Paths to Self-Organized Criticality

We present a pedagogical introduction to self-organized criticality (SOC), unraveling its connections with nonequilibrium phase transitions. There are several paths from a conventional critical point to SOC. They begin with an absorbing-state phase transition (directed percolation is a familiar example), and impose supervision or driving on the system; two commonly used methods are extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved density. Other paths to SOC, in driven interfaces, the Bak-Sneppen model, and self-organized directed percolation, are also examined. We review the status of experimental realizations of SOC in light of these observations.Comment: 23 pages + 2 figure

Non-triviality of a discrete Bak-Sneppen evolution model

Consider the following evolution model, proposed in \cite{BS} by Bak and Sneppen. Put $N$ vertices on a circle, spaced evenly. Each vertex represents a certain species. We associate with each vertex a random variable, representing the `state' or `fitness' of the species, with values in $[0,1]$. The dynamics proceeds as follows. Every discrete time step, we choose the vertex with minimal fitness, and assign to this vertex, and to its two neighbours, three new independent fitnesses with a uniform distribution on $[0,1]$. A conjecture of physicists, based on simulations, is that in the stationary regime, the one-dimensional marginal distributions of the fitnesses converges, when $N \to \infty$, to a uniform distribution on $(f,1)$, for some threshold $f<1$. In this paper we consider a discrete version of this model, proposed in \cite{BK}. In this discrete version, the fitness of a vertex can be either 0 or 1. The system evolves according to the following rules. Each discrete time step, we choose an arbitrary vertex with fitness 0. If all the vertices have fitness 1, then we choose an arbitrary vertex with fitness 1. Then we update the fitnesses of this vertex and of its two neighbours by three new independent fitnesses, taking value 0 with probability $0<q<1$, and 1 with probability $p=1-q$. We show that if $q$ is close enough to one, then the mean average fitness in the stationary regime is bounded away from 1, uniformly in the number of vertices. This is a small step in the direction of the conjecture mentioned above, and also settles a conjecture mentioned in \cite{BK}. Our proof is based on a reduction to a continuous time particle system

Dynamic Critical approach to Self-Organized Criticality

A dynamic scaling Ansatz for the approach to the Self-Organized Critical (SOC) regime is proposed and tested by means of extensive simulations applied to the Bak-Sneppen model (BS), which exhibits robust SOC behavior. Considering the short-time scaling behavior of the density of sites ($\rho(t)$) below the critical value, it is shown that i) starting the dynamics with configurations such that $\rho(t=0) \to 0$ one observes an {\it initial increase} of the density with exponent $\theta = 0.12(2)$; ii) using initial configurations with $\rho(t=0) \to 1$, the density decays with exponent $\delta = 0.47(2)$. It is also shown that he temporal autocorrelation decays with exponent $C_a = 0.35(2)$. Using these, dynamically determined, critical exponents and suitable scaling relationships, all known exponents of the BS model can be obtained, e.g. the dynamical exponent $z = 2.10(5)$, the mass dimension exponent $D = 2.42(5)$, and the exponent of all returns of the activity $\tau_{ALL} = 0.39(2)$, in excellent agreement with values already accepted and obtained within the SOC regime.Comment: Rapid Communication Physical Review E in press (4 pages, 5 figures

Evolution of economic entities under heterogeneous political/environmental conditions within a Bak-Sneppen-like dynamics

A model for economic behavior, under heterogeneous spatial economic conditions is developed. The role of selection pressure in a Bak-Sneppen-like dynamics with entity diffusion on a lattice is studied by Monte-Carlo simulation taking into account business rule(s), like enterprise - enterprise short range location "interaction"(s), business plan(s) through spin-offs or merging and enterprise survival evolution law(s). It is numerically found that the model leads to a sort of phase transition for the fitness gap as a function of the selection pressure.Comment: 6 figures. to be published in Physica

Replicating financial market dynamics with a simple self-organized critical lattice model

We explore a simple lattice field model intended to describe statistical properties of high frequency financial markets. The model is relevant in the cross-disciplinary area of econophysics. Its signature feature is the emergence of a self-organized critical state. This implies scale invariance of the model, without tuning parameters. Prominent results of our simulation are time series of gains, prices, volatility, and gains frequency distributions, which all compare favorably to features of historical market data. Applying a standard GARCH(1,1) fit to the lattice model gives results that are almost indistinguishable from historical NASDAQ data.Comment: 20 pages, 33 figure

Regimes of self-organized criticality in the atmospheric convection

Large scale organization in ensembles of events of atmospheric convection can be generated by the combined effect of forcing and of the interaction between the raising plumes and the environment. Here the "large scale" refers to the space extension that is larger or comparable with the basic resolved cell of a numerical weather prediction system. Under the action of external forcing like heating individual events of convection respond to the slow accumulation of vapor by a threshold-type dynamics. This is due to the a time-scale separation, between the slow drive and the fast convective response, expressed as the "quasi-equilibrium". When there is interaction between the convection plumes, the effect is a correlated response. We show that the correlated response have many of the characteristics of the self-organized criticality (SOC). It is suggested that from the SOC perspective, a description of the specific dynamics induced by "quasi-equilibrium" can be provided by models of "punctuated equilibrium". Indeed the Bak-Sneppen model is able to reproduce (within reasonable approximation) two of the statistical results that have been obtained in observations on the organized convection. We also give detailed derivation of the equations connecting the probabilities of the states in the update sequence of the Bak-Sneppen model with $K=2$ random neighbors. This analytical framework allows the derivation of scaling laws for the size of avalanches, a result that gives support to the SOC interpretation of the observational data.Comment: Text prepared for the Report of COST ES0905 collaboration (2014). Latex 45 page