calculated for the whole phase diagram, both outside and inside the liquid-gas mixed phase region. It is shown that the scaled variance of these fluctuations remains finite within the mixed phase and goes to infinity at the critical point. The GCE formulation of the VDW equation of state can be also an important step for its application to a statistical description of hadronic systems, where numbers of different particle species are usually not conserved.