Rare Events in Reaction-Diffusion Systems Field-theoretical Approximations and Monte Carlo Simulations

Abstract

Rare events are events that have near-zero probability of occurring. Despite their apparent irrelevance, when they do occur, they can have substantial, and even catastrophic, repercussions. In this thesis, we study rare events in reaction-diffusion systems, a class of mathematical models that finds various applications in physics and life sciences. We employ both theoretical and computational methods to determine the tails of the probability distribution describing the state of system. Firstly, we follow existing literature to derive a quantum-mechanical description for entirely classical reaction-diffusion systems, called the Doi-Peliti formalism. We express the time evolution of the systems as a Feynman path integral, which we then evaluate at the saddle point to obtain a semiclassical approximation for the probability distribution, and a closed-form leading-order expression for the tails. Secondly, we tailor a lesser-known Monte Carlo algorithm for rare probability estimation, called adaptive multilevel splitting, to compute the probability distribution of reaction-diffusion processes. We derive some theoretical results regarding its efficiency, discuss practical implementation choices, and benchmark its performance against well-understood examples. Lastly, we compare the semiclassical approximation to the computational results, determining under which conditions the former succeeds or fails.