Rare Events in Reaction-Diffusion Systems Field-theoretical Approximations and Monte Carlo Simulations
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Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Biotechnology (MPBIO), MSc
Publicerad
2025
Författare
Manoni, Edoardo Maria
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
rare events, reaction-diffusion, Doi-Peliti, semiclassical approximation, Monte Carlo method, adaptive multilevel splitting.