Density relaxation in conserved Manna sandpiles

TL;DR

This paper investigates how density perturbations relax in one-dimensional conserved Manna sandpiles, revealing diffusive behavior far from criticality and anomalous diffusion near criticality, with analytical and simulation results supporting these findings.

Contribution

It provides a detailed analysis of density relaxation dynamics in Manna sandpiles, including the derivation of relaxation times and scaling forms near criticality, which was not previously characterized.

Findings

Diffusive relaxation with  scaling far from criticality

Anomalous diffusion with z<2 near criticality

Analytical scaling form for density profile evolution

Abstract

We study relaxation of long-wavelength density perturbations in one dimensional conserved Manna sandpile. Far from criticality where correlation length $\xi$ is finite, relaxation of density profiles having wave numbers $k \rightarrow 0$ is diffusive, with relaxation time $\tau_R \sim k^{-2}/D$ with $D$ being the density-dependent bulk-diffusion coefficient. Near criticality with $k \xi \gsim 1$, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as $\tau_R \sim k^{-z}$, with the dynamical exponent $z=2-(1-\beta)/\nu_{\perp} < 2$, where $\beta$ is the critical order-parameter exponent and and $\nu_{\perp}$ is the critical correlation-length exponent. Relaxation of initially localized density profiles on infinite critical background exhibits a self-similar structure. In this case, the asymptotic scaling form of the time-dependent density profile is analytically calculated: we find that, at long times $t$, the width $\sigma$ of the density perturbation grows anomalously, i.e., $\sigma \sim t^{w}$, with the growth exponent $\omega=1/(1+\beta) > 1/2$. In all cases, theoretical predictions are in reasonably good agreement with simulations.