  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 last century, Smoluchowski reaction
theory has been applied widely in the physical, chemical, environmental and,
more recently, the biological sciences. Despite the widespread utility of the
Smoluchowski theory, it only describes the rates of second order reactions and
is inadequate for the description of higher order reactions for which there is
no equivalent method for theoretical investigation. In this paper, we derive a
generalised Smoluchowski framework in which we define what should be meant by
proximity in this 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 using 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.
