Random Walks and Mesoscopic Reaction-Transport Equations

Random Walks and Mesoscopic Reaction-Transport Equations 

As discussed in Sect. 2.1, the standard reaction-diffusion equation for the particle density p x,t) has the form 

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. For example, we can obtain (3.1) from the mesoscopic integro-differential equation 

These are the mean-field equations for the density of particles that follow a continuous-time random walk (CTRW). Each random step of a particle is characterized by a waiting time and a jump length, which are distributed according to the joint 

Mendez et at., Reaction-Transport Systems, Springer Series in Synergetics, DOI 10.1007/978-3-642-11443-4 3, Springer-Verlag Berlin Heidelberg 2010 

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Random Walks and Mesoscopic Reaction-Transport Equations Non-Markov CTRW... 

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