Power laws and logarithmic oscillations in diffusion processes on infinite countable ultrametric spaces

TL;DR

This paper derives exact solutions for diffusion equations on infinite ultrametric spaces, revealing power-law and log-periodic behaviors in the asymptotic solutions under certain conditions.

Contribution

It provides a general analytical solution for ultrametric diffusion problems, including reactions, and identifies conditions for power-law and log-periodic asymptotics.

Findings

Exact solutions for ultrametric diffusion equations.

Conditions for power-law asymptotics.

Log-periodic oscillations in solutions.

Abstract

It is shown that if the initial condition of the Cauchy problem for the diffusion equation on a general infinite countable ultrametric space is spherically symmetric with respect to some point, then this problem has an exact analytical solution. A general solution of this problem is presented for pure ultrametric diffusion, as well as for ultrametric diffusion with a reaction sink concentrated at the center of spherical symmetry. Conditions on the ultrametric and the distribution of the number of states in ultrametric spheres are found that lead at large times to the asymptotic behavior of the solutions obtained in the form of a power law modulated by a bounded function that is log-periodic under some additional conditions.