Absorbing Phase Transition Research Articles

Time-Dependent Properties of Sandpiles

Bak, Tang, and Wiesenfeld (BTW) proposed the theory of self-organized criticality (SOC), and sandpile models, to connect “1/f” noise, observed in systems in a diverse natural setting, to the fractal spatial structure. We review some of the existing works on the problem of characterizing time-dependent properties of sandpiles and try to explore if the BTW's original ambition has really been fulfilled. We discuss the exact hydrodynamic structure in a class of conserved stochastic sandpiles, undergoing a non-equilibrium absorbing phase transition. We illustrate how the hydrodynamic framework can be used to capture long-ranged spatio-temporal correlations in terms of large-scale transport and relaxation properties of the systems. We particularly emphasize certain interesting aspects of sandpiles—the transport instabilities, which emerge through the threshold-activated nature of the dynamics in the systems. We also point out some open issues at the end.

Absorbing phase transition with a continuously varying exponent in a quantum contact process: A neural network approach

Phase transitions in dissipative quantum systems are intriguing because they are induced by the interplay between coherent quantum and incoherent classical fluctuations. Here, we investigate the crossover from a quantum to a classical absorbing phase transition arising in the quantum contact process (QCP). The Lindblad equation contains two parameters, $\omega$ and $\kappa$, which adjust the contributions of the quantum and classical effects, respectively. We find that in one dimension when the QCP starts from a homogeneous state with all active sites, there exists a critical line in the region $0 \le \kappa < \kappa_*$ along which the exponent $\alpha$ (which is associated with the density of active sites) decreases continuously from a quantum to the classical directed percolation (DP) value. This behavior suggests that the quantum coherent effect remains to some extent near $\kappa=0$. However, when the QCP in one dimension starts from a heterogeneous state with all inactive sites except for one active site, all the critical exponents have the classical DP values for $\kappa \ge 0$. In two dimensions, anomalous crossover behavior does not occur, and classical DP behavior appears in the entire region of $\kappa \ge 0$ regardless of the initial configuration. Neural network machine learning is used to identify the critical line and determine the correlation length exponent. Numerical simulations using the quantum jump Monte Carlo technique and tensor network method are performed to determine all the other critical exponents of the QCP.

Diffusive epidemic process in 3D: a two-species reaction–diffusion phase transition

By the use of Monte Carlo simulation we study the critical behavior of a three-dimensional stochastic lattice model describing a diffusive epidemic propagation process. In this model, healthy (A) and sick (B) individuals diffuse on the lattice with diffusion constants d A and d B , respectively, and undergo reactions B → A and A + B → 2B. We determine the absorbing phase transition between a steady reactive state and a vacuum state. We obtained the order parameter, order parameter fluctuations, correlation length and their critical exponents by the use of steady state and short-time dynamics simulations. We studied three different diffusion regimes: the case of species A diffusing much slower than species B (d A ≪ d B ), the case of species with equal diffusion constants (d A = d B ) and the case of species A diffusing much faster than species B (d A ≫ d B ). We found only second order transition for all three cases. We did not identify any signal of first order transition for the case d A > d B as predicted by field theory in first order approximation.

Constraint relaxation leads to jamming.

Adding transitions to an equilibrium system increases the activity. Naively, one would expect this to hold also in out-of-equilibrium systems. We demonstrate, using relatively simple models, how adding transitions to an out of equilibrium system may in fact reduce the activity and even cause it to vanish. This surprising effect is caused by adding heretofore forbidden transitions into less and less active states. We investigate six related kinetically constrained lattice gas models, some of which behave as naively expected while others exhibit this nonintuitive behavior. These models exhibit an absorbing state phase transition, which is also affected by the added transitions. We introduce a semi-mean-field approximation describing the models, which agrees qualitatively with our numerical simulation.

