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The solution to the Fokker-Planck equation can be thought of as a macroscopic description of the dynamics of a diffusion process. Various results make this heuristic more precise - Ito integration, the Feynman-Kac theorem and so on.

Do reaction-diffusion equations have a similar probabilistic interpretation? If so, where can I read about this?

Many thanks.

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2 Answers 2

Many reaction–diffusion systems can be interpreted as deterministic scaling limits of interacting particle systems, obtained in much the same way as the Fokker–Planck equation is obtained from a random walk (effectively a non-interacting particle system). Indeed, I would say that this is the reason why reaction–diffusion equations are useful for modelling real world systems involving, at a fundamental level, stochastic interactions of discrete entities.

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Interesting, thanks. –  Simon Aug 30 '11 at 19:21

The interpretation is similar to the diffusive dynamics in a sense that FP description of RD systems likewise operates with a probability distribution which evolves in time due to fluxes brought about by diffusion and reactions. The nature of these fluxes need to be reflected in the FP operator. Also just like in the case of ordinary diffusion the validity of FP picture for RD systems rests on the plausibility of spatial and temporal coarse graining. E.g one demands that 1) system is very large so that one can approximate change of otherwise discrete molecular numbers with differentials of smooth and continuously changing probability distributions and 2) Reactions are of Markovian nature, meaning that thermal motion erases the microscopic dynamics fairly quickly relative to the reactive events and diffusion.

I would say the best source to read up on this topic is Van Kampen's book (Stochastic Processes in Physics and Chemistry) which is the standard citing reference in many research papers and books.

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