Persistent random walk
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The persistent random walk is a modification of the random walk model.
A population of particles are distributed on a line, with constant speed c 0 {\displaystyle c_{0}}, and each particle's velocity may be reversed at any moment. The reversal time is exponentially distributed as e − t / τ / τ {\displaystyle e^{-t/\tau }/\tau }, then the population density n {\displaystyle n} evolves according to( 2 τ − 1 ∂ t + ∂ t t − c 0 2 ∂ x x ) n = 0 {\displaystyle (2\tau ^{-1}\partial _{t}+\partial _{tt}-c_{0}^{2}\partial _{xx})n=0}which is the telegrapher's equation.