Exact Growth Dynamics of One-Dimensional Run-and-Tumble Particles with a Localised Fertile Site

Yao, Xueqi (2025) Exact Growth Dynamics of One-Dimensional Run-and-Tumble Particles with a Localised Fertile Site PhD thesis, University of Liverpool.

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Abstract

This thesis examines non-equilibrium growth in one-dimensional active matter systems, with run-and-tumble particles (RTPs) serving as a minimal theoretical setting. The aim is to construct an exact analytical description of particle-number non-conserving dynamics in active systems and to apply it to a fertile-site growth problem that lies outside the scope of equilibrium statistical mechanics. After introducing run-and-tumble dynamics and the analytical tools needed to address non-equilibrium transport, the thesis develops an exact solution for a model of RTPs interacting with a single localised fertile site. Particles undergo persistent motion with stochastic velocity reversals and reproduce upon crossing the fertile region, which is characterised by a smooth fertility function. Using Laplace transform and self-consistency techniques, an exact relation governing the density at the fertile site is obtained, from which exponential population growth follows with a growth rate independent of initial conditions. Although the total population grows without bound, the spatial density profile, when normalised by the population size, approaches a stationary form at long times. This stationary profile shows a robust local minimum at the fertile site, indicating a qualitative departure from passive diffusive growth models. More generally, the results show how persistent motion reshapes non-equilibrium growth processes and gives rise to spatial structures that do not occur in passive systems.

Item Type:	Thesis (PhD)
Divisions:	Faculty of Science & Engineering Faculty of Science & Engineering > School of Physical Sciences Faculty of Science & Engineering > School of Physical Sciences > Mathematical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	23 Apr 2026 10:50
Last Modified:	24 Apr 2026 08:08
DOI:	10.17638/03197695
Supervisors:	Gresnigt, Niels Yuan, Linglong Konstantopoulos, Panagiotis Meng, Jia
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3197695
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