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 $\ensuremath{\xi}$ is finite, relaxation of density profiles having wave numbers $k\ensuremath{\rightarrow}0$ is diffusive, with relaxation time ${\ensuremath{\tau}}_{R}\ensuremath{\sim}{k}^{\ensuremath{-}2}/D$ with $D$ being the density-dependent bulk-diffusion coefficient. Near criticality with $k\ensuremath{\xi}\ensuremath{\gtrsim}1$, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as ${\ensuremath{\tau}}_{R}\ensuremath{\sim}{k}^{\ensuremath{-}z}$, with the dynamical exponent $z=2\ensuremath{-}(1\ensuremath{-}\ensuremath{\beta})/{\ensuremath{\nu}}_{\ensuremath{\perp}}<2$, where $\ensuremath{\beta}$ is the critical order-parameter exponent and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ 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 $\ensuremath{\sigma}$ of the density perturbation grows anomalously, $\ensuremath{\sigma}\ensuremath{\sim}{t}^{w}$, with the growth exponent $\ensuremath{\omega}=1/(1+\ensuremath{\beta})>1/2$. In all cases, theoretical predictions are in reasonably good agreement with simulations.