Hydrodynamics, density fluctuations and universality in conserved stochastic sandpiles

TL;DR

This paper uncovers a hydrodynamic framework for conserved stochastic sandpiles, revealing universal relations and critical scaling behaviors near the phase transition, and showing that the Manna sandpile belongs to a distinct universality class from directed percolation.

Contribution

It introduces a hydrodynamic structure with an Einstein relation for CSSs and derives new scaling relations, distinguishing the universality class of the Manna sandpile from directed percolation.

Findings

Hydrodynamic structure with Einstein relation in CSSs

Scaling relations for density fluctuations near criticality

Manna sandpile belongs to a distinct universality class

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).