On completeness of description of an equilibrium canonical ensemble by reduced s-particle distribution function

TL;DR

This paper demonstrates that in a classical equilibrium canonical ensemble, the full Gibbs distribution can be uniquely reconstructed from the reduced s-particle distribution function, which contains complete information about the system.

Contribution

It establishes that the reduced s-particle distribution function fully characterizes the canonical ensemble, providing a new way to represent the Gibbs distribution as a convergent power series.

Findings

Full Gibbs distribution can be expressed as a convergent power series in the reduced s-particle distribution function.

Reduced distribution functions of order less than s do not contain complete information about the system.

A linear term of the power series expansion is explicitly calculated.

Abstract

In this article it is shown that in a classical equilibrium canonical ensemble of molecules with $s$-body interaction full Gibbs distribution can be uniquely expressed in terms of a reduced s-particle distribution function. This means that whenever a number of particles $N$ and a volume $V$ are fixed the reduced $s$-particle distribution function contains as much information about the equilibrium system as the whole canonical Gibbs distribution. The latter is represented as an absolutely convergent power series relative to the reduced $s$-particle distribution function. As an example a linear term of this expansion is calculated. It is also shown that reduced distribution functions of order less than $s$ don't possess such property and, to all appearance, contain not all information about the system under consideration.