  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.
