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 $ρ$. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation $σ^2(ρ) = χ(ρ)/D(ρ)$, which connects bulk-diffusion coefficient $D(ρ)$, conductivity $χ(ρ)$ and mass-fluctuation, or scaled variance of subsystem mass, $σ^2(ρ)$. Consequently, density large-deviations are governed by an equilibriumlike chemical potential $μ(ρ) \sim \ln a(ρ)$ where $a(ρ)$ is the activity in the system. Using the above hydrodynamics, we derive two scaling relations: As $Δ= (ρ- ρ_c) \rightarrow 0^+$, $ρ_c$ being the critical density, (i) the mass-fluctuation $σ^2(ρ) \sim Δ^{1-δ}$ with $δ=0$ and (ii) the dynamical exponent $z = 2 + (β-1)/ν_{\perp}$, expressed in terms of two static exponents $β$ and $ν_{\perp}$ for activity $a(ρ) \sim Δ^β$ and correlation length $ξ\sim Δ^{-ν_{\perp}}$, 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).