Absorbing-state transition in a coevolution model with node and link states in an adaptive network: network fragmentation transition at criticality

We consider a general model in which there is a coupled dynamics of node states and link states in a network. This coupled dynamics coevolves with dynamical changes of the topology of the network caused by a link rewiring mechanism. Such coevolution model features the interaction of the local dynamics of node and link states with the nonlocal dynamics of link rewiring in a random network. The coupled dynamics of the states of the nodes and the links produces by itself an absorbing phase transition which is shown to be robust against the link rewiring mechanism. However, the dynamics of the network gives rise to significant physical changes, specially in the limit in which some links do not change state but are always rewired. Firstly a network fragmentation occurs at the critical line of the absorbing-state transition, and only at this line, so that fragmentation is a manifestation of criticality. Secondly, in the active phase of the absorbing-state transition, finite-size fluctuations take the system to a single network component consensus phase, while other configurations are possible in the absence of rewiring. In addition, this phase is reached after a survival time that scales linearly with system size, while the survival time scales exponentially with system size when there is no rewiring. A social interpretation of our results contribute to the description of processes of emergence of social fragmentation and polarization.

Coupled two-species model for the pair contact process with diffusion.

The contact process with diffusion (PCPD) defined by the binary reactions B+B→B+B+B,B+B→∅ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles B and particle pairs A are dynamically coupled according to the stochastic reaction processes B+B→A,A→A+B,A→∅, and A→B+B, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, A particles rapidly go extinct, effectively leaving the B species to undergo pure diffusion-limited pair annihilation kinetics B+B→∅. At criticality, both A and B densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension d_{c}=2. In one dimension, the critical exponents for the B species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the A species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.

Critical phenomena in the presence of symmetric absorbing states: A microscopic spin model with tunable parameters.

The Langevin description of systems with two symmetric absorbing states yields a phase diagram with three different phases (disordered and active, ordered and active, absorbing) separated by critical lines belonging to three different universality classes (generalized voter, Ising, and directed percolation). In this paper we present a microscopic spin model with two symmetric absorbing states that has the property that the model parameters can be varied in a continuous way. Our results, obtained through extensive numerical simulations, indicate that all features of the Langevin description are encountered for our two-dimensional microscopic spin model. Thus the Ising and direction percolation lines merge into a generalized voter critical line at a point in parameter space that is not identical to the classical voter model. A vast range of different quantities are used to determine the universality classes of the order-disorder and absorbing phase transitions. The investigation of time-dependent quantities at a critical point belonging to the generalized voter universality class reveals a more complicated picture than previously discussed in the literature.

Phase transition behaviour in yeast and bacterial populations under stress.

Non-equilibrium phase transitions from survival to extinction have recently been observed in computational models of evolutionary dynamics. Dynamical signatures predictive of population collapse have been observed in yeast populations under stress. We experimentally investigate the population response of the budding yeast Saccharomyces cerevisiae to biological stressors (temperature and salt concentration) in order to investigate the system's behaviour in the vicinity of population collapse. While both conditions lead to population decline, the dynamical characteristics of the population response differ significantly depending on the stressor. Under temperature stress, the population undergoes a sharp change with significant fluctuations within a critical temperature range, indicative of a continuous absorbing phase transition. In the case of salt stress, the response is more gradual. A similar range of response is observed with the application of various antibiotics to Escherichia coli, with a variety of patterns of decreased growth in response to antibiotic stress both within and across antibiotic classes and mechanisms of action. These findings have implications for the identification of critical tipping points for populations under environmental stress.

Order-parameter critical exponent of absorbing phase transitions in one-dimensional systems with two symmetric absorbing states.

Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β of absorbing phase transitions in systems with two symmetric absorbing states. The value of β was conjectured to be 13/14≈0.93 and Monte Carlo simulation studies in the literature have repeatedly reproduced values consistent with the conjecture. In this paper, we systematically estimate β by analyzing the effective exponent after finding how strong corrections to scaling are. We show that the widely accepted numerical value of β is not correct. Rather, we obtain β=1.020(5) from different models with two symmetric absorbing states.

