Avalanche Dynamics in Quenched Random Directed Sandpile Models

We numerically investigate the quenched random directed sandpile models which are local, conservative and Abelian. A local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site plays an important role when all the nearest-neighbouring sites of the above-mentioned site topple for once. The quenched model has the same critical exponents with the Abelian deterministic directed sandpile model when the local flow balance exists, otherwise the critical exponents of this quenched model and the annealed Abelian random directed sandpile model are the same. These results indicate that the presence or absence of this local flow balance determines the universality class of the Abelian directed sandpile model.     

Asymmetrically coupled directed percolation systems

We introduce a dynamical model of coupled directed percolation systems with two particle species. The two species A and B are coupled asymmetrically in that A particles branch B particles, whereas B particles prey on A particles. This model may describe epidemic spreading controlled by reactive immunization agents. We study nonequilibrium phase transitions with attention focused on the multicritical point where both species undergo the absorbing phase transition simultaneously. In one dimension, we find that the inhibitory coupling from B to A is irrelevant and the model belongs to the unidirectionally coupled directed percolation class. On the contrary, a mean-field analysis predicts that the inhibitory coupling is relevant and a new universality appears with a variable dynamic exponent. Numerical simulations on small-world networks confirm our predictions.     

Flowing Sand—A Possible Physical Realization of Directed Percolation

A simple model for flowing sand on an inclined plane is introduced. The model is related to recent experiments by Douady and Daerr and reproduces some of the experimentally observed features. Avalanches of intermediate size appear to be compact, placing the critical behavior of the model into the universality class of compact directed percolation. On very large scales, however, the avalanches break up into several branches, leading to a crossover from compact to ordinary directed percolation. Thus, systems of flowing granular matter on an inclined plane could serve as a first physical realization of directed percolation.     

Non-equilibrium critical phenomena and phase transitions into absorbing states

This review addresses recent developments in non-equilibrium 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. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic non-equilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.     

Dynamics of Boolean networks: an exact solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.     

The field theory approach to percolation processes

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous non-equilibrium active to absorbing state phase transitions whose asymptotic features are governed, respectively, by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions d_c = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder.     

Annealed disorder, rare regions, and local moments: A novel mechanism for metal-insulator transitions

It is shown that, for noninteracting electron systems, annealed magnetic disorder leads to a new mechanism, and a new universality class, for a metal-insulator transition. The transition is driven by a vanishing of the thermodynamic density susceptibility rather than by localization effects. The critical behavior in d = 2+epsilon dimensions is determined, and the underlying physics is discussed. It is further argued that annealed magnetic disorder, in addition to underlying quenched disorder, describes local magnetic moments in electronic systems.     

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling p) and deterministic critical slope processes with internal correlation time t_c equal to the avalanche lifetime, in model A, and ,in model B. In both cases nonuniversal scaling properties of avalanche distributions are found for , where is related to directed percolation threshold in d=3. Distributions of avalanche durations for are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of p. At a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at approaches the parity conserving universality class in model A, and the mean-field universality class in model B. We also estimate roughness exponent at the transition. Received: 29 May 1998 / Revised: 8 September 1998 / Accepted: 10 September 1998     

Dynamic critical properties of a one-dimensional probabilistic cellular automaton

Dynamic properties of a one-dimensional probabilistic cellular automaton are studied by Monte Carlo simulation near a critical point which marks a second-order phase transition from an active state to an effectively unique absorbing state. Values obtained for the dynamic critical exponents indicate that the transition belongs to the universality class of directed percolation. Finally the model is compared with a previously studied one to show that a difference in the nature of the absorbing states places them in different universality classes. Received: 6 February 1998 / Revised and Accepted: 17 February 1998     

Nonperturbative renormalization-group study of reaction-diffusion processes

We generalize nonperturbative renormalization group methods to nonequilibrium critical phenomena. Within this formalism, reaction-diffusion processes are described by a scale-dependent effective action, the flow of which is derived. We investigate branching and annihilating random walks with an odd number of offspring. Along with recovering their universal physics (described by the directed percolation universality class), we determine their phase diagrams and predict that a transition occurs even in three dimensions, contrarily to what perturbation theory suggests.     

