Absorbing Phase Transition Research Articles

Universality classes of absorbing phase transitions in generic branching-annihilating particle systems with nearest-neighbor bias.

We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one-dimensional ring, where the walkers hop to their nearest neighbor with a bias ε. For ε=0, three universality classes-directed percolation (DP), parity-conserving (PC), and pair contact process with diffusion (PCPD)-are typically observed in such systems. We find that the introduction of ε does not change the DP universality class but alters the other two universality classes. For nonzero ε, the PCPD class crosses over to DP, and the PC class changes to a new universality class.

Hyperuniformity of generalized random organization models.

Studies of random organization models of monodisperse (i.e., identical) spherical particles have shown that a hyperuniform state is achievable when the system goes through an absorbing phase transition to a critical state. Here we investigate to what extent hyperuniformity is preserved when the model is generalized to particles with a size distribution and/or nonspherical shapes. We begin by examining binary disks in two dimensions and demonstrate that their critical states are hyperuniform as two-phase media, but not hyperuniform nor multihyperuniform as point patterns formed by the particle centroids. We further confirm the generality of our findings by studying particles with a continuous size distribution. Finally, to study the effect of rotational degrees of freedom, we extend our model to noncircular particles, namely, hard rectangles with various aspect ratios, including the hard-needle limit. Although these systems exhibit only short-range orientational order, hyperuniformity is still preserved. Our analysis reveals that the redistribution of the "mass" of the particles rather than the particle centroids is central to this dynamical process. The consideration of the "active volume fraction" of generalized random organization models may help to resolve which universality class they belong to and hence may lead to a deeper theoretical understanding of absorbing-state models. Our results suggest that general particle systems subject to random organization can be a robust way to fabricate a wide class of hyperuniform states of matter by tuning the structures via different particle-size and -shape distributions. This in turn potentially enables the creation of multifunctional hyperuniform materials with desirable optical, transport, and mechanical properties.

Reversible to irreversible transitions in periodically driven skyrmion systems

We examine skyrmions driven periodically over random quenched disorder and show that there is a transition from reversible motion to a state in which the skyrmion trajectories are chaotic or irreversible. We find that the characteristic time required for the system to organize into a steady reversible or irreversible state exhibits a power law divergence near a critical ac drive period, with the same exponent as that observed for reversible to irreversible transitions in periodically sheared colloidal systems, suggesting that the transition can be described as an absorbing phase transition in the directed percolation universality class. We compare our results to the behavior of an overdamped system and show that the Magnus term enhances the irreversible behavior by increasing the number of dynamically accessible orbits. We discuss the implications of this work for skyrmion applications involving the long time repeatable dynamics of dense skyrmion arrays.

How to experimentally probe universal features of absorbing phase transitions using steady state

We propose experimentally feasible ways to probe universal features of absorbing phase transitions from two different approaches, both based on numerical validations. On one hand, we numerically study a probability distribution of duration/length of intervals of local inactive state in quasi-steady state, which has been very commonly used in experiments, in a case of the contact process. We show that the distributions obey the universal scaling ansatz expected from the phenomenological scaling argument, but that care must be taken in order to suppress a bias caused by censoring due to a finite observation window. To demonstrate the latter point, we compare the distributions for the temporal intervals estimated through conventional histograms with those through an estimator, which properly takes account of censoring and sampling bias. On the other hand we also propose that, if a system is subject to uniform advection, as is often the case with flowing systems, a correlation length and a correlation time near the transition point can be easily quantified by supplying the system with an active boundary condition. In order to support our proposal, we introduce a new model whose advection strength can be arbitrarily controlled. The results of numerical simulations on our model suggest that a correlation time, which is difficult to measure through the interval distributions without the aforementioned bias, can be measured through characteristic decay length of an order parameter. Crossovers between two different power-law behaviors are also identified in this case, and the universal scaling ansätze associated with the crossovers are discussed.

