# [049] Propagation-Dispersion Equation

A propagation-dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the hydrodynamic limit of the first visit equation, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time-delayers. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker–Planck equation) as it describes a dispersion process in time (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The temporal dispersion coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.

Recommended citation: Jean Pierre Boon, Patrick Grosfils, and James F. Lutsko, "Propagation-Dispersion Equation", *J. Stat. Phys.*, **113**, 527 (2003)

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