Scaling and criticality in a phenomenological renormalization group

We present a systematic study to test a recently introduced phenomenological renormalization group, proposed to coarse-grain data of neural activity from their correlation matrix. The approach allows, at least in principle, to establish whether the collective behavior of the network of spiking neurons is described by a non-Gaussian critical fixed point. We test this renormalization procedure in a variety of models focusing in particular on the contact process, which displays an absorbing phase transition at $\lambda = \lambda_c$ between a silent and an active state. We find that the results of the coarse-graining do not depend on the presence of long-range interactions, but some scaling features persist in the super-critical system up to a distance of $10\%$ from $\lambda_c$. Our results provide insights on the possible subtleties that one needs to consider when applying such phenomenological approaches directly to data to infer signatures of criticality.

Absorbing phase transitions in a non-conserving sandpile model

We introduce and study a non-conserving sandpile model, the autonomously adapting sandpile (AAS) model, for which a site topples whenever it has two or more grains, distributing three or two grains randomly on its neighboring sites, respectively with probability p and . The toppling process is independent of the actual number of grains zi of the toppling site, as long as . For a periodic lattice the model evolves into an inactive state for small p , with the number of active sites becoming stationary for larger values of p . In one and two dimensions we find that the absorbing phase transition occurs for and .The symmetry of bipartite lattices allows states in which all active sites are located alternatingly on one of the two sublattices, A and B, respectively for even and odd times. We show that the AB-sublattice symmetry is spontaneously broken for the AAS model, an observation that holds also for the Manna model. One finds that a metastable AB-symmetry conserving state is transiently observable and that it has the potential to influence the width of the scaling regime, in particular in two dimensions.The AAS model mimics the behavior of integrate-and-fire neurons which propagate activity independently of the input received, as long as the threshold is crossed. ing from regular lattices, one can identify sites with neurons and consider quenched networks of neurons connected to a fixed number G of other neurons, with G being drawn from a suitable distribution. The neuronal activity is then propagated to G other neurons. The AAS model is hence well suited for theoretical studies of nearly critical brain dynamics. We also point out that the waiting-time distribution allows an avalanche-free experimental access to criticality.

Influence of quenched disorder on discontinuous absorbing phase transition

Influence of quenched disorder on critical behavior of the deterministic conserved threshold transfer process (DCTTP) was investigated. DCTTP is a variant of the conserved threshold transfer process modified such that active-particle hopping is deterministic and has been reported to exhibit discontinuous phase transition (Lee and Lee 2009 Phys. Rev. E 80 011106) in all considered dimensions, i.e. one, two, and four dimensions. Two disorder types were considered. First, while particle hopping on ordered sites is deterministic, hopping on disordered sites is subject to be stochastic, i.e. active particles on disordered sites hop to randomly selected neighboring sites. Second, disorder is assumed to be random dilution of lattice sites, i.e. particles cannot occupy disordered sites. The order parameter was found to be continuous for disorder of stochastic hopping, i.e. the first-order transition was rounded to the second-order transition for any small amount of disorder, whereas the transition remained first order but the critical exponent associated with the power law decrease of active-site densities varied for considerable amount of lattice site dilution. These results are consistent with predictions established for equilibrium n-vector spin systems.

Critical behavior near the reversible-irreversible transition in periodically driven vortices under random local shear

