Anomalous diffusion for a class of systems with two conserved quantities

TL;DR

This paper investigates super-diffusive behavior in one-dimensional energy-volume conserving models, demonstrating super-diffusivity through simulations and rigorous proofs for harmonic cases, and explores effects of stochastic noise on ergodicity.

Contribution

It introduces a new class of deterministic models with conserved quantities, analyzes their super-diffusive properties, and extends the study to stochastic modifications with proven results for harmonic potentials.

Findings

Models exhibit super-diffusive behavior in simulations.

Adding stochastic noise preserves super-diffusivity.

Rigorous proof of super-diffusivity for harmonic potentials.

Abstract

We introduce a class of one dimensional deterministic models of energy-volume conserving interfaces. Numerical simulations show that these dynamics are genuinely super-diffusive. We then modify the dynamics by adding a conservative stochastic noise so that it becomes ergodic. System of conservation laws are derived as hydrodynamic limits of the modified dynamics. Numerical evidence shows these models are still super-diffusive. This is proven rigorously for harmonic potentials.