Scale-free energy dissipation and dynamic phase transition in stochastic sandpiles

TL;DR

This paper investigates the scaling properties of energy dissipation and the nature of the phase transition in a stochastic sandpile model, revealing new exponents and universality class characteristics.

Contribution

It introduces new scaling exponents for energy distribution and characterizes a novel universality class for the phase transition in stochastic sandpiles.

Findings

Energy distribution exhibits p-dependent scaling exponents.

The phase transition belongs to a new universality class.

Critical exponents violate hyperscaling relations.

Abstract

We study numerically scaling properties of the distribution of cumulative energy dissipated in an avalanche and the dynamic phase transition in a stochastic directed cellular automaton [B. Tadi\'c and D. Dhar, Phys. Rev. Lett. {\bf 79}, 1519 (1997)] in d=1+1 dimensions. In the critical steady state occurring for the probability of toppling $p\ge p^\star$= 0.70548, the dissipated energy distribution exhibits scaling behavior with new scaling exponents $\tau_E $ and D_E for slope and cut-off energy, respectively, indicating that the sandpile surface is a fractal. In contrast to avalanche exponents, the energy exponents appear to be p- dependent in the region $p^\star \le p <1$, however the product $(\tau_E-1)D_E$ remains universal. We estimate the roughness exponent of the transverse section of the pile as $\chi =0.44\pm 0.04$. Critical exponents characterizing the dynamic phase transition at $p^\star $ are obtained by direct simulation and scaling analysis of the survival probability distribution and the average outflow current. The transition belongs to a new universality class with the critical exponents $\nu_\| =\gamma =1.22 \pm 0.02$, $\beta =0.56\pm 0.02$ and $\nu_\bot = 0.761 \pm 0.029$, with apparent violation of hyperscaling. Generalized hyperscaling relation leads to $\beta + \beta ^\prime = (d-1)\nu_\bot $, where $\beta ^\prime = 0.195 \pm 0.012$ is the exponent governed by the ultimate survival probability.