Non-commutative geometry and irreversibility

TL;DR

This paper introduces a $q$-calculus-based kinetic model for diffusion on hierarchical ultrametric spaces, revealing inherent irreversibility in the motion along the hierarchy levels.

Contribution

It develops a novel $q$-calculus framework to describe diffusion on ultrametric spaces, linking hierarchical structure to irreversibility in dynamics.

Findings

Diffusion is modeled on a hierarchical lattice using $q$-calculus.

The observable 'quasi-position' corresponds to hierarchy levels.

Motion along the hierarchy is inherently irreversible.

Abstract

A kinetics built upon $q$-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the "quasi-position" whose eigenvalues are the levels of the hierarchy, corresponding to the volume ofphase space available to the system at any given time. Motion along the lattice of quasi-positions is irreversible.