= Logarithmic diffusion
{title2=$u_t=\kappa\Delta\log u$}
For a positive density $u$, logarithmic diffusion is the <nonlinear diffusion equation> $u_t=\kappa\Delta\log u$, equivalently $u_t=\kappa\operatorname{div}(u^{-1}\nabla u)$. Its <diffusion coefficient> is inversely proportional to the density. In one dimension, a mass-$M$ <self-similar solution> is
$$
u(x,t)=\frac{2\kappa t}{x^2+(2\pi\kappa t/M)^2}.
$$
Its integral is $M$, and substitution verifies the equation. This is $M$ times a <Cauchy distribution> density of scale $2\pi\kappa t/M$. It tends weakly to $M$ times the <Dirac delta distribution> as $t\downarrow0$; the slow algebraic tails illustrate the enhanced diffusion where density is small.
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