On one-to-one correspondence of Gibbs distribution and reduced two-particle distribution function

TL;DR

This paper demonstrates that in an equilibrium classical system with two-body interactions, the full Gibbs distribution can be uniquely reconstructed from the reduced two-particle distribution function, establishing a one-to-one correspondence.

Contribution

It proves the unique expressibility of the Gibbs distribution in terms of the reduced two-particle distribution function and provides a convergent power series expansion for it.

Findings

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

The first-order linear term of the expansion is explicitly calculated.

Gibbs distribution can also be represented using the first-order reduced distribution and pair correlation functions.

Abstract

In this article it is shown that in an equilibrium classical canonical ensemble of molecules with two-body interaction and external field full Gibbs distribution can be uniquely expressed in terms of a reduced two-particle distribution function. This means that while a number of particles $N$ and a volume $V$ are fixed the reduced two-particle distribution function contains as much information about the equilibrium system as the whole canonical distribution. The latter is represented as an absolutely convergent power series relative to the reduced two-particle distribution function. As an example a linear term of this expansion is calculated. It is also shown that Gibbs distribution function can de expressed in terms of reduced distribution function of the first order and pair correlation function.That is the later two functions contain the whole information about system under consideration.