The inverse Henderson problem in the thermodynamic limit
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Abstract
Particle systems interacting via some Hamiltonian are often described via so-called Gibbs measures.
However, in practice it is not possible to measure or calculate these interactions directly from experiments
or simulation data. What, however, is possible is to calculate certain statistics of the number and
relative positions of particles from spectroscopy or simulations. This of course gives rise to mathematical
questions: „Given these point statistics, how do we recover the interaction Hamiltonian and is it unique?“
In an often cited paper Henderson proved, for the case of a finite volume box at fixed density, i.e. a
canonical ensemble, that any such solution is necessarily unique and in the ’80s Chayes and Chayes
showed that under broad conditions a solution exists. However their results do not immediately extend
to the thermodynamic limit, i.e. the infinite volume case. The goal of this work is to extend known results
for the finite volume to the thermodynamic limit.
The thesis is structured in the following way. We start with a brief introduction to the setting we are
working in and some historic background for the problem. In the second chapter we give a rigorous
introduction to the mathematical background, i.e. the theory of point processes and in particular Gibbs
point processes. Chapters 3 to 5 consist of the main results of this work, namely three peer-reviewed
articles.
In Chapter 3 and 4 we discuss the inverse Henderson problem in the thermodynamic limit. We start
by discussing Henderson’s theorem and proving the thermodynamic limit version of it. Following, we
use the Gibbs variational principle to rewrite the inverse Henderson problem as a minimization problem
of the specific relative entropy in the thermodynamic limit. Using our version of Henderson’s theorem
established before, we then show that this functional is strictly convex, and thus that any minimizer has
to be unique. We furthermore show that for any given pair potential and any particle density below the
close-packing density, there always exists a chemical potential and at least one Gibbs measure for these
parameters realizing this density. We can then reformulate the minimization problem in the canonical
setting, i.e. work at fixed density. Lastly, on an appropriate space of perturbations, we calculate the
derivatives of the relative entropy functional and show the connections to the well-known inverse Monte-
Carlo scheme when using Newton’s method for the minimization of the relative entropy functional.
In Chapter 5 we take a different approach: Assuming we are given all correlation functions of some Gibbs
measure, we construct an expansion for the chemical potential associated to this measure. We show that
the conditions for convergence are indeed very mild, as we can show the assumptions are satisfied in a
variety of cases. In particular the results also hold for some models with many-body interactions.