When many-particle (vortex) assemblies with disordered distribution are subjected to a periodic shear with a small amplitude {boldsymbol{d}}, the particles gradually self-organize to avoid next collisions and transform into an organized configuration. We can detect it from the time-dependent voltage {boldsymbol{V}}{boldsymbol{(}}{boldsymbol{t}}{boldsymbol{)}} (average velocity) that increases towards a steady-state value. For small {boldsymbol{d}}, the particles settle into a reversible state where all the particles return to their initial position after each shear cycle, while they reach an irreversible state for {boldsymbol{d}} above a threshold {{boldsymbol{d}}}_{{boldsymbol{c}}}. Here, we investigate the general phenomenon of a reversible-irreversible transition (RIT) using periodically driven vortices in a strip-shaped amorphous film with random pinning that causes local shear, as a function of {boldsymbol{d}}. By measuring {boldsymbol{V}}{boldsymbol{(}}{boldsymbol{t}}{boldsymbol{)}}, we observe a critical behavior of RIT, not only on the irreversible side, but also on the reversible side of the transition, which is the first under random local shear. The relaxation time {boldsymbol{tau }}{boldsymbol{(}}{boldsymbol{d}}{boldsymbol{)}} to reach either the reversible or irreversible state shows a power-law divergence at {{boldsymbol{d}}}_{{boldsymbol{c}}}. The critical exponent is determined with higher accuracy and is, within errors, in agreement with the value expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. As {boldsymbol{d}} is decreased down to the intervortex spacing in the reversible regime, {boldsymbol{tau }}{boldsymbol{(}}{boldsymbol{d}}{boldsymbol{)}} deviates downward from the power-law relation, reflecting the suppression of intervortex collisions. We also suggest the possibility of a narrow smectic-flow regime, which is predicted to intervene between fully reversible and irreversible flow.

One-dimensional Absorbing phase transition in the fermionic parity-conserving particle process with second neighbors branching

By means of Monte Carlo procedure this article analyzes a stochastic process in one-dimensional model. The parity-conserving particle process with infinite annihilation and branching of the two offspring to the second neighbors, each one at each side. The model was firstly considered in Takayasu and Tretyakov (1991) and Jensen (1994) but branching of four offspring to four first neighbors. It has also been solved in Zhong and ben-Avraham (1995) with branching of the two particles to the first neighbors, each one at each side, but with finite annihilation. Now the model has been studied with infinite annihilation in presence of only second neighbors branching of two particles, one to left and the other to the right. Simulations were performed on a regular one dimension lattice in order to determine the threshold of absorbing phase transition. From steady state simulations and finite time scaling analysis, it is shown that the transition exists and critical exponents are obtained. The exponents are found to differ from those of the directed percolation (DP) universality class and are the same found in Takayasu and Tretyakov (1991), Jensen (1994) and Zhong and ben-Avraham (1995). It is also calculated the short-time relaxation dynamical critical exponents and checked the consistence of the related scaling relation.

Absorbing phase transition in a parity-conserving particle process on a Sierpinski carpet fractal

We study a stochastic lattice model with parity-conserving particle process using a Monte Carlo procedure. We perform simulations on a Sierpinski carpet fractal with dimension . We calculate the critical exponents at the threshold of the absorbing phase transition at the known value for the critical diffusion p c = 1 (Cardy and Tauber 1996 Phys. Rev. Lett. 77 4780). Using finite-size and finite-time scaling analysis we calculate the critical exponents at p c = 1 and below, where a finite density of particles is found in the long-time limit. From dynamic simulations we calculate the dynamical exponents Z, , , and , and they are found to differ from the mean-field values, as well as the stationary exponent . We check the consistence of the results with the hyperscaling relation.

Entropy Governed by the Absorbing State of Directed Percolation.

We investigate the informational aspect of (1+1)-dimensional directed percolation, a canonical model of a nonequilibrium continuous transition to a phase dominated by a single special state called the "absorbing" state. Using a tensor network scheme, we numerically calculate the time evolution of state probability distribution of directed percolation. We find a universal relaxation of Rényi entropy at the absorbing phase transition point as well as a new singularity in the active phase, slightly but distinctly away from the absorbing transition point. At the new singular point, the second-order Rényi entropy has a clear cusp. There we also detect a singular behavior of "entanglement entropy," defined by regarding the probability distribution as a wave function. The entanglement entropy vanishes below the singular point and stays finite above. We confirm that the absorbing state, though its occurrence is exponentially rare in the active phase, is responsible for these phenomena. This interpretation provides us with a unified understanding of time evolution of the Rényi entropy at the critical point as well as in the active phase.

