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 $\xi$ is finite, relaxation of density profiles having wave numbers $k\to 0$ is diffusive, with relaxation time ${\tau }_R\sim k^{-2}/D$ 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 ${\tau }_R\sim k^{-z}$, with the dynamical exponent $z=2-(1-\beta )/{\nu }_{\perp }<2$, where $\beta$ is the critical order-parameter exponent and and ${\nu }_{\perp }$ 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 $\sigma$ of the density perturbation grows anomalously, i.e., $\sigma \sim t^w$, with the growth exponent $\omega =1/(1+\beta )>1/2$. In all cases, theoretical predictions are in reasonably good agreement with simulations.

15 pages, 10 figures

References in corpus (7)

• Critical phenomena in atmospheric precipitation
• Hyperuniformity of critical absorbing states
• Hyperuniformity and its Generalizations
• An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media
• Fixed-Energy Sandpiles Belong Generically to Directed Percolation
• Confirming and extending the hypothesis of universality in sandpiles
• Activity, diffusion, and correlations in a two-dimensional conserved stochastic sandpile