Density relaxation in conserved Manna sandpiles

Abstract

We study relaxation of long-wavelength density perturbations in a one-dimensional conserved Manna sand-pile. Far from criticality where correlation length xi is finite, relaxation of density profiles having wave numbers k -> 0 is diffusive, with relaxation time tau(R) - k(-2)/D with D being the density-dependent bulk-diffusion co-efficient. Near criticality with k xi greater than or similar to 1, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as tau(R) - k(-z), with the dynamical exponent z = 2 - (1 - beta)/v(perpendicular to) < 2, where beta is the critical order-parameter exponent and v(perpendicular to) is the critical correlation-length exponent. Relaxation of initially localized density profiles on an 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 a of the density perturbation grows anomalously, sigma - t(w), with the growth exponent omega = 1/(1 + beta) > 1/2. In all cases, theoretical predictions are in reasonably good agreement with simulations.