Density relaxation in conserved Manna sandpiles

Abstract

We study relaxation of long-wavelength density perturbations in one dimensional conserved Manna sandpile. Far from criticality where correlation length $ξ$ is finite, relaxation of density profiles having wave numbers $k \rightarrow 0$ is diffusive, with relaxation time $τ_R \sim k^{-2}/D$ with $D$ being the density-dependent bulk-diffusion coefficient. Near criticality with $k ξ\gsim 1$, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as $τ_R \sim k^{-z}$, with the dynamical exponent $z=2-(1-β)/ν_{\perp} < 2$, where $β$ is the critical order-parameter exponent and and $ν_{\perp}$ 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 $σ$ of the density perturbation grows anomalously, i.e., $σ\sim t^{w}$, with the growth exponent $ω=1/(1+β) > 1/2$. In all cases, theoretical predictions are in reasonably good agreement with simulations.