First Principles Modeling of Bimolecular Reactions with Diffusion

We consider three approaches to modeling A + B → C irreversible reactions in natural media: 1) a discretized diffusion-reaction equation (DRE), 2) a particle tracking (PT) scheme in which reaction occurs if and only if an A and B particle pair are within a fixed distance, r (the "reaction radius"), and 3) a PT scheme using an alternative to the fixed reaction radius: a collocation probability distribution derived directly from first principles. Each approach has advantages. In some cases a discretized DRE may be the most computationally efficient method. For PT simulations, robust codes exist based on use of a fixed reaction radius. And finally, collocation probabilities may be derived directly from the Fick's Law constant, D, which is a well-established property for most species. In each approach, a single parameter governs the 'promiscuity' of the reaction (i.e. the thermodynamic favorability of reaction, predicated on the particles being locally well mixed). For the DRE, fixed-reaction-radius PT, and collocation-based PT, these parameters are, respectively: a second-order decay rate, r, and D. We established a number of new results enhancing these approaches and relating them to each other (and to nature). In particular, a thought experiment concerning a simple system in which the predictions of each approach can be computed analytically was used to derive formulas establishing a universal one-to-one correspondence among each of the governing parameters. We thus showed the conditions for equivalence of the three approaches, and grounded both the DRE approach and the fixed-radius PT approach in the Fick's Law D. We further showed that the existing collocation-based PT theory is based on a probability distribution that is only correct for infinitesimally small times, but which can be modified to be accurate for larger times by means of continuous time random walk analysis and first-passage probability distributions. Finally, we employed a novel mathematical approach to adapt this into a workable collocation-based particle tracking technique.