Stochastic simulation of binary annihilation reactions within q-analysis formalism

TL;DR

This paper introduces a novel stochastic simulation algorithm based on the generalized q formalism to model binary annihilation reactions, accounting for system heterogeneity and nonextensivity, extending beyond Gillespie's traditional approach.

Contribution

The authors develop a new stochastic simulation method using q-parameterized probabilities, enabling analysis of reactions in heterogeneous media and linking nonextensiveness to reaction kinetics.

Findings

The q parameter influences reaction rates and order.

Different behaviors observed for samples with varying heterogeneity.

Empirical relations between reaction order and heterogeneity are proposed.

Abstract

Although Gillespie's algorithm is justified under a set of axioms based on the assumption of homogeneity of the system, many chemical systems deviate from this assumption, as is the case for reactions taking place in low-mobility media. Using instead the generalized q formalism, we propose a new stochastic simulation algorithm by redefining the probability with which a mu reaction occurs between t + tao and t + tao + delta tao as P(tao q, mu) = a mu exp q(-a0 tao q), taking into account the separation of the natural exponential by the q parameter. Our algorithm has been implemented in the study of binary annihilation reactions, demonstrating a wider amplitude within the range of established physicochemical reactions, being the stochastic Gillespie scheme and the classical deterministic approach, particular cases of this new proposal. The effect of the nonextensivity parameter, q, on the reaction rate is analyzed and its relationship with the reaction order, n, and the heterogeneity parameter, h, is determined for two different reactant concentrations in the annihilation reaction. Different behaviors of these parameters are observed for the two types of samples, especially as q moves away from 1, confirming that quasi-second order reactions occur when reactant concentrations are similar and quasi-first order reactions when they are different. Empirical expressions between the classical reaction order and the degree of heterogeneity are proposed. The results obtained allow us to associate the behavior of sub and supra Arrhenius kinetics reported in other different reactions with the degree of non-extensiveness of the reaction.