2025/10/08 by Vasili Baranau, Baranau, Vasili
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Diffusion and Search Dynamics #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2510.07571
openalex publication_date 2025/10/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
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(x, t) at every time t, under certain restrictions. Specifically, we focus on a case when the rate of the Kullback-Leibler divergence DKL is fixed. If Δ(x, t, dt) = (∂ f)/( ∂ t) dt is the pdf change at a time step dt, we maximize the differential entropy H[f + Δ] under the restriction DKL(f + Δ|| f) = A2 dt2, A = const > 0. It leads to the following equation: (∂ f)/( ∂ t) = - κf (lnf - ∫ f lnf dx), with κ= \fracA√ ∫ f ln2f dx - ( ∫ f lnf dx )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 (-∞; ∞) and [0; ∞), respectively, both with variance ∼ e2 A t, i.e. diffusion is highly anomalous. We numerically demonstrate for sigmoid-like functions on a segment that the entropy change rate (d H)/(d t) produced by such an optimal "diffusion" is, as expected, higher than produced by the "classical" diffusion.