In this paper we first give a brief overview of Monte Carlo simulation results obtained by applying the Damage Spreading method. We analyse the transition between a state where the damage becomes healed (the frozen phase) and a regime where the damage spreads arriving at a finite (stationary) value (the damaged phase), when a control parameter is finely tuned. These kinds of transitions are actually true irreversible phase transitions themselves, and the issue of their universality class is also discussed.
Subsequently, the attention is focused on the propagation of damage in magnetic systems placed in confined geometries, such as the Abraham's Model, the standard Ising magnet (Glauber dynamics) and the corner geometry. The influence of interfaces between magnetic domains of different orientation on the spreading of the perturbation is also discussed, showing that the presence of interfaces enhances the propagation of the damage. Furthermore, the critical transition between propagation and nonpropagation of the damage is discussed. In all cases, the determined critical exponents suggest that the DS transition does not belong to the universality class of Directed Percolation, unlike many other systems exhibiting irreversible phase transitions. This result reflects the dramatic influence of interfaces on the propagation of perturbations in magnetic systems.
We present a general definition of damage spreading in a pair of models. Using this general framework, one can define damage spreading in an objective manner, that does not depend on the particular dynamic procedure that is being used. The formalism is applied to the Domany-Kinzel cellular automaton in one dimension; the active phase of this model is shown to consist of three sub-phases, characterized by different damage-spreading properties.
Damage spreading for 2D Ising cluster dynamics is investigated numerically by using random numbers in a way that conforms with the notion of submitting the two evolving replicas to the same thermal noise. Two damage spreading transitions are found; damage does not spread either at low or high temperatures. We determine the critical exponents at the high-temperature transition point, which seem consistent with directed percolation.
We investigate the spreading of damage in the three-dimensional Ising model by means of large-scale Monte-Carlo simulations. Within the Glauber dynamics we use different rules for the order in which the sites are updated. We find that the stationary damage values and the spreading temperature are different for different update order. In particular, random update order leads to larger damage and a lower spreading temperature than regular order. Consequently, damage spreading in the Ising model is non-universal not only with respect to different update algorithms (e.g. Glauber vs. heat-bath dyna…
An Ising model for damage spreading with a probability of damage healing ($q = 1 - p$) is proposed and studied by means of Monte Carlo simulations. In the limit $p \to 1$ the new model is mapped to the standard Ising model. It is found that, for temperatures above the Onsager critical temperature ($T_{C}$), there exist a no trivial finite value of $p$ that sets the critical point ($p_{c}$) for the onset of damage spreading. It is found that $p_{c}$ depends on $T$, defining a critical curve at the border between damage spreading and damage healing. Transitions along such curve are found to belo…
We investigate how a quenched random field influences the damage spreading transition in kinetic Ising models. To this end we generalize a recent master equation approach and derive an effective field theory for damage spreading in random field systems. This theory is applied to the Glauber Ising model with a bimodal random field distribution. We find that the random field influences the spreading transition by two different mechanisms with opposite effects. First, the random field favors the same particular direction of the spin variable at each site in both systems which reduces the damage. …
We present two new results regarding damage spreading in ferromagnetic Ising models. First, we show that a damage spreading transition can occur in an Ising chain that evolves in contact with a thermal reservoir. Damage heals at low temperature and spreads for high T. The dynamic rules for the system's evolution for which such a transition is observed are as legitimate as the conventional rules (Glauber, Metropolis, heat bath). Our second result is that such transitions are not always in the directed percolation universality class.
Using the concept of the Boolean derivative we study damage spreading for one dimensional elementary cellular automata and define their maximal Lyapunov exponent. A random matrix approximation describes quite well the behavior of ``chaotic'' rules and predicts a directed percolation-type phase transition. After the introduction of a small noise elementary cellular automata reveal the same type of transition.
We study the behavior under perturbations of different versions of Bak-Sneppen (BS) model in 1 + 1 dimension. We focus our attention on the damage-spreading features of the BS model with both random as well as deterministic updating, with sequential as well as parallel updating. In addition, we compute analytically the asymptotic plateau reached by the distance after the growing phase.
We study the spreading of damage 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 into de Domany-Kinzel 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 and the one where it does not.
