Frechet differentiability of molecular distribution functions II. The Ursell function

TL;DR

This paper proves the Frechet differentiability of the Ursell function in a grand canonical ensemble of classical particles, providing explicit formulas for its derivative under certain conditions.

Contribution

It establishes the Frechet differentiability of the Ursell function with respect to pair potentials and explicitly computes its derivative in the thermodynamic limit.

Findings

Proves Frechet differentiability of the Ursell function in $L^1$ norm.

Provides explicit formula for the derivative of the pair distribution function.

Analyzes the effect of perturbations in pair potentials on correlation functions.

Abstract

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the thermodynamical limit of the Ursell or pair correlation function. For certain admissible perturbations of the pair potential and sufficiently small activity we rigorously establish Frechet differentiability of the Ursell function in the $L^1$ norm. Furthermore, concerning the thermodynamical limit of the pair distribution function we explicitly compute its Fr\'echet derivative as a sum of a multiplication operator and an integral operator.