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 (21)

• Crackling Noise
• Critical phenomena in atmospheric precipitation
• The abelian sandpile and related models
• Critical brain networks
• Hyperuniformity of critical absorbing states
• 25 Years of Self-Organized Criticality: Concepts and Controversies
• Self-organized criticality as an absorbing-state phase transition
• Driving, conservation and absorbing states in sandpiles
• Hyperuniformity and its Generalizations
• Critical behavior of a one-dimensional fixed-energy stochastic sandpile
• An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media
• Fixed-Energy Sandpiles Belong Generically to Directed Percolation
• Sandpiles with height restrictions
• The Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model
• Confirming and extending the hypothesis of universality in sandpiles
• Power spectrum of mass and activity fluctuations in a sandpile
• Probability distribution of residence times of grains in models of ricepiles
• Diffusion in stochastic sandpiles
• Hydrodynamics, density fluctuations and universality in conserved stochastic sandpiles
• Probability distribution of residence-times of grains in sandpile models
• Activity, diffusion, and correlations in a two-dimensional conserved stochastic sandpile

Cited by in corpus (5)

• Dynamic correlations in the conserved Manna sandpile
• Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry
• How far do Activated Random Walkers spread from a single source?
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Generic power laws in higher-dimensional lattice models with multidirectional hopping