Maximum entropy methods as the bridge between macroscopic and microscopic theory

TL;DR

This paper explores the singular potential as a maximum entropy function linking macroscopic and microscopic theories, establishing its properties and applications in free-energy functionals, with implications for mean-field models and variational analysis.

Contribution

It introduces the singular potential's properties, including convexity and boundary behavior, and demonstrates its application in relating microscopic and macroscopic free-energy functionals.

Findings

The singular potential is strictly convex and blows up at the boundary of admissible moments.

An equivalence between microscopic and macroscopic free-energy functionals is established.

Taylor approximations often fail to preserve the shape of the singular potential.

Abstract

This paper investigates a function of macroscopic variables known as the singular potential, building on previous work by Ball and Majumdar. The singular potential is a function of the admissible statistical averages of probability distributions on a state space, defined so that it corresponds to the maximum possible entropy given known observed statistical averages, although non-classical entropy-like objective functions will also be considered. First the set of admissible moments must be established, and under the conditions presented in this work the set is open, bounded and convex allowing a description in terms of supporting hyperplanes, which provides estimates on the development of singularities for related probability distributions. Under appropriate conditions it is shown that the singular potential is strictly convex, as differentiable as the microscopic entropy and blows up uniformly as the macroscopic variable tends to the boundary of the set of admissible moments. Applications of the singular potential are then discussed, and particular consideration will be given to certain free-energy functionals typical in mean-field theory, demonstrating an equivalence between certain microscopic and macroscopic free-energy functionals. This allows statements about L^1-local minimisers of Onsager's free energy to be obtained which cannot be given by two-sided variations, and overcomes the need to ensure local minimisers are bounded away from zero and infinity before taking bounded variations. The analysis also permits the definition of a dual order parameter for which Onsager's free energy allows an explicit representation. Also the difficulties in approximating the singular potential by everywhere defined functions, in particular by polynomials, are addressed with examples demonstrating the failure of the Taylor approximation to preserve shape properties of the singular potential.