Exact closed equation for reduced equilibrium distribution functions of the many-particle system

TL;DR

This paper derives an exact closed integrodifferential equation for the s-particle equilibrium distribution function in a many-particle system, using a projection operator approach and an expansion in particle density.

Contribution

It introduces a novel exact equation for reduced distribution functions and provides a linear approximation solution, advancing the theoretical understanding of many-particle equilibrium states.

Findings

Derived an exact integrodifferential equation for s-particle distributions.

Proposed a density expansion method for the equation kernel.

Obtained a linear approximation solution for the distribution functions.

Abstract

An exact closed equation for s - particle equilibrium distribution function (s<N) of the system of N>>1 interacting particles is obtained. This integra-differential {\beta} - convolution equation ({\beta}=1/k_{B}T) follows from the Bloch equation for the canonical distribution function by applying the projection operator integrating off the coordinates of N-s irrelevant particles. The method of expansion of the obtained equation kernel in the particle density n is suggested. The solution to this equation in the linear in n approximation for the kernel is found.