Frechet differentiability of molecular distribution functions I. $L^\infty$ analysis

TL;DR

This paper rigorously proves Frechet differentiability of molecular distribution functions in classical particle ensembles with respect to pair potentials, under certain conditions, using $L^ abla$ analysis.

Contribution

It establishes the Frechet differentiability of molecular distribution functions in the $L^ abla$ norm for classical particles, extending understanding of potential-function relationships.

Findings

Proves Frechet differentiability for bounded domains.

Extends results to the thermodynamical limit.

Validates differentiability under small perturbations of potentials.

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 corresponding molecular distribution functions. For certain admissible perturbations of the pair potential and sufficiently small activity we rigorously establish Frechet differentiability with respect to the supremum norm in the image space -- both for bounded domains and in the thermodynamical limit.