Dynamic correlations in the conserved Manna sandpile
Anirban Mukherjee, Punyabrata Pradhan
Abstract
We study dynamic correlations for current and mass, as well as the associated power spectra, in the one-dimensional conserved Manna sandpile. We show that, in the thermodynamic limit, the variance of cumulative bond current up to time $T$ grows subdiffusively as ${T}^{1/2\ensuremath{-}\ensuremath{\mu}}$ with the exponent $\ensuremath{\mu}\ensuremath{\ge}0$ depending on the density regimes considered and, likewise, the power spectra of current and mass at low frequency $f$ varies as ${f}^{1/2+\ensuremath{\mu}}$ and ${f}^{\ensuremath{-}3/2+\ensuremath{\mu}}$, respectively. Our theory predicts that, far from criticality, $\ensuremath{\mu}=0$ and, near criticality, $\ensuremath{\mu}=(\ensuremath{\beta}+1)/2{\ensuremath{\nu}}_{\ensuremath{\perp}}z>0$ with $\ensuremath{\beta}, {\ensuremath{\nu}}_{\ensuremath{\perp}}$, and $z$ being the order parameter, correlation length, and dynamic exponents, respectively. The anomalous suppression of fluctuations near criticality signifies a ``dynamic hyperuniformity,'' characterized by a set of fluctuation relations, in which current, mass, and tagged-particle displacement fluctuations are shown to have a precise quantitative relationship with the density-dependent activity (or its derivative). In particular, the relation, ${\mathcal{D}}_{s}(\overline{\ensuremath{\rho}})=a(\overline{\ensuremath{\rho}})/\overline{\ensuremath{\rho}}$, between the self-diffusion coefficient ${\mathcal{D}}_{s}(\overline{\ensuremath{\rho}})$, activity $a(\overline{\ensuremath{\rho}})$ and density $\overline{\ensuremath{\rho}}$ explains a previous simulation observation [Eur. Phys. J. B 72, 441 (2009)] that, near criticality, the self-diffusion coefficient in the Manna sandpile has the same scaling behavior as the activity.