Reaction–Diffusion Kinetics in Growing Domains

Abstract

Reaction-diffusion models have been used over decades to study biological systems. In this context, evolution equations for probability distribution functions and the associated stochastic differential equations have nowadays become indispensable tools. In population dynamics, say, such approaches are utilized to study many systems, e.g., colonies of microorganisms or ecological systems. While the majority of studies focus on the case of a static domain, the time-dependent case is also important, as it allows one to deal with situations where the domain growth takes place over time scales that are relevant for the computation of reaction rates and of the associated reactant distributions. Such situations are indeed frequently encountered in the field of developmental biology, notably in connection with pattern formation, embryo growth or morphogen gradient formation. In this chapter, we review some recent advances in the study of pure diffusion processes in growing domains. These results are subsequently taken as a starting point to study the kinetics of a simple reaction-diffusion process, i.e., the encounter-controlled annihilation reaction. The outcome of the present work is expected to pave the way for the study of more complex reaction-diffusion systems of possible relevance in various fields of research.

Statistical Physics Of Opinion Formation: Is it a SPOOF?

“…In this sense, the system is homogeneous and without fluctuations. In the context of modeling chemical reactions, systems under mean-field treatment are said to be well-stirred since rapid mixing destroys correlations and makes chemical compounds evenly spread [103,104]. In the case of opinion dynamics and voter models, we talk about well-mixed populations by analogy [45-47, 51, 105].…”

Encounter-controlled coalescence and annihilation on a one-dimensional growing domain

“…IV, we discuss the behavior of the annihilation reaction on a uniformly growing domain. As already mentioned, in this paper we go well beyond previous results that were restricted to the concentration in the special case where the time growth of the medium is described by a power law [46]. Here, we consider the case of an arbitrary time growth of the medium, and discuss the behavior of the concentration and of the IPDF (as well as of the closely related two-point correlation function) over the whole time domain.…”

“…Returning to the specific case of the diffusion-limited A+A→A and A+A→ ∅ reactions in one dimension, one may ask how their well-known phenomenology changes if one allows for a medium expansion, and, specifically, whether the IPDF method remains a suitable tool to study such changes. In a very recent work about the kinetics of the annihilation reaction on a 1d domain which grows according to a power law [46], we have (partially) answered the second question in the affirmative. In the present work we carry out a detailed analysis of the coalescence reaction by means of the IPDF method (conveniently generalized to deal with growing media), and also extend the study of the annihilation reaction performed in Ref.…”

“…In the present work we carry out a detailed analysis of the coalescence reaction by means of the IPDF method (conveniently generalized to deal with growing media), and also extend the study of the annihilation reaction performed in Ref. [46] along these lines. More specifically, we give the exact solution for the coalescence reaction on a growing 1d domain.…”

“…In doing so, we do not limit ourselves to the behavior of the global particle concentration (as done in Ref. [46] for the special case of the 1d annihilation reaction with a power-law domain growth and a random initial condition), but also consider the influence of the type of domain growth and initial condition on the particle concentration and on the associated survival probability, as well as the behavior of correlation functions characterizing the onset of spatial ordering in the system (in all cases, exact expressions for the full spatial and temporal dependence are derived). While the obtained expressions are similar to those for the static case, in the case of a sufficiently fast domain growth, the physical behavior may become very different.…”