Stationary and dynamical critical behavior of the three-dimensional Diffusive Epidemic Process

We investigate the critical behavior of a stochastic lattice model describing the hopping of two species that undergo reactions B→A and A+B→2B. We simulate the model defined on a 3D lattice and determine the threshold of the absorbing phase transition to the state at which species B is extinct. Using steady state and short-time dynamics simulations, we calculate the order parameter, order parameter fluctuations, correlation length and their critical exponents. We focus in the case of species A diffusing much faster than species B. We did not find signatures of a first-order transition that has been conjectured in the literature. We report a continuous transition with the perpendicular correlation exponent ν⊥≈0.61(6), in agreement with ν⊥=2∕D. Also, for this diffusion regime, we estimate β∕ν⊥=1.48, very close to the ϵ expansion prediction β∕ν⊥=D−ε∕82 (ε=4−D) for the regime of equally diffusing particles.

Symmetric conserved-mass aggregation model with a mass threshold

We study a symmetric conserved-mass aggregation (SCA) model with a mass threshold mc(>1). In the SCA model, each site may have any integer mass m, which is active for and inactive otherwise. For active masses, random toppling of a unit of mass takes place with a rate λ. An active mass randomly moves to the one of the nearest neighbors with a unit rate and aggregates with a mass on the target site. The SCA model undergoes absorbing phase transition (APT) depending on λ. The SCA model shares the same features as Manna class, so its critical behaviors are expected to belong to Manna class, if APTs occur. For mc = 2, it is shown that the SCA model exhibits the same kinetics of for and at without any transitions, where . For , APT is shown to occur at a critical . The critical decay is similar to that of previous studies for random IC. However, for regular IC, the critical decay exponent agrees with the known results of natural IC, which are close to the value of the directed percolation (DP) class. Off-critical scaling exponents are also shown to be consistent with previous results of natural and regular ICs which are close to DP values. The present study suggests a possibility that the interplay of the diffusion of active sites and the random toppling processes can lead to the same critical decay for both regular and natural ICs unlike the decay of models without the interplay.

Dynamic critical properties of non-equilibrium Potts models with absorbing states

We present extensive numerical simulations of a family of non-equilibrium Potts models with absorbing states that allows for a variety of scenarios, depending on the number of spin states and the range of the spin–spin interactions. These scenarios encompass a voter critical point, a discontinuous transition as well as the presence of both a symmetry-breaking phase transition and an absorbing phase transition. While we also investigate standard steady-state quantities, our emphasis is on time-dependent quantities that provide insights into the transient properties of the models.

Mesoscopic description of the equal-load-sharing fiber bundle model

One aim of the equal load sharing fiber bundle model is to describe the critical behavior of failure events. One way of accomplishing this, is through a discrete recursive dynamics. We introduce a continuous mesoscopic equation catching the critical behavior found through recursive dynamics. It allows us to link the model with the unifying framework of absorbing phase transitions traditionally used in the study of non-equilibrium phase transitions. Moreover, it highlights the analogy between equal load sharing and spinodal nucleation. Consequently, this work is a first step towards the quest of a field theory for fiber bundle models.

Crossover from Jamming to Clogging Behaviours in Heterogeneous Environments

Jamming describes a transition from a flowing or liquid state to a solid or rigid state in a loose assembly of particles such as grains or bubbles. In contrast, clogging describes the ceasing of the flow of particulate matter through a bottleneck. It is not clear how to distinguish jamming from clogging, nor is it known whether they are distinct phenomena or fundamentally the same. We examine an assembly of disks moving through a random obstacle array and identify a transition from clogging to jamming behavior as the disk density increases. The clogging transition has characteristics of an absorbing phase transition, with the disks evolving into a heterogeneous phase-separated clogged state after a critical diverging transient time. In contrast, jamming is a rapid process in which the disks form a homogeneous motionless packing, with a rigidity length scale that diverges as the jamming density is approached.

Evidence of a Critical Phase Transition in Purely Temporal Dynamics with Long-Delayed Feedback.

Experimental evidence of an absorbing phase transition, so far associated with spatiotemporal dynamics, is provided in a purely temporal optical system. A bistable semiconductor laser, with long-delayed optoelectronic feedback and multiplicative noise, shows the peculiar features of a critical phenomenon belonging to the directed percolation universality class. The numerical study of a simple, effective model provides accurate estimates of the transition critical exponents, in agreement with both theory and our experiment. This result pushes forward a hard equivalence of nontrivial stochastic, long-delayed systems with spatiotemporal ones and opens a new avenue for studying out-of-equilibrium universality classes in purely temporal dynamics.

