The functional of the grand partition function for the investigation of the liquid-gas critical point

Abstract

An approach for the description of a spatially homogeneous system of interacting particles in the vicinity of the critical point is proposed. An explicit expression for the functional integral form of the partition function in the grand canonical ensemble is obtained. The functional integral is defined on a set of the collective variables (CV) {ϱ k }. The variable ϱ k corresponds to the density vibration mode with the wave vector k. It is shown that in the vicinity of the critical point, for all k larger than some B, the CV are distributed with the Gaussian measure density. In the long wave region, k < B, the surfaces of the cumulants consisting of the correlation functions of some reference system (the reference system is a model system of particles with repulsive pair interactions) possess wide plateaus in the vicinity of the point k = 0. It is shown that due to these two facts the homogeneous system can be put in correspondance with a certain lattice system with spacing c = π B . The problem of the calculation of the partition function can be reduced to the calculation of the functional for a three-dimensional Ising model with an external field.