Representation of the grand partition function of the cell model: The state equation in the mean-field approximation

TL;DR

This paper develops a method to calculate the grand partition function for a fluid cell model with interacting particles, deriving a state equation in mean-field approximation that describes phase behavior across temperature ranges.

Contribution

It introduces an exact integration procedure for the grand partition function of the cell model, distinct from the Ising model, and derives a comprehensive state equation applicable across critical temperatures.

Findings

Pressure increases continuously above critical temperature.

Pressure isotherms show horizontal parts below critical temperature.

Derived state equation describes phase transitions in the cell model.

Abstract

The method to calculate the grand partition function of a particle system, in which constituents interact with each other via potential, that include repulsive and attractive components, is proposed. The cell model, which was introduced to describe critical phenomena and phase transitions, is used to provide calculations. The exact procedure of integration over particle coordinates and summation over number of particles is proposed. As a result, an evident expression for the grand partition function of the fluid cell model is obtained in the form of multiple integral over collective variables. As it can be seen directly from the structure of the transition jacobian, the present multiparticle model appeared to be different from the Ising model, which is widely used to describe fluid systems. The state equation, which is valid for wide temperature ranges both above and below the critical one, is derived in mean-field approximation. The pressure calculated for the cell model at temperatures above the critical one is found to be continuously increasing function of temperature and density. The isotherms of pressure as a function of density have horizontal parts at temperatures below the critical one.