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 $\Delta = (\rho - \rho_c) \rightarrow 0^+$, $\rho_c$ 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).