Suppose that $p(x,t)$ be population density of some species at time $t$ in position $x$. Then the spatial movement is modeled by the Laplacian with diffusion coefficient $k$. I want to know, why Laplacian came into the picture here?
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Robert Lewis
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Manoj Kumar
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4The short answer is: discretize space, calculate the flux through a segment of width $dx$, write an expression for total population in that segment, then let $dx \rightarrow 0$. The result is the laplacian. The long answer is long. – Sort of Damocles Dec 30 '19 at 05:11
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3It arises under the assumption of Fick's first law, i.e. that the rate of diffusive flow of the species is proportional to the density gradient. Wikipedia has a derivation as well, although it's less intuitive than dbx's suggestion to consider the rate of in/out-flow for an infinitesimal volume. – Dec 30 '19 at 05:18
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@dbx when we write the expression for the total population, do we need to differentiate it? – Manoj Kumar Dec 30 '19 at 05:19
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1The Laplacian is a local averaging at every point in the domain, which is exactly what diffusion is. – Matthew Cassell Dec 30 '19 at 14:49
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Thanks @mattos , that makes more sense to me. – Manoj Kumar Dec 30 '19 at 15:51
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1This question contains the answer, even if it is not exactly a duplicate. – Giuseppe Negro Jan 01 '20 at 23:04