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, profiles having wave numbers $k\ensuremath{\rightarrow}0$ diffusive, with time ${\ensuremath{\tau}}_{R}\ensuremath{\sim}{k}^{\ensuremath{-}2}/D$ $D$ being the density-dependent bulk-diffusion coefficient. Near $k\ensuremath{\xi}\ensuremath{\gtrsim}1$, bulk diffusivity diverges and transport becomes anomalous; accordingly, varies as ${\ensuremath{\tau}}_{R}\ensuremath{\sim}{k}^{\ensuremath{-}z}$, dynamical exponent $z=2\ensuremath{-}(1\ensuremath{-}\ensuremath{\beta})/{\ensuremath{\nu}}_{\ensuremath{\perp}}<2$, $\ensuremath{\beta}$ critical order-parameter ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ correlation-length exponent. Relaxation initially localized on an infinite background exhibits self-similar structure. In this case, asymptotic scaling form time-dependent profile analytically calculated: we find that, at long times $t$, width $\ensuremath{\sigma}$ perturbation grows anomalously, $\ensuremath{\sigma}\ensuremath{\sim}{t}^{w}$, growth $\ensuremath{\omega}=1/(1+\ensuremath{\beta})>1/2$. all cases, theoretical predictions are reasonably good agreement simulations.