Chemical Continuous Time Random Walks
Accurately simulating reactive transport through heterogeneous media requires resolving the spatial scales at which mixing takes place. This typically requires a fine spatial discretization (Eulerian methods) or large numbers of particles (Lagrangian methods), leading to prohibitively expensive simulations for large-scale transport. We explore a different approach and consider the question: In heterogeneous chemically reactive systems, is it possible to describe the evolution of macroscopic reactant concentrations without explicitly resolving the spatial transport? Traditional Kinetic Monte Carlo methods, such as the Gillespie algorithm , model chemical reactions as random walks in particle number space, without the introduction of spatial coordinates. The inter-reaction times are exponentially distributed under the assumption that the system is well mixed. In real systems, transport limitations lead to incomplete mixing and decreased reaction efficiency. We introduce an arbitrary inter-reaction time distribution, which may account for the impact of incomplete mixing. The resulting process defines an inhomogeneous continuous time random walk in particle number space, from which we derive a generalized chemical master equation and formulate a generalized Gillespie algorithm . We then determine the modified chemical rate laws for different inter-reaction time distributions, which describe the macroscopic reaction kinetics. We trace Michaelis–Menten-type kinetics back to finite-mean delay times, and predict time-nonlocal macroscopic reaction kinetics as a consequence of broadly distributed delays. Non-Markovian kinetics exhibit weak ergodicity breaking and show key features of reactions under local non-equilibrium.