Dynamic correlations in the conserved Manna sandpile

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 $T^{1/2-\mu }$ with the exponent $\mu \ge 0$ depending on the density regimes considered and, likewise, the power spectra of current and mass at low frequency $f$ varies as $f^{1/2+\mu }$ and $f^{-3/2+\mu }$, respectively; our theory predicts that, far from criticality, $\mu =0$ and, near criticality, $\mu =(\beta +1)/2{\nu }_{\perp }z>0$ with $\beta$, ${\nu }_{\perp }$ 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, it's derivative). In particular, the relation, ${\mathcal{D}}_s(\bar{\rho })=a(\bar{\rho })/\bar{\rho }$, between the self-diffusion coefficient ${\mathcal{D}}_s(\bar{\rho })$, activity $a(\bar{\rho })$ and density $\bar{\rho }$ explains a previous simulation observation [Eur. Phys. J. B \textbf{72}, 441 (2009)] that, near criticality, the self-diffusion coefficient in the Manna sandpile has the same scaling behavior as the activity.

arXiv commentTypo corrected in the rhs of eq. (24)

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• Hyperuniformity in mass transport processes with center-of-mass conservation: Some exact results
• Dynamic fluctuations of current and mass in nonequilibrium mass transport processes
• Current fluctuations in the symmetric zero-range process below and at critical density
• Anomalous current fluctuations and mobility-driven clustering