Hydrodynamics, density fluctuations and universality in conserved stochastic sandpiles

Abstract

We study conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density $\rho$. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation ${\sigma }^2(\rho )=\chi (\rho )/D(\rho )$, which connects bulk-diffusion coefficient $D(\rho )$, conductivity $\chi (\rho )$ and mass-fluctuation, or scaled variance of subsystem mass, ${\sigma }^2(\rho )$. Consequently, density large-deviations are governed by an equilibriumlike chemical potential $\mu (\rho )\sim \ln a(\rho )$ where $a(\rho )$ is the activity in the system. Using the above hydrodynamics, we derive two scaling relations: As $\Delta =(\rho -{\rho }_c)\to 0^{+}$, ${\rho }_c$ being the critical density, (i) the mass-fluctuation ${\sigma }^2(\rho )\sim {\Delta }^{1-\delta }$ with $\delta =0$ and (ii) the dynamical exponent $z=2+(\beta -1)/{\nu }_{\perp }$, expressed in terms of two static exponents $\beta$ and ${\nu }_{\perp }$ for activity $a(\rho )\sim {\Delta }^{\beta }$ and correlation length $\xi \sim {\Delta }^{-{\nu }_{\perp }}$, respectively. Our results imply that conserved Manna sandpile, a well studied variant of the CSS, belongs to a distinct universality - {\it not} that of directed percolation (DP), which, without any conservation law as such, does not obey scaling relation (ii).

12 pages, 7 figures