# Hyperbolic Diffusion Functionals on a Ring with Finite Velocity

## TL;DR

This paper explores diffusion on a ring lattice using non-local master equations and classical theories to analyze Fisher information and entropy trends over time.

## Contribution

The study introduces a novel approach to analyzing diffusion using non-local master equations and identifies power-law decay patterns in Fisher information and complexity.

## Key findings

- Fisher information decays as t−ν with ν=2 for short times and ν=1 for long times.
- Similar power-law trends were observed for complexity and Fisher information related to Shannon entropy.
- Small rings converge to a uniform distribution for long times.

## Abstract

I study a lattice with periodic boundary conditions using a non-local master equation that evolves over time. I investigate different system regimes using classical theories like Fisher information, Shannon entropy, complexity, and the Cramér–Rao bound. To simulate spatial continuity, I employ a large number of sites in the ring and compare the results with continuous spatial systems like the Telegrapher’s equations. The Fisher information revealed a power-law decay of t−ν, with ν=2 for short times and ν=1 for long times, across all jump models. Similar power-law trends were also observed for complexity and the Fisher information related to Shannon entropy over time. Furthermore, I analyze toy models with only two ring sites to understand the behavior of the Fisher information and Shannon entropy. As expected, a ring with a small number of sites quickly converges to a uniform distribution for long times. I also examine the Shannon entropy for short and long times.