Abstract
Bak, Tang, and Wiesenfeld (BTW) proposed the theory of self-organized criticality (SOC), and sandpile models, to connect “1/f” noise, observed in systems in a diverse natural setting, to the fractal spatial structure. We review some of the existing works on the problem of characterizing time-dependent properties of sandpiles and try to explore if the BTW's original ambition has really been fulfilled. We discuss the exact hydrodynamic structure in a class of conserved stochastic sandpiles, undergoing a non-equilibrium absorbing phase transition. We illustrate how the hydrodynamic framework can be used to capture long-ranged spatio-temporal correlations in terms of large-scale transport and relaxation properties of the systems. We particularly emphasize certain interesting aspects of sandpiles—the transport instabilities, which emerge through the threshold-activated nature of the dynamics in the systems. We also point out some open issues at the end.
Highlights
More than three decades ago, Bak, Tang, and Wiesenfeld (BTW) proposed the theory of “selforganized criticality” (SOC) [1], and sandpile models [2, 3], as an explanation of the physical origin of the spatio-temporal scale invariance in natural systems found around us [4,5,6]
Hwa and Kardar [55] argued that the 1/f noise could arise in the system when the individual avalanches, which are created at different space points, are allowed to overlap with each other and to develop nontrivial correlations in that process
The hydrodynamic equations, and the corresponding dynamic renormalization group (RG) theory, proposed by Hwa and Kardar [55, 57] was successful to some extent in explaining the emergence of scale invariance, and in extracting the critical exponents of sandpiles in a general hydrodynamic framework, based on symmetries and conservation laws
Summary
More than three decades ago, Bak, Tang, and Wiesenfeld (BTW) proposed the theory of “selforganized criticality” (SOC) [1], and sandpile models [2, 3], as an explanation of the physical origin of the spatio-temporal scale invariance in natural systems found around us [4,5,6]. The origin of the subtle longtime correlations in 1/f noise has lacked a general theoretical understanding so far In their original paper [1], BTW attempted to provide a universal mechanism of the 1/f noise through the interplay between scale-invariant spatial and temporal structures, which can develop in slowly driven spatially extended dynamical systems observed in nature. Later several stochastic variants of the BTW model, such as the two celebrated models—the Manna sandpile [17] and the Oslo ricepile [18], with stochastic toppling rules, were introduced It has been debated whether the SOC systems can be thought of as ones spontaneously evolving, or “self-organizing.” toward criticality, especially when the driving rate itself is “tuned” to zero [19]. We end the article with concluding remarks and some open issues
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