The Abelian Manna model on various lattices in one and two dimensions

Abstract

We perform a high-accuracy moment analysis of the avalanche size, duration and area distribution of the Abelian Manna model on eight two-dimensional and four one-dimensional lattices. The results provide strong support to establish universality of exponents and moment ratios across different lattices and a good survey for the strength of corrections to scaling which are notorious in the Manna universality class. The results are compared against previous work done on Manna model, Oslo model and directed percolation. We also confirm hypothesis of various scaling relations.

25 Years of Self-organized Criticality: Concepts and Controversies

“…In fact, later studies make it abundantly clear that the average value of the dynamical variable (the local degree of freedom subject to interaction, see below) in SOC systems is normally well away from the threshold. For example, in two dimensions, the sandpile model has been conjectured (Grassberger quoted by Dhar 2006, finally confirmed analytically by Caracciolo and Sportiello 2012) to have average height of 17/8 = 2.125, well below the threshold of 3, and the Abelian Manna Model (Manna 1991;Dhar 1999b) with threshold 1 in one dimension has average height 0.9488(5) (Dickman et al 2001), expected to drop to 1/2 with increasing dimension (Huynh et al 2011).…”

“…• Firstly, robust, reproducible finite size scaling without dependence on any control parameter or details of the definition of the model (Dickman et al 2002), such as the underlying lattice structure (Huynh et al 2011). • Secondly, spatio-temporal correlations, which were initially measured through integrated observables (avalanche size, duration, area, radius of gyration etc.).…”

Continuous transforming the BTW to the Manna model

“…It has been shown that the power spectrum in both models decay as P(f ) ∼ 1/f α , with α = 1.59 ± 0.05 for BTW model and α = 1.77 ± 0.05 for the Manna model [26] which confirms this result [25]. Many papers discussed this result [19][20][21]23,[27][28][29].…”

Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

“…It is widely accepted that the BTW model has a multiscaling behaviour [16,17] due to its complete toppling balance [18] and positive auto-correlation in avalanche wave series [19,20], whereas the SSM does not show such correlation and consequently follows finite size scaling (FSS) ansatz. Recent numerical studies of the SSM have been carried out not only on various regular lattices of integer dimension but also on various kind of deterministic fractal lattices [21,22,23] and the results confirm the existence of robust FSS behaviour of the SSM across different regular, as well as fractal lattices though the universality class depends on the space or fractal dimension of these lattices. The fractal lattices considered for such studies were deterministic, the properties of SSM as well as BTW on random fractal lattices are yet to be studied.…”