We study the damage spreading phenomena in two different ways in a opinion dynamics model introduced recently. This kinetic exchange type model is characterized by a fraction $q$ of negative interactions and shows the presence of an order-disorder transition at $q_c$. In the traditional method, two replicas of the population are considered in which the opinion of all the agents are identical initially except for a single agent. The systems are then allowed to evolve identically. In the other method, the initial opinions are identical for all agents but the two replicas are evolved independentl…
We investigate two-dimensional Ising systems with multisping interactions of three- (m=3) and four-body terms (m=4). The application of a new type of finite-size algorithm of de Oliveira allow us to clearly distinguish a first-order transition (in the m=4 case) from a continuous one (in the m=3 one). We also study the damage spreading in these systems. In this study, a dynamical phenomenon is observed to occur at a critical point separating a chaotic phase from a frozen one. However, the width of the interval where this transition happens does not yield a conclusive evidence about the order of…
The Fortuin-Kasteleyn and heat-bath damage spreading temperatures T_{FK}(p) and T_{ds}(p) are studied on random bond Ising models of dimension two to five and as functions of the ferromagnetic interaction probability p; the conjecture that T_{ds}(p) ~ T_{FK}(p) is tested. It follows from a statement by Nishimori that in any such system exact coordinates can be given for the intersection point between the Fortuin-Kasteleyn T_{FK}(p) transition line and the Nishimori line, [p_{NL,FK},T_{NL,FK}]. There are no finite size corrections for this intersection point. In dimension three, at the intersec…
We implement the damage spreading technique on 2-dimensional isotropic and anisotropic Bak-Sneppen models. Our extensive numerical simulations show that there exists a power-law sensitivity to the initial conditions at the statistically stationary state (self-organized critical state). Corresponding growth exponent $α$ for the Hamming distance and the dynamical exponent $z$ are calculated. These values allow us to observe a clear data collapse of the finite size scaling for both versions of the Bak-Sneppen model. Moreover, it is shown that the growth exponent of the distance in the isotropic a…
We investigate how the time evolution of different kinetic Ising models depends on the initial conditions of the dynamics. To this end we consider the simultaneous evolution of two identical systems subjected to the same thermal noise. We derive a master equation for the time evolution of a joint probability distribution of the two systems. This equation is then solved within an effective-field approach. By analyzing the fixed points of the master equation and their stability we identify regular and chaotic phases.
We studied damage spreading in a Driven Lattice Gas (DLG) model as a function of the temperature $T$, the magnitude of the external driving field $E$, and the lattice size. The DLG model undergoes an order-disorder second-order phase transition at the critical temperature $T_c(E)$, such that the ordered phase is characterized by high-density strips running along the direction of the applied field; while in the disordered phase one has a lattice-gas-like behaviour. It is found that the damage always spreads for all the investigated temperatures and reaches a saturation value $D_{sat}$ that depe…
We study the transient behavior of damage propagation in the two-dimensional spin-$1$ Blume-Capel model using Monte Carlo simulations with Metropolis dynamics. We find that, for a particular region in the second-order transition regime of the crystal field--temperature phase diagram of the model, the average Hamming distance decreases exponentially with time in the weakly damaged system. Additionally, its rate of decay appears to depend linearly on a number of Hamiltonian parameters, namely the crystal field, temperature, applied magnetic field, but also on the amount of damage. Finally, a com…
We have studied the damage spreading (defined in the text) in the 'sandpile' model of self organised criticality. We have studied the variations of the critical time (defined in the text) and the total number of sites damaged at critical time as a function of system size. Both shows the power law variation.
Exact results are derived relating quantities computable by the so-called damage spreading method and thermodynamic functions for the N-color Ashkin-Teller model. The results are valid for any ergodic dynamics. Since we restrict our analysis to the ferromagnetic case the results are also valid for any translational invariant lattice. The derived relations should be used in order to determine numerically the N-color Ashkin-Teller critical exponents with better accuracy and less computational efforts than standard Monte Carlo simulations.
