Pair-correlation effects in many-body systems

Abstract

The laws of nature encompass the small, the large, the few, and the many. In this thesis we are concerned with classical (i.e. not quantum) many-body systems, which refers to any microscopic or macroscopic system that contains a large number of interacting entities. The nearest-neighbor Ising model, originally developed in 1920 by Wilhelm Lenz, forms a cornerstone in our theoretical understanding of collective effects in classical many-body systems, and is to date a paradigm for statistical physics. Despite its elegant and simplistic description, exact analytical results in dimensions equal and larger than two are difficult to obtain. Therefore, much work has been done to con- struct methods that allow for approximate, yet accurate, analytical solutions. One of these methods is the Bethe-Guggenheim approximation, originally developed independently by Hans Bethe and Edward Guggenheim in 1935. This approximation goes beyond the well- known mean field approximation, and explicitly accounts for pair correlations between the spins in the Ising model. In this thesis, we embark on a journey to exploit the full capacity of the Bethe-Guggenheim approximation, in non-uniform and non-equilibrium settings. After we formally introduce the original Bethe-Guggenheim approximation for uniform systems, we will extend its scope to non-uniform systems, and derive a Cahn-Hilliard free energy functional. Here we find that the one-dimensional equilibrium concentration profile undergoes a delocalization-induced broadening transition at interaction strengths near and above the thermal energy. The broadening transition arises from a decreasing amplitude of capillary wave fluctuations, and is not accounted for in the mean field approximation. Finally, going beyond equilibrium properties, we also study the kinetics of the Ising model in and out of equilibrium. First, based on firm theoretical and experimental evidence, we con- struct an Ising-like minimal model for interacting cellular adhesion molecules. Here we find a dynamical critical point where adhesion cluster dissolution and formation are the fastest, and undergo a qualitative change in dynamics. Second, we demonstrate the existence of a finite-time dynamical phase transition for disordering quenches in the nearest-neighbor Ising model. Starkly different from the mean field approximation, the time at which the dynamical phase transition occurs, is bounded from below by a speed limit. Altogether, in this thesis, we unveil the non-trivial and a priori non-intuitive effects of pair correlations in the classical nearest-neighbor Ising model, which are taken into account in the Bethe-Guggenheim approximation, and neglected in the mean field approximation.