Stochastic processes in mesoscale physics and the early universe

Abstract

This thesis is dedicated to the study of stochastic processes; non-deterministic physical phenomena that can be well described by classical physics. The stochastic processes we are interested in are akin to Brownian Motion and can be described by an overdamped Langevin equation comprised of a deterministic drift term and a random noise term. Because of the random noise term, one must solve the Langevin equation many times to make physical predictions for what would happen on average. As Langevin equations have such wide applications this thesis is split into two parts where we examine them in two very different, yet connected, contexts. In Part I we examine stochastic processes in the Mesoscale. For us this means that the Langevin equation is driven by thermal noise, with amplitude proportional to the temperature T and there exists a genuine equilibrium thermal state. We outline how the Langevin equation can be reformulated using techniques from Quantum Field Theory as a path integral that is closely related to SuperSymmetric Quantum Mechanics. We apply a technique known as the Functional Renormalisation Group (FRG) which allows us to coarse-grain in temporal scales. We describe how the FRG can be used to compute correlation functions as the system relaxes towards equilibrium. In particular we describe how to obtain effective equations of motion for the average position, variance and covariance of a particle evolving in highly non-trivial potentials and verify their accuracy by comparison to direct numerical simulations. In this way we outline a novel procedure for describing the behaviour of stochastic processes without having to resort to time consuming numerical simulations. In Part II we turn to the Early Universe and in particular examine stochastic processes occurring during a period of accelerated expansion known as inflation. This inflationary period is driven by a scalar field called the inflaton which also obeys a Langevin equation. In this context however the noise term does not come from thermal fluctuations but from inherently quantum fluctuations that are stretched to cosmological scales by the expansion of the universe. These quantum fluctuations then act as the seed for the formation of all large scale structure. We review how to compute inflationary perturbations and focus on a formalism known as Stochastic Inflation. We outline how the Hamilton-Jacobi formulation of Stochastic Inflation allows one to move beyond the simplest inflationary approximation, slow-roll, and discuss an interesting period known as ultra-slow roll. A period of ultra-slow roll is generally needed to form very large density perturbations and we outline how to compute the full probability distribution function of curvature perturbations for a plateau in the inflationary potential using heat kernel techniques. We use this to study the formation of Primordial Black Holes. These extreme objects are formed in the early universe, before the first galaxies, and are a possible Dark Matter candidate. We finish this thesis by applying the techniques developed in Part I to a spectator field during inflation. We confirm that the FRG techniques can compute cosmologically relevant observables such as the power spectrum and spectral tilt. We also extend the FRG formalism so that it can be used to solve first-passage time problems and verify that it gives the correct prediction for the average time taken for a field (or particle) to overcome a barrier in the potential.