Absorbing phase transition in the coupled dynamics of node and link states in random networks

We present a stochastic dynamics model of coupled evolution for the binary states of nodes and links in a complex network. In the context of opinion formation node states represent two possible opinions and link states represent positive or negative relationships. Dynamics proceeds via node and link state update towards pairwise satisfactory relations in which nodes in the same state are connected by positive links or nodes in different states are connected by negative links. By a mean-field rate equations analysis and Monte Carlo simulations in random networks we find an absorbing phase transition from a dynamically active phase to an absorbing phase. The transition occurs for a critical value of the relative time scale for node and link state updates. In the absorbing phase the order parameter, measuring global order, approaches exponentially the final frozen configuration. Finite-size effects are such that in the absorbing phase the final configuration is reached in a characteristic time that scales logarithmically with system size, while in the active phase, finite-size fluctuations take the system to a frozen configuration in a characteristic time that grows exponentially with system size. There is also a class of finite-size topological transition associated with group splitting in the network of these final frozen configurations.

Absorbing phase transition in the three dimensional fermionic parity-conserving particle process

By use of a Monte Carlo procedure we analyze a stochastic lattice model with parity-conserving particle process, previously considered in Takayasu and Tretyakov (1991). We perform simulations on a regular three-dimensional (3D) lattice in order to determine the threshold of absorbing phase transition. A finite-time scaling analysis is employed to calculate the critical exponents at the critical diffusion probability pc, below which a finite density of particles is developed in the long-time limit. From steady state simulations and finite-time scaling analysis we obtained these critical exponents and found them to differ from those of the 3D directed percolation (DP) universality class. We also calculated the short-time relaxation dynamical critical exponents and checked the consistence with the hyperscaling relation.

Effect of Thermal Noise on Conserved Lattice Gas

We present a modified conserved lattice gas (CLG) model in two dimensions with a thermal noise parameter, P = exp(−1/T), where T is the effective temperature. In particular, we focus on the interplay of thermal noise P and the total density of particles ρ in the CLG model. The universality class of the case of 0 <P < 1 can be compared with P = 0 and P =1 cases. We discuss the critical behavior of absorbing phase transitions in the generalized CLG (GCLG) model by two well-defined order parameters: one is the density of active particles (a pair of consecutive particles) and the other is the density of energy-loss particles. While the former is useful at P = 1 (T = ∞)asthe typical one in nonequilibrium absorbing phase transitions, the latter is suggested as the best at P = 0 (T = 0) due to the oscillatory behavior in the localized active phase. Based on extensive numerical tests, we find that the density of energy-loss particles can also be a good indicator even for finite P values. Finally, we propose a schematic phase diagram of the GCLG as P varies.

Hyperuniformity and absorbing phase transition in continuous media with a conserved field

The nonequilibrium absorbing phase transition of interacting particles in two-dimensional continuous media is investigated via Monte Carlo simulations. Starting from random and natural homogeneous distributions of equi-sized spherical particles (disks) in an off-lattice continuous medium, particles that overlap with other particles are considered to be active and isolated particles are considered inactive. Active particles tend to repel overlapping particles. In a steady state, the particle distribution is found to be hyperuniform with a larger hyperuniformity exponent than the lattice Manna model. The critical exponents that characterize the power laws of the steady-state densities and the critical active-particle densities are calculated using both the sequential and parallel updates; they are found to be consistent, within the statistical margin of error, with those of the lattice Manna model but are distinctly different from those of the directed percolation universality class. The influence of impurities is also investigated by adding quenched point-impurity particles, and it is found that the impurity has a negligible effect on the critical behavior, supporting the universality between the continuum model and lattice Manna model. The results are in disagreement with the recent observations.