Non-equilibrium statistical field theory for classical particles: Initially correlated grand canonical ensembles

TL;DR

This paper extends non-equilibrium statistical field theory for classical particles from canonical to grand canonical ensembles, incorporating initial correlations and developing a diagrammatic approach for cumulant calculations.

Contribution

It introduces a grand canonical formulation with diagrammatic methods and cluster expansion, enabling analysis of initial correlations in classical particle systems.

Findings

Derived the grand canonical generating functional for correlated initial conditions.

Developed a diagrammatic representation and cluster expansion for particle number summation.

Established theorems for cumulants of density and response fields.

Abstract

It was recently shown by Bartelmann et al. how correlated initial conditions can be introduced into the statistical field theory for classical particles pioneered by Das and Mazenko. In this paper we extend this development from the canonical to the grand canonical ensemble for a system satisfying statistical homogeneity and isotropy. We do this by translating the probability distribution for the initial phase space coordinates of the particles into an easy diagrammatic representation and then using a variant of the Mayer cluster expansion to sum over particle numbers. The grand canonical generating functional is then used in a structured approach to the derivation of the non-interacting cumulants of the two core collective fields, the density $\rho$ and the response field $B$. As a side-product we find several theorems pertaining to these cumulants which will be useful when investigating the interacting regime of the theory in future work.