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Register a new userSystem-size independence of a large deviation function for frequency of events in a one-dimensional forest-fire model with a single ignition site
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Authors
Tetsuya Mitsudo
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It is found that a large deviation function for frequency of events of size not equal to the system size in the one dimensional forest-fire model with a single ignition site at an edge is independent of the system size, by using an exact decomposition of the modified transition matrix of a master equation. An exchange in the largest eigenvalue of the modified transition matrix may not occur in the model.
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Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.
We present a driven diffusive model which we call the Bus Route Model. The model is defined on a one-dimensional lattice, with each lattice site having two binary variables, one of which is conserved (``buses) and one of which is non-conserved (``passengers). The buses are driven in a preferred direction and are slowed down by the presence of passengers who arrive with rate lambda. We study the model by simulation, heuristic argument and a mean-field theory. All these approaches provide strong evidence of a transition between an inhomogeneous ``jammed phase (where the buses bunch together) and a homogeneous phase as the bus density is increased. However, we argue that a strict phase transition is present only in the limit lambda -> 0. For small lambda, we argue that the transition is replaced by an abrupt crossover which is exponentially sharp in 1/lambda. We also study the coarsening of gaps between buses in the jammed regime. An alternative interpretation of the model is given in which the spaces between ``buses and the buses themselves are interchanged. This describes a system of particles whose mobility decreases the longer they have been stationary and could provide a model for, say, the flow of a gelling or sticky material along a pipe.
The standard Large Deviation Theory (LDT) represents the mathematical counterpart of the Boltzmann-Gibbs factor which describes the thermal equilibrium of short-range Hamiltonian systems, the velocity distribution of which is Maxwellian. It is generically applicable to systems satisfying the Central Limit Theorem (CLT). When we focus instead on stationary states of typical complex systems (e.g., classical long-range Hamiltonian systems), both the CLT and LDT need to be generalized. Specifically, when the N->infinity attractor in the space of distributions is a Q-Gaussian related to a Q-generalized CLT (Q=1 recovers Gaussian attractors), we expect the LDT probability distribution to approach a q-exponential (where q=f(Q) with f(1)=1, thus recovering the standard LDT exponential distribution) with an argument proportional to N, consistently with thermodynamics. We numerically verify this conjectural scenario for the standard map, the coherent noise model for biological extinctions and earthquakes, the Ehrenfest dog-flea model, and the random-walk avalanches.
Many sociological networks, as well as biological and technological ones, can be represented in terms of complex networks with a heterogeneous connectivity pattern. Dynamical processes taking place on top of them can be very much influenced by this topological fact. In this paper we consider a paradigmatic model of non-equilibrium dynamics, namely the forest fire model, whose relevance lies in its capacity to represent several epidemic processes in a general parametrization. We study the behavior of this model in complex networks by developing the corresponding heterogeneous mean-field theory and solving it in its steady state. We provide exact and approximate expressions for homogeneous networks and several instances of heterogeneous networks. A comparison of our analytical results with extensive numerical simulations allows to draw the region of the parameter space in which heterogeneous mean-field theory provides an accurate description of the dynamics, and enlights the limits of validity of the mean-field theory in situations where dynamical correlations become important.
Forest fire models may be interpreted as a simple model for earthquake occurrence by translating trees and fire into stressed segments of a fault and their rupture, respectively. Here we adopt a twodimensional forest-fire model in continuous time, and focus on the temporal changes of seismicity and the b-value. We find the b-value change and seismic quiescence prior to large earthquakes by stacking many sequences towards large earthquakes. As the magnitude-frequency relation in this model is directly related to the cluster-size distribution, decrease of the b-value can be explained in terms of the change in the cluster-size distribution. Decrease of the b-value means that small clusters of stressed sites aggregate into a larger cluster. Seismic quiescence may be attributed to the decrease of stressed sites that do not belong to percolated clusters.
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