We investigate the sensitivity of the time evolution of a kinetic Ising model with Glauber dynamics against the initial conditions. To do so we apply the "damage spreading" method, i.e., we study the simultaneous evolution of two identical systems subjected to the same thermal noise. We derive a master equation for the joint probability distribution of the two systems. We then solve this master equation within an effective-field approximation which goes beyond the usual mean-field approximation by retaining the fluctuations though in a quite simplistic manner. The resulting effective-field the…
The propagation of damage on the square Ising lattice with a corner geometry is studied by means of Monte Carlo simulations. It is found that, just at $T=T_f (h)$ (critical temperature of the filling transition) the damage initially propagates along the interface of the competing domains, according to a power law given by $D(t) \propto t^η$. The value obtained for the dynamic exponent ($η^{*} = 0.89(1)$) is in agreement with that corresponding to the wetting transition in the slit geometry (Abraham Model) given by $η^{WT} = 0.91(1)$. However, for later times the propagation crosses to a new re…
The technique of damage spreading is used to study the phase diagram of the easy axis anisotropic Heisenberg antiferromagnet on two geometrically frustrated lattices. The triangular and kagome systems are built up from triangular units that either share edges or corners respectively. The triangular lattice undergoes two sequential Kosterlitz-Thouless transitions while the kagome lattice undergoes a glassy transition. In both cases, the phase boundaries obtained using damage spreading are in good agreement with those obtained from equilibrium Monte Carlo simulations.
The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at $T= \infty$ and magnetization $M=0$, an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization $M_0$ in one of the configurations upon quenching the system at $T_C$, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the dama…
We study damage-spreading in the ferromagnetic Ising model on small world networks using Monte Carlo simulation with Glauber dynamics. The damage spreading temperature $T_d$ is determined as a function of rewiring probability $p$ for small world networks obtained by rewiring the 2D square and 3D cubic lattices. We find that the damage for different values of $p$ collapse onto master curves when plotted against a rescaled temperature and that the distance between $T_d$ and the critical temperature $T_c$ increases with $p$. We argue that when using the Ising model to study social systems, it is …
We examine the problem of damage spreading in the off-equilibrium mode coupling equations. The study is done for the spherical $p$-spin model introduced by Crisanti, Horner and Sommers. For $p>2$ we show the existence of a temperature transition $T_0$ well above any relevant thermodynamic transition temperature. Above $T_0$ the asymptotic damage decays to zero while below $T_0$ it decays to a finite value independent of the initial damage. This transition is stable in the presence of asymmetry in the interactions. We discuss the physical origin of this peculiar phase transition which occurs as…
We study damage spreading among the opinions of a system of agents, subjected to the dynamics of the Krause-Hegselmann consensus model. The damage consists in a sharp change of the opinion of one or more agents in the initial random opinion configuration, supposedly due to some external factors and/or events. This may help to understand for instance under which conditions special shocking events or targeted propaganda are able to influence the results of elections. For agents lying on the nodes of a Barabasi-Albert network, there is a damage spreading transition at a low value epsilon_d of the…
In this paper, we relate the coupling of Markov chains, at the basis of perfect sampling methods, with damage spreading, which captures the chaotic nature of stochastic dynamics. For two-dimensional spin glasses and hard spheres we point out that the obstacle to the application of perfect-sampling schemes is posed by damage spreading rather than by the survey problem of the entire configuration space. We find dynamical damage-spreading transitions deeply inside the paramagnetic and liquid phases, and show that critical values of the transition temperatures and densities depend on the coupling …
The propagation of damage in a confined magnetic Ising film, with short range competing magnetic fields ($h$) acting at opposite walls, is studied by means of Monte Carlo simulations. Due to the presence of the fields, the film undergoes a wetting transition at a well defined critical temperature $T_w(h)$. In fact, the competing fields causes the occurrence of an interface between magnetic domains of different orientation. For $T < T_w(h)$ ($T > T_w(h)$) such interface is bounded (unbounded) to the walls, while right at $T_w(h)$ the interface is essentially located at the center of the film. I…
