Non-Archimedean Reaction-Ultradiffusion Equations and Complex Hierarchic Systems

TL;DR

This paper introduces non-Archimedean reaction-ultradiffusion equations linked to complex hierarchic systems, extending classical models to p-adic spaces and revealing unique diffusion mechanisms in ultrametric settings.

Contribution

It develops the first mathematical framework for non-Archimedean reaction-ultradiffusion equations and connects them to physical models of complex hierarchic systems.

Findings

Equations are p-adic analogs of classical phase separation models.

Solutions describe density profiles in ultrametric spaces.

Diffusion mechanisms differ fundamentally from classical cases.

Abstract

We initiate the study of non-Archimedean reaction-ultradiffusion equations and their connections with models of complex hierarchic systems. From a mathematical perspective, the equations studied here are the p-adic counterpart of the integro-differential models for phase separation introduced by Bates and Chmaj. Our equations are also generalizations of the ultradiffusion equations on trees studied in the 80's by Ogielski, Stein, Bachas, Huberman, among others, and also generalizations of the master equations of the Avetisov et al. models, which describe certain complex hierarchic systems. From a physical perspective, our equations are gradient flows of non-Archimedean free energy functionals and their solutions describe the macroscopic density profile of a bistable material whose space of states has an ultrametric structure. Some of our results are p-adic analogs of some well-known results in the Archimedean settting, however, the mechanism of diffusion is completely different due to the fact that it occurs in an ultrametric space.