What is the most optimal diffusion?

Abstract

What is the fastest possible "diffusion"? A trivial answer would be "a process that converts a Dirac delta-function into a uniform distribution infinitely fast". Below, we consider a more reasonable formulation: a process that maximizes differential entropy of a probability density function (pdf) $f(\vec{x},t)$ at every time $t$, under certain restrictions. Specifically, we focus on a case when the rate of the Kullback-Leibler divergence $D_{\text{KL}}$ is fixed. If $\Delta (\vec{x},t,dt)=\frac{\partial f}{\partial t}dt$ is the pdf change at a time step $dt$, we maximize the differential entropy $H[f+\Delta ]$ under the restriction $D_{\text{KL}}(f+\Delta ||f)=A^{2}d{t}^2$, $A=\text{const}>0$. It leads to the following equation: $\frac{\partial f}{\partial t}=-\kappa f(\ln f-\int f\ln fd\vec{x})$, with $\kappa =\frac{A}{\sqrt{\int f{\ln }^{2}fd\vec{x}-{\left(\int f\ln fd\vec{x}\right)}^2}}$. Notably, this is a non-local equation, so the process is different from the Itô diffusion and a corresponding Fokker-Planck equation. We show that the normal and exponential distributions are solutions to this equation, on $(-\infty ;\infty )$ and $[0;\infty )$, respectively, both with $\text{variance}\sim e^{2At}$, i.e. diffusion is highly anomalous. We numerically demonstrate for sigmoid-like functions on a segment that the entropy change rate $\frac{dH}{dt}$ produced by such an optimal "diffusion" is, as expected, higher than produced by the "classical" diffusion.