We investigate the spreading of damage in Ising models with Kawasaki spin-exchange dynamics which conserves the magnetization. We first modify a recent master equation approach to account for dynamic rules involving more than a single site. We then derive an effective-field theory for damage spreading in Ising models with Kawasaki spin-exchange dynamics and solve it for a two-dimensional model on a honeycomb lattice. In contrast to the cases of Glauber or heat-bath dynamics, we find that the damage always spreads and never heals. In the long-time limit the average Hamming distance approaches t…
The damage spreading (DS) transitions of two one-dimensional stochastic cellular automata suggested by Grassberger (A and B) and the kinetic Ising model of Menyhárd (NEKIM) have been investigated on the level of kinks and spins. On the level of spins the parity conservation is not satisfied and therefore studying these models provides a convenient tool to understand the dependence of DS properties on symmetries. For the model B the critical point and the DS transition point is well separated and directed percolation damage spreading transition universality was found for spin damage as well as …
We investigate the damage spreading effect in the Fortuin-Kasteleyn random cluster model for 2- and 3-dimensional grids with periodic boundary. For 2D the damage function has a global maximum at $p=\sqrt{q}/(1+\sqrt{q})$ for all $q>0$ and also local maxima at $p=1/2$ and $p=q/(1+q)$ for $q\lesssim 0.75$. For 3D we observe a local maximum at $p=q/(1+q)$ for $q\lesssim 0.46$ and a global maximum at $p=1/2$ for $q\lesssim 4.5$. The chaotic phase of the model's $(p,q)$-parameter space is where the coupling time is of exponential order and we locate points on its boundary. For 3-dimensional grids t…
The short-time and long-time dynamics of the Bak-Sneppen model of biological evolution are investigated using the damage spreading technique. By defining a proper Hamming distance measure, we are able to make it exhibits an initial power-law growth which, for finite size systems, is followed by a decay towards equilibrium. In this sense, the dynamics of self-organized critical states is shown to be similar to the one observed at the usual critical point of continuous phase-transitions and at the onset of chaos of non-linear low-dimensional dynamical maps. The transient, pre-asymptotic and asym…
We study the dynamical behavior of disordered many-particle systems with long-range Coulomb interactions by means of damage-spreading simulations. In this type of Monte-Carlo simulations one investigates the time evolution of the damage, i.e. the difference of the occupation numbers of two systems, subjected to the same thermal noise. We analyze the dependence of the damage on temperature and disorder strength. For zero disorder the spreading transition coincides with the equilibrium phase transition, whereas for finite disorder, we find evidence for a dynamical phase transition well below the…
The damage spreading transition: a hierarchy of renormalization group fixed points
Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院及巴黎文理研究大学，索邦大学，巴黎西岱大学）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（理论科学国际中心，塔塔基础研究所）
We investigate numerically and theoretically the precursory intermittent activity characterizing the preliminary phase of damage accumulation prior to failure of quasi-brittle solids. We use a minimal but thermodynamically consistent model of damage growth and localization developed by Berthier et al. (2017). The approach accounts for both microstructural disorder and non-local interactions and permits inferring a complete scaling description of the spatio-temporal structure of failure precursors. By developing a theoretical model of damage growth in disordered elasto-damageable specimen, we d…
We study a (1+1) dimensional probabilistic cellular automaton that is closely related to the Domany-Kinzel (DKCA), but in which the update of a given site depends on the state of {\it three} sites at the previous time step. Thus, compared with the DKCA, there is an additional parameter, $p_3$, representing the probability for a site to be active at time $t$, given that its nearest neighbors and itself were active at time $t-1$. We study phase transitions and critical behavior for the activity {\it and} for damage spreading, using one- and two-site mean-field approximations, and simulations, fo…
Using information of the ground state topology and the damage spreading technique we show a Griffiths-like phase is present in the three-dimensional $\pm J$ Edwards-Anderson spin glass model. With spin-flipping dynamics damage spreads for temperatures larger than $T_{\mathrm{g}}$, the glass transition temperature. With spin-orienting dynamics and for temperatures in $T_{\mathrm{g}} < T < T_{\mathrm{d}}$, damage spreads over a finite region of the system, composed of finite clusters of ferromagnetic character. $T_{\mathrm{d}}$ is the spin-orienting damage critical temperature, which is of the s…
