Density relaxation in conserved Manna sandpiles

We study relaxation of long-wavelength density perturbations in a one-dimensional conserved Manna sandpile. Far from criticality where correlation length ξ is finite, relaxation of density profiles having wave numbers k→0 is diffusive, with relaxation time τR~k^-2/D with D being the density-dependent bulk-diffusion coefficient. Near criticality with kξ1, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as τR~k^-z, with the dynamical exponent z=2-(1-β)/ν_⊥<2, where β is the critical order-parameter exponent and ν_⊥ 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 σ of the density perturbation grows anomalously, σ~tʷ, with the growth exponent ω=1/(1+β)>1/2. In all cases, theoretical predictions are in reasonably good agreement with simulations.