RandomWalks and Mesoscopic Reaction–Transport Equations

Abstract

As discussed in Sect. 2.1, the standard reaction–diffusion equation for the particle density ρ(x, t) has the form has the form $$\frac{\partial \rho }{\partial t}=D\Delta \rho +F(\rho ),\qquad \mathbf{x}\in \mathbb{R}^{3}.$$ (3.1) This equation is an example of a macroscopic reaction–transport equation that can be obtained in the long-time large-scale limit of mesoscopic equations. Recall that the mesoscopic approach is based on the idea that one can introduce mean-field equations for the particle density involving a detailed description of the movement of particles on the microscopic level. At the same time, random fluctuations around the mean behavior can be neglected due to a large number of individual particles.