Percolation theory characterizing the robustness of a network has applications ranging from biology, to epidemic spreading, and complex infrastructures. Percolation theory, however, only concern the typical response of a infinite network to random damage of its nodes while in real finite networks, fluctuations are observable. Consequently for finite networks there is an urgent need to evaluate the risk of collapse in response to rare configurations of the initial damage. Here we build a large deviation theory of percolation characterizing the response of a sparse network to rare events. This g…
Local measurements in quantum systems are projective operations which act to counteract the spread of quantum entanglement. Recent work has shown that local, random measurements applied to a generic volume-law entanglement generating many-body system are able to force a transition into an area-law phase. This work shows that projective operations can also force a similar classical phase transition; we show that local projections in a chaotic system can freeze information dynamics. In rough analogy with measurement-induced phase transitions, this is characterized by an absence of information sp…
We have performed computer simulations of Kauffman's automata on several graphs such as the regular square lattice and invasion percolation clusters in order to investigate phase transitions, radial distributions of the mean total damage (dynamical exponent $z$) and propagation speeds of the damage when one adds a damaging agent, nicknamed "strange man". Despite the increase in the damaging efficiency, we have not observed any appreciable change at the transition threshold to chaos neither for the short-range nor for the small-world case on the square lattices when the strange man is added in …
Random Threshold Networks with sparse, asymmetric connections show complex dynamical behavior similar to Random Boolean Networks, with a transition from ordered to chaotic dynamics at a critical average connectivity $K_c$. In this type of model - contrary to Boolean Networks - propagation of local perturbations (damage) depends on the in-degree of the sites. $K_c$ is determined analytically, using an annealed approximation, and the results are confirmed by numerical simulations. It is shown that the statistical distributions of damage spreading near the percolation transition obey power-laws, …
This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed.…
In this paper we perform numerical simulations to study Kauffman cellular automata (KCA) on quasiperiod lattices. In particular, we investigate phase transition, magnetic entropy and propagation speed of the damage on these lattices. Both the critical threshold parameter $p_{c}$ and the critical exponents are estimated with good precision. In order to investigate the increase of statistical fluctuations and the onset of chaos in the critical region of the model, we have also defined a magnetic entropy to these systems. It is seen that the magnetic entropy behaves in a different way when one pa…
The synchronization transition between two coupled replicas of spatio-temporal chaotic systems in 2+1 dimensions is studied as a phase transition into an absorbing state - the synchronized state. Confirming the scenario drawn in 1+1 dimensional systems, the transition is found to belong to two different universality classes - Multiplicative Noise (MN) and Directed Percolation (DP) - depending on the linear or nonlinear character of damage spreading occurring in the coupled systems. By comparing coupled map lattice with two different stochastic models, accurate numerical estimates for MN in 2+1…
New results are reported of the computer simulations on the magnetic behaviour of magnetic arrays of nanoscopic dots, placed in cells of the square lattice. We show that the remanence magnetization $M_r$ decreases with the array size. For arrays 50x50, we investigate also the stability of the magnetic structure of an array in an oscillating magnetic field. The damage spreading technique reveals that this stability increases with the standard deviation $σ$ of the switching field of individual elements of the array. On the other hand, $M_r$ decreases with $σ$. An optimalization of the system (la…
We study the phase diagram and the critical behavior of a one-dimensional radius-1 two-state totalistic probabilistic cellular automaton having two absorbing states. This system exhibits a first-order phase transition between the fully occupied state and the empty state, two second-order phase transitions between a partially occupied state and either the fully occupied state or the empty state, and a second-order damage-spreading phase transition. It is found that all the second-order phase transitions have the same critical behavior as the directed percolation model. The mean-field approximat…
Emergence of cooperation in self-centered individuals has been a major puzzle in the study of evolutionary ethics. Reciprocal altruism is one of explanations put forward and prisoner's dilemma has been a paradigm in this context. Emergence of cooperation was demonstrated for prisoner's dilemma on a lattice with synchronous update [Nature, 359, 826 (1992)]. However, the cooperation disappeared for asynchronous update and the general validity of the conclusions was questioned [PNAS, 90, 7716 (1993)]. Neither synchronous nor asynchronous updates are realistic for natural systems. In this paper, w…
