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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 6

### A discrete stochastic formulation for reversible bimolecular reactions via diffusion encounter

Pages: 1741 – 1772

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n6.a13

#### Authors

#### Abstract

The classical models for irreversible diffusion-influenced reactions can be derived by introducing absorbing boundary conditions to over-damped continuous Brownian motion (BM) theory. As there is a clear corresponding stochastic process, the mathematical description takes both Kolmogorov forward equation for the evolution of the probability distribution function and the stochastic sample trajectories. This dual description is a fundamental characteristic of stochastic processes and allows simple particle-based simulations to accurately match the expected statistical behavior. However, in the traditional theory using the back-reaction boundary condition to model reversible reactions with geminate recombinations, several subtleties arise: It is unclear what the underlying stochastic process is, which causes complications in producing accurate simulations; and it is non-trivial how to perform an appropriate discretization for numerical computations. In this work, we derive a discrete stochastic model that recovers the classical models and their boundary conditions in the continuous limit. In the case of reversible reactions, we recover the back-reaction boundary condition, unifying the back-reaction approach with those of current simulation packages. Furthermore, all the complications encountered in the continuous models become trivial in the discrete model. Our formulation brings to attention the question: With computations in mind, can we develop a discrete reaction kinetics model that is more fundamental than its continuous counterpart?

#### Keywords

stochastic reaction-diffusion, diffusion-influenced reactions, reversible reactions, chemical kinetics, Markov chain, Brownian motion, absorption boundary, back-reaction boundary

#### 2010 Mathematics Subject Classification

60J10, 60J22, 60J50, 60J70, 65C35, 65C40, 92C40