arXiv subjects

Chris D Greenman

Publications and source records attributed to Chris D Greenman.

6 recordsLinked to original sources

arXiv2026

Renormalisation for Reaction-Diffusion Systems with Non-Local Interactions

Models of reaction diffusion processes usually employ discrete lattice models with particles interacting at the same site, resulting in localized reactions in the continuum limit. Here, various non-local interactions are considered, and two features reported. Firstly, it is shown that sufficiently non-local interactions will regulate ultra-violet divergences that perturbative methods with local interactions produce. However, in asymptotic regimes, infra-red divergences persist and ultra-violet divergences can reappear. Renormalisation methods are shown to report the same universal behaviour as local interactions at critical points. Secondly, the renormalisation group can be interpreted as a space-time-field rescaling that preserves action structure. This can be used to extract solutions to Callan-Symanzik equations directly without having to solve (or construct) the equation. These observations are exemplified for two paradigm models; annihilation $A_p+A_q\rightarrow ϕ$, and this process paired with branching, birth and death.

math-ph↗

arXiv2023

Reaction Diffusion Systems and Extensions of Quantum Stochastic Processes

Reaction diffusion systems describe the behaviour of dynamic, interacting, particulate systems. Quantum stochastic processes generalise Brownian motion and Poisson processes, having operator valued Itô calculus machinery. Here it is shown that the three standard noises of quantum stochastic processes can be extended to model reaction diffusion systems, the methods being exemplified with spatial birth-death processes. The usual approach for these systems are master equations, or Doi-Peliti path integration techniques. The machinery described here provide efficient analyses for many systems of interest, and offer an alternative set of tools to investigate such problems.

math-ph↗

arXiv2021

Time Series Path Integral Expansions for Stochastic Processes

A form of time series path integral expansion is provided that enables both analytic and numerical temporal effect calculations for a range of stochastic processes. Birth-death processes with linear rates are analysed via coherent state Doi-Peliti techniques. The $\mathfrak{su}(1,1)$ Lie algebra is utilised to capture quadratic rate birth-death processes. The techniques are also adapted to diffusion processes. All methods rely on finding a suitable reproducing kernel associated with the underlying algebra to perform the expansion. The resulting series differ from those found in standard Dyson time series field theory techniques.

cond-mat.stat-mech↗

arXiv2019

Duality relations between spatial birth-death processes and diffusions in Hilbert space

Spatially dependent birth-death processes can be modelled by kinetic models such as the BBGKY hierarchy. Diffusion in infinite dimensional systems can be modelled with Brownian motion in Hilbert space. In this work Doi field theoretic formalism is utilised to establish dualities between these classes of processes. This enables path integral methods to calculate expectations of duality functions. These are exemplified with models ranging from stochastic cable signalling to jump-diffusion processes.

math-ph↗

arXiv2016

A Path Integral Approach to Age Dependent Branching Processes

Age dependent population dynamics are frequently modeled with generalizations of the classic McKendrick-von Foerster equation. These are deterministic systems, and a stochastic generalization was recently reported in [1,2]. Here we develop a fully stochastic theory for age-structured populations via quantum field theoretical Doi-Peliti techniques. This results in a path integral formulation where birth and death events correspond to cubic and quadratic interaction terms. This formalism allows us to efficiently recapitulate the results in [1,2], exemplifying the utility of Doi-Peliti methods. Furthermore, we find that the path integral formulation for age-structured moments has an exact perturbative expansion that explicitly relates to the hereditary structure between correlated individuals. These methods are then generalized with a binary fission model of cell division.

physics.bio-ph↗

arXiv2016

A hierarchical kinetic theory of birth, death, and fission in age-structured interacting populations

We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete probability density for the population-size-age-chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behaviour. Our approach utilizes mixed type, multi-dimensional probability distributions similar to those employed in the study of gas kinetics, with terms that satisfy BBGKY-like equation hierarchies.

math.PR↗