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Register a new userSelf-organised criticality in stochastic sandpiles: Connection to directed percolation
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Pradeep Kumar Mohanty
Publication date
2014
fields
Physics
and research's language is
English
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We introduce a stochastic sandpile model where finite drive and dissipation are coupled to the activity field. The absorbing phase transition here, as expected, belongs to the directed percolation (DP) universality class. We focus on the small drive and dissipation limit, i.e. the so-called self-organised critical (SOC) regime and show that the system exhibits a crossover from ordinary DP scaling to a dissipation-controlled scaling which is independent of the underlying dynamics or spatial dimension. The new scaling regime continues all the way to the zero bulk drive limit suggesting that the corresponding SOC behaviour is only DP, modified by the dissipation-controlled scaling. We demonstrate this for the continuous and discrete Manna model driven by noise and bulk dissipation.
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These lectures provide an introduction to the directed percolation and directed animals problems, from a physicists point of view. The probabilistic cellular automaton formulation of directed percolation is introduced. The planar duality of the diode-resistor-insulator percolation problem in two dimensions, and relation of the directed percolation to undirected first passage percolation problem are described. Equivalence of the $d$-dimensional directed animals problem to $(d-1)$-dimensional Yang-Lee edge-singularity problem is established. Self-organized critical formulation of the percolation problem, which does not involve any fine-tuning of coupling constants to get critical behavior is briefly discussed.
A simple one-dimensional microscopic model of the depinning transition of an interface from an attractive hard wall is introduced and investigated. Upon varying a control parameter, the critical behaviour observed along the transition line changes from a directed-percolation to a multiplicative-noise type. Numerical simulations allow for a quantitative study of the multicritical point separating the two regions, Mean-field arguments and the mapping on a yet simpler model provide some further insight on the overall scenario.
We study a hierarchy of directed percolation (DP) processes for particle species A, B, ..., unidirectionally coupled via the reactions A -> B, ... When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta^{(k)} which are markedly reduced at each hierarchy level k >= 2. We compute the fluctuation corrections to beta^{(2)} to O(epsilon = 4-d) using field-theoretic renormalization group techniques. Monte Carlo simulations are employed to determine the new exponents in dimensions d <= 3.
Self organisation provides an elegant explanation for how complex structures emerge and persist throughout nature. Surprisingly often, these structures exhibit remarkably similar scale-invariant properties. While this is sometimes captured by simple models that feature a critical point as an attractor for the dynamics, the connection to real-world systems is exceptionally hard to test quantitatively. Here we observe three key signatures of self-organised criticality in the dynamics of a driven-dissipative gas of ultracold atoms: (i) self-organisation to a stationary state that is largely independent of the initial conditions, (ii) scale-invariance of the final density characterised by a unique scaling function, and (iii) large fluctuations of the number of excited atoms (avalanches) obeying a characteristic power-law distribution. This establishes a well-controlled platform for investigating self-organisation phenomena and non-equilibrium criticality with unprecedented experimental access to the underlying microscopic details of the system.
We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a non-trivial p_c. On the augmented triangular lattice we find, by extensive numerical simulation, that the directed rigidity percolation transition belongs to the same universality class as directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e. the spreading of rigidity from finite clusters close to a non-rigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.
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