Criticality of Parasitic Disease Transmission in a Diffusive Population

Through using the methods of finite-size effect and short time dynamic scaling, we study the critical behavior of parasitic disease spreading process in a diffusive population mediated by a static vector environment. Through comprehensive analysis of parasitic disease spreading we find that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. We determine the critical population density, above which the system reaches an epidemic spreading stationary state. We also perform a scaling analysis to determine the order parameter and critical relaxation exponents. The results show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.     

Strong disorder fixed point in absorbing-state phase transitions

The effect of quenched disorder on nonequilibrium phase transitions in the directed percolation universality class is studied by a strong disorder renormalization group approach and by density matrix renormalization group calculations. We show that for sufficiently strong disorder the critical behavior is controlled by a strong disorder fixed point and in one dimension the critical exponents are conjectured to be exact: beta=(3-sqrt[5])/2 and nu( perpendicular )=2. For disorder strengths outside the attractive region of this fixed point, disorder dependent critical exponents are detected. Existing numerical results in two dimensions can be interpreted within a similar scenario.     

Revisiting the nonequilibrium phase transition of the triplet-creation model

The nonequilibrium phase transition in the triplet-creation model is investigated using critical spreading and the conservative diffusive contact process. The results support the claim that at high enough diffusion the phase transition becomes discontinuous. As the diffusion probability increases the critical exponents change continuously from the ordinary directed percolation (DP) class to the compact directed percolation (CDP). The fractal dimension of the critical cluster, however, switches abruptly between those two universality classes. Strong crossover effects in both methods make it difficult, if not impossible, to establish the exact location of the tricritical point.     

New universality classes in “Percolative” dynamics

We study a site analogue of directed percolation. Random trajectories are generated and their critical behavior is studied. The critical behavior corresponds to that of simple percolation in some of the parameter space, but elsewhere the exponents reveal new universality classes. As a byproduct, we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice.     

Universality classes in the random-storage sandpile model

The avalanche statistics in a stochastic sandpile model where toppling takes place with a probability p is investigated. The limiting case p=1 corresponds to the Bak-Tang-Wiesenfeld (BTW) model with a deterministic toppling rule. Based on the moment analysis of the distribution of avalanche sizes we conclude that for 0相似文献   

Critical behavior of a random diode network

We study the percolation properties of a random diode network (RDN) which contains two kinds of directed bonds on a square lattice. This network is a special case of the random insulation-resistor-diode network. Both Monte Carlo simulations and series expansion for the percolation probability show that an estimated critical exponent, beta=0.1794+/-0.008, is different from known values for a conventional insulation-resistor-diode network. RDN belongs to neither the isotropic percolation universality class nor to the directed percolation universality, which we attribute to a difference of symmetry breakdown around the critical point.     

Minimum spanning trees on random networks

We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Because of this geometric universality, we are able to characterize the energy of MST using a scaling distribution [P(epsilon)] found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to P(epsilon). We discuss the relationship to invasion percolation, to the directed polymer in a random media, to uniform spanning trees, and also the implications for the broader issue of universality in disordered systems.     

Breakdown of universality in transitions to spatiotemporal chaos.

We show that the transition from laminar to active behavior in extended chaotic systems can vary from a continuous transition in the universality class of directed percolation with infinitely many absorbing states to what appears as a first-order transition. The latter occurs when finite lifetime nonchaotic structures, called "solitons," dominate the dynamics. We illustrate this scenario in an extension of the deterministic Chaté-Manneville coupled map lattice model and in a soliton including variant of the stochastic Domany-Kinzel cellular automaton.     

Phase diagram of optimal paths

We show that by choosing appropriate distributions of the randomness the search for optimal paths links diverse problems of disordered media, such as directed percolation, invasion percolation, and directed and nondirected spanning polymers. We also introduce a simple and efficient algorithm, which solves the d-dimensional model numerically in O(N(1+df/d)) steps, where df is the fractal dimension of the path. Using extensive simulations in two dimensions, we identify the phase boundaries of the directed polymer universality class. A new strong-disorder phase occurs where the optimum paths are self-affine with parameter-dependent scaling exponents. Furthermore, the phase diagram contains directed and nondirected percolation as well as the directed random walk models at specific points and lines.     

Are damage spreading transitions generically in the universality class of directed percolation?

We present numerical evidence for the fact that the damage spreading transition in the Domany-Kinzel automaton found by Martinset al. is in the same universality class as directed percolation. We conjecture that also other damage spreading transitions should be in this universality class, unless they coincide with other transitions (as in the Ising model with Glauber dynamics) and provided the probability for a locally damaged state to become healed is not zero.