Smoluchowski reaction kinetics for reactions of any order

In 1917, Marian von Smoluchowski presented a simple mathematical description of diffusion-controlled reactions on the scale of individual molecules. His model postulated that a reaction would occur when two reactants were sufficiently close and, more specifically, presented a succinct relationship between the relative proximity of two reactants at the moment of reaction and the macroscopic reaction rate. Over the past century, the Smoluchowski reaction theory has been applied widely in the physical, chemical, environmental, and, more recently, biological sciences. Despite the widespread utility of the Smoluchowski theory, it only describes the rates of second order reactions and is inadequate for describing higher order reactions for which there is no equivalent method for theoretical investigation. In this paper, we derive a generalized Smoluchowski framework in which we define proximity in the context when more than two reactants are involved. We derive the relationship between the macroscopic reaction rate and the critical proximity at which a reaction occurs for higher order reactions. Using this theoretical framework and through numerical experiments, we explore various peculiar properties of multimolecular diffusion-controlled reactions which, due to there being no other numerical method of this nature, have not been previous reported.