Spatio-temporal correlations in the Manna model in one, three and five dimensions

Abstract

Although the paradigm of criticality is centred around spatial correlations and their anomalous scaling, not many studies of Self-Organised Criticality (SOC) focus on spatial correlations. Often, integrated observables, such as avalanche size and duration, are used, not least as to avoid complications due to the unavoidable lack of translational invariance. The present work is a survey of spatio-temporal correlation functions in the Manna Model of SOC, measured numerically in detail in d = 1, 3 and 5 dimensions and compared to theoretical results, in particular relating them to "integrated" observables such as avalanche size and duration scaling, that measure them indirectly. Contrary to the notion held by some of SOC models organising into a critical state by re-arranging their spatial structure avalanche by avalanche, which may be expected to result in large, non-trivial, system-spanning spatial correlations in the quiescent state (between avalanches), correlations of inactive particles in the quiescent state have a small amplitude that does not increase with the system size, although they display (noisy) power law scaling over a range linear in the system size. Selforganisation, however, does take place as the (one-point) density of inactive particles organises into a particular profile that is asymptotically independent of the driving location, also demonstrated analytically in one dimension. Activity and its correlations, on the other hand, display non-trivial long-ranged spatio-temporal scaling with exponents that can be related to established results, in particular avalanche size and duration exponents. The correlation length and amplitude are set by the system size (confirmed analytically for some observables), as expected in systems displaying finite size scaling. In one dimension, we find some surprising inconsistencies of the dynamical exponent. A (spatially extended) mean field theory is recovered, with some corrections, in five dimensions.

Field-theoretic approach to the universality of branching processes

“…For a fixed parameter set, we first rescale time τ = t/T and then average all the avalanche profiles irrespectively of T . Finally, to achieve convergence to a shape comparable across parameter settings, we normalise the result by area [38],…”

Critical density of the Abelian Manna model via a multitype branching process

“…Tab. I shows a comparison between this result and the numerical values found by simulations [9,17,18] on lattices in dimensions d ∈ {1, 2, 3, 5}. While our estimate Eq.…”

“…Firstly, the total number of particles in the system during avalanching may be reduced due to dissipation at open boundaries, and the number of singly occupied sites may decrease because of this as well as because of growing activity. As a result, branching is suppressed globally, yet this effect is weak, as only a few sites are affected [24]. Secondly, We ignore longranged anti-correlations [22,23] in the quiescent state of the AMM, which, however, appear to be rather weak albeit algebraic [24].…”

“…As a result, branching is suppressed globally, yet this effect is weak, as only a few sites are affected [24]. Secondly, We ignore longranged anti-correlations [22,23] in the quiescent state of the AMM, which, however, appear to be rather weak albeit algebraic [24]. Building on the mapping we construct here, it would be interesting for future work to establish a rigorous lower bound of the critical density in the AMM by associating the activity with some critical population dynamics, for example via the coupling method [45,46].…”

“…The evolution from one quiescent state to another quiescent state by adding a particle at a site and letting an avalanche complete is a Markov process. Because of the arXiv:1902.07793v2 [cond-mat.stat-mech] 25 Sep 2019 finiteness of the state space of quiescent configurations and assuming accessibility of all states (but see [24]), the probability to find the AMM in a particular quiescent state approaches a unique, stationary, strictly positive value. The analysis below is concerned solely with the the stationary state of the AMM.…”