Stochastic dynamics of coupled systems and damage spreading

Abstract

We study the damage spreading in the one-dimensional Ising model by means of the stochastic dynamics resulting from coupling the system and its replica by a family of algorithms that interpolate between the heat bath and the Hinrichsen-Domany algorithms. At high temperatures the dynamics is exactly mapped to the DomanyKinzel probabilistic cellular automaton. Using a mean-field approximation and Monte Carlo simulations we find the critical line that separates the phase where the damage spreads from the one where it does not.

The threshold of coexistence and critical behaviour of a predator–prey cellular automaton

“…The critical behavior of this model has been extensively studied and it belongs to the directed percolation class [26,27,28]. Other examples are the Domany-Kinzel cellular automaton [35] and the spreading of damage transitions [27,36,37]. Among the models with more then one absorbing state, perhaps the most known is the ZiffGullari-Barshad model [38] for the reaction of oxidation of carbon monoxide over a catalytic surface; it also has dynamic critical exponents [24] consistent with the ones associated to the directed percolation class.…”

Stable oscillations of a predator–prey probabilistic cellular automaton: a mean-field approach

“…In this way we can have an infinite set of coupled equations for the correlations which is equivalent to the evolution equation for the probability P ( ) (η), described in equation ( 1) for the automaton. The scope of the dynamic mean-field approximation consists in the truncation of this infinite set of coupled equations [36][37][38][39].…”

“…as expected for a Hopf bifurcation [19] and shown in figure 8. The transition line c = c 2 from the oscillating phase to the nonoscillating phase can either be obtained by using the criterion given by equation (42) or by analyzing the eigenvalues associated with the map given by the set of equations ( 33)- (37). This last criterion means to find the points of phase diagram such that the real part of the dominant complex eigenvalue equals 1.…”

“…In a very clear manner, Durrett and Levin [13] have pointed out that the modeling of population dynamics systems which are spatially distributed by interacting particle systems [13,[33][34][35] is the appropriate theoretical approach that is able to give the more complete description of the problem. We include in this approach probabilistic cellular automata (PCA) [14,[35][36][37], which will concern us here. We refer to interacting particle systems and PCA as stochastic lattice models.…”

Anisotropic probabilistic cellular automaton for a predator-prey system

“…The evolution equations for the pair correlations depend on clusters of three and four sites. Following the approach of the pair approximation [25][26][27][28] we write the clusters of three and four sites as products of pair correlations and one-site correlations. However, we must be careful in doing this approximation because we are maintaining the spatial anisotropy dependence of the one and two site correlations.…”