Stochastic simulations of reaction-diffusion processes

In this educational project I will attempt to explore and better learn the theory and methods behind stochastic simulation of reaction-diffusion processes.

This work is (almost) entirely based on what has already been done in [1]. In the explanations (again, heavily drawn from [1]) I will write some comments on probability, information theory, and statistical mechanics, topics I am currently studying. Warning: they may be completely wrong.

The article is divided into the following sections (which follow the same structure as the article mentioned above). There will also be some Rust codes that implement the algorithms and help visualize the results.

Stochastic simulation of chemical reactions

Exploration of stochastic methods for the modelling of (spatially homogeneous) systems of chemical reactions through the following examples.

Stochastic simulation of degradation

Consider the single chemical reaction of degradation

$$ A\overset{k}{\rightarrow}\emptyset $$

where

• $A$ is the chemical species of interest
• $k$ is the constant rate of the reaction defined so that $k dt$ gives the probability that a randomly chosen molecule of chemical species A reacts (is degraded) during the time interval $[t, dt)$
• $\emptyset$ denotes chemical species which are of no further interest in what follows

In order to simulate it, it is used firstly a naive stochastic simulation algorithm (SSA) and then an improvement of it.

References

[1] Erban, Radek and Chapman, Jonathan and Maini, Philip (2007). A practical guide to stochastic simulations of reaction-diffusion processes. arXiv. 10.48550/ARXIV.0704.1908.