Applications of Modern Statistical Mechanics

Statistical Mechanics describes the macroscopic behavior of a system through the analysis of its microscopic components. It is therefore a framework to move from a probabilistic, high-dimensional description of a system to its macroscopic description through averaging. This approach, now one of the pillars of physics, has seen successes in other fields, such as statistics or mathematical finance. This broadening of the applications of statistical physics has opened new avenues of research in the field itself. Ideas from information theory, differential geometry, and approximate computation are making their way into modern statistical physics. The research presented in this dissertation straddles this boundary: we start by showing how concepts from statistical physics can be applied to statistical learning, then show how modern statistical physics can provide insights into molecular transport phenomena. The first three chapters focus on building an understanding of statistical learning as a thermodynamic relaxation process in a high-dimensional space: in the same way that a statistical mechanical system is composed of a large number of particles relaxing to their equilibrium distribution, a statistical learning system is a parametric function whose optimal parameters minimize an empirical loss. We present this process as a trajectory in a high-dimensional probability Riemannian manifold, and show how this conceptual framework can lead to practical improvements in learning algorithms for large scale neural networks. The second part of this thesis focuses on two applications of modern statistical mechanics to molecular transport. First, I propose a statistical mechanical interpretation of metabolon formation through cross-diffusion, a generalization of the reaction-diffusion framework to multiple reacting species with non-diagonal terms in the diffusion matrix. These theoretical results are validated by experimental results obtained using a microfluidic system. Second, I demonstrate how fluctuation analysis in motility assays can allow us to infer nanoscale properties from microscale measurements. I accomplish this using computational Langevin dynamics simulations and show how this setup can be used to simplify the testing of theoretical non-equilibrium statistical mechanics hypotheses.