Abstract
An exact solution to the discrete diffusion equation allows for accurate predictions of how the probabilities of reaction diffusion processes evolve over time.
Highlights
While random walks have their root in the 17th century analysis of games of chance [1], they are associated with Pearson who introduced them in 1905 [2], many of their properties had already been elucidated in 1900 by Bachelier [3] whose work remained largely unknown until the 1950s [4]
lattice random walks (LRW) are a special class of Markov chains [14]; they were popularized in the 1920s by Pólya’s seminal work [15,16] on the dimensionality dependence of the recurrence probability, that is, the probability that a random walker on an infinite space lattice eventually returns to its starting point
The diffusion equation is one of a small set of fundamental equations that has left a legacy across a vast number of disciplines
Summary
While random walks have their root in the 17th century analysis of games of chance [1], they are associated with Pearson who introduced them in 1905 [2], many of their properties had already been elucidated in 1900 by Bachelier [3] whose work remained largely unknown until the 1950s [4]. Another quantity directly related to the propagator generating function is the number of distinct sites visited. The 1D case can be found analytically [see Eqs. (D1) and (D3), respectively, for reflecting and periodic boundaries]
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