Statistical physics is concerned with deducing the largescale (macroscopic) behaviour of a system given the microscopic rules of interaction between its constituents. There are many systems outside the traditional remit of physics which involve emergent phenomena arising from the complex interactions between many individuals. In the past few decades, ideas from from statistical physics (both mathematical techniques and the underlying philosophy) have been borrowed and applied to such systems. This thesis contains a collection of works which apply ideas from statistical physics to a selection of current problems in developmental biology and ecology. The thesis is centred around three main mathematical themes: intrinsic noise in individualbased systems, nonMarkovian (memory) effects and quenched disorder. The combination of any two of these concepts presents a significant mathematical challenge as well as an interesting array of physical phenomena. First, the possibility of replicating the stationary behaviour of a nonMarkovian reactiondiffusion system (with anomalous diffusion) with a normally diffusing system is demonstrated. This provides a meaningful way to define the effective diffusion coefficients of the nonMarkovian reactiondiffusion system. Pathintegral approaches for calculating the intrinsic noise in nonMarkovian systems are then presented. Such methods facilitate a systemsize expansion when one cannot easily construct a master equation for the system of interest. The results of this analysis are applied in two cases: (i) The intrinsic fluctuations in particle number in a reactiondiffusion system with anomalous transport are calculated. This is used to demonstrate that Turing pattern formation is made more easily realisable by a combination of noise and subdiffusion. (ii) The fluctuations in mRNA and protein numbers (whose production is subject to time delays) in a gene regulatory model is quantified. Such fluctuations are shown to produce persistent noisy cycles of gene expression. The effects of intercell signalling on these cycles, with reference to the cycles' utility as a cellular clock, are then studied. Finally, the effect of dispersal (diffusion) on stability in a model of a complex ecosystem is investigated. The seminal models of Robert May and Alan Turing are combined within a randommatrix theoretical framework to demonstrate that dispersal can be (perhaps counterintuitively) a destabilising influence. Just as subdiffusion and intrinsic noise are found to increase the range of parameters for which the Turing instability exists, the amount of complexity in the interactions between species is found to have a similar effect.
Date of Award  1 Aug 2020 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Tobias Galla (Supervisor) & Ahsan Nazir (Supervisor) 

 Diffusion
 Path integral
 Turing patterns
 Systemsize expansion
 Ecosystem stability
 NonMarkovian
 Reactiondiffusion
 Stochastic
 Gene regulation
 Subdiffusion
 Statistical physics
The statistical physics of systems with disorder, intrinsic noise and nonMarkovian dynamics
Baron, J. (Author). 1 Aug 2020
Student thesis: Phd