Series expansion for a stochastic sandpile

Abstract

Using operator algebra, we extend the series for the activity density in a one-dimensional stochastic sandpile with fixed particle density p, the first terms of which were obtained via perturbation theory [R. Dickman and R. Vidigal, J. Phys. A 35, 7269 (2002)]. The expansion is in powers of the time; the coefficients are polynomials in p. We devise an algorithm for evaluating expectations of operator products and extend the series to O(t 16 ). Constructing Padé approximants to a suitably transformed series, we obtain predictions for the activity that compare well against simulations, in the supercritical regime.

Asymptotic Behavior of the Order Parameter in a Stochastic Sandpile

“…In figure 4 we compare equation ( 31) and the results of Monte Carlo [19] simulations using systems of up to 800 sites. (For each p value, simulations are performed for various system sizes and the results extrapolated to the infinite-size limit.)…”

“…Recently, a time-dependent perturbation theory based on the path-integral formalism was derived for a stochastic sandpile [18]. In [19] the series expansion for the one-dimensional case was extended using operator methods.…”

“…(Any configuration devoid of active sites is absorbing, i.e., no futher evolution of the system is possible once such a configuration is reached.) In this work, as in [18,19], we adopt a toppling rate of n(n − 1) at a site having n particles, which leads us to define the order parameter as ρ = n(n − 1) . While this choice of rate represents a slight departure from the usual definition (in which all active sites have the same toppling rate), it leads to a much simpler evolution operator, and should yield the same scaling properties [18].…”

Steady State of Stochastic Sandpile Models

“…the 1-dimensional Oslo rice pile model, but a straightforward direct depth-first calculation of the exact probabilities of different configurations in the steady state takes O(exp(L 3 )) steps where L is the system length [9]. While the exact values of the critical exponents have been conjectured for (1 + 1) dimensional directed Manna model [10,11], the prototypical undirected Manna model in one dimension has resisted an exact solution so far [12,13,14]. In higher dimensions, most of the studies are only numerical, and deal with the estimation of the critical exponents of avalanche distribution, and the universality class of the model [4,15,16,17,18,19,20,21].…”

Approximation scheme for master equations: Variational approach to multivariate case

“…The analogy of the master equation to quantum descriptions has been introduced by Doi, and several authors have developed the formalism [6,7,8]. The field theoretic approach has revealed the anomalous kinetics in reaction-diffusion systems incorporating the renormalization group method [9,10], and the analytical scheme has been applied to various phenomena [11,12,13,14,15]. In addition, the variational method based on the second quantization description developed by Sasai and Wolynes [2] is hopeful in order to investigate the fluctuation and discrete effects in the small size systems.…”