Absorbing phase transitions in deterministic fixed-energy sandpile models.

We investigate the origin of the difference, which was noticed by Fey etal. [Phys. Rev. Lett. 104, 145703 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.145703], between the steady state density of an Abelian sandpile model (ASM) and the transition point of its corresponding deterministic fixed-energy sandpile model (DFES). Being deterministic, the configuration space of a DFES can be divided into two disjoint classes such that every configuration in one class should evolve into one of absorbing states, whereas no configurations in the other class can reach an absorbing state. Since the two classes are separated in terms of toppling dynamics, the system can be made to exhibit an absorbing phase transition (APT) at various points that depend on the initial probability distribution of the configurations. Furthermore, we show that in general the transition point also depends on whether an infinite-size limit is taken before or after the infinite-time limit. To demonstrate, we numerically study the two-dimensional DFES with Bak-Tang-Wiesenfeld toppling rule (BTW-FES). We confirm that there are indeed many thresholds. Nonetheless, the critical phenomena at various transition points are found to be universal. We furthermore discuss a microscopic absorbing phase transition, or a so-called spreading dynamics, of the BTW-FES, to find that the phase transition in this setting is related to the dynamical isotropic percolation process rather than self-organized criticality. In particular, we argue that choosing recurrent configurations of the corresponding ASM as an initial configuration does not allow for a nontrivial APT in the DFES.

Partial inertia induces additional phase transition in the majority vote model.

Explosive (i.e., discontinuous) transitions have aroused great interest by manifesting in distinct systems, such as synchronization in coupled oscillators, percolation regime, absorbing phase transitions, and more recently, the majority-vote model with inertia. In the latter, the model rules are slightly modified by the inclusion of a term depending on the local spin (an inertial term). In such a case, Chen etal. [Phys Rev. E 95, 042304 (2017)2470-004510.1103/PhysRevE.95.042304] have found that relevant inertia changes the nature of the phase transition in complex networks, from continuous to discontinuous. Here we give a further step by embedding inertia only in vertices with degree larger than a threshold value 〈k〉k^{*}, 〈k〉 being the mean system degree and k^{*} the fraction restriction. Our results, from mean-field analysis and extensive numerical simulations, reveal that an explosive transition is presented in both homogeneous and heterogeneous structures for small and intermediate k^{*}'s. Otherwise, a large restriction can sustain a discontinuous transition only in the heterogeneous case. This shares some similarities with recent results for the Kuramoto model [Phys. Rev. E 91, 022818 (2015)PLEEE81539-375510.1103/PhysRevE.91.022818]. Surprisingly, intermediate restriction and large inertia are responsible for the emergence of an extra phase, in which the system is partially synchronized and the classification of phase transition depends on the inertia and the lattice topology. In this case, the system exhibits two phase transitions.

Universality of continuous and discrete conserved Manna models in one dimension

The critical behaviour of an absorbing phase transition for the conserved continuous Manna model (CCMM) was investigated in one dimension via extensive Monte Carlo simulations, using sequential updates from the uniform initial states. The discrete conserved Manna model (DCMM) was also revisited and previous data, together with newly generated data, were reanalysed and compared with those for the CCMM. All critical exponents in the two models appear to be consistent to within statistical error, whereas the off-critical scaling function takes a different form in each model. The results differed by less than 10% from those of the directed percolation (DP) universality class. To examine the existence of the Griffith phase, additional simulations were carried out on a stripe of lattice sites, 20% of which were diluted. While the non-universal power laws were reported for the models in the DP class, there was no sign of the Griffith phase in either model. Thus, we conclude that both the DCMM and CCMM belong to a universality class that is different from the DP class. We examine the possibility of weak universality between the discrete and continuous models, whereby the critical exponents are similar, but the off-critical scaling functions are different.

Symmetry breaking by heating in a continuous opinion model.

We study the critical behavior of a continuous opinion model, driven by kinetic exchanges in a fully connected population. Opinions range in the real interval [-1,1], representing the different shades of opinions against and for an issue under debate. Individuals' opinions evolve through pairwise interactions, with couplings that are typically positive, but a fraction p of negative ones is allowed. Moreover, a social temperature parameter T controls the tendency of the individual responses toward neutrality. Depending on p and T, different collective states emerge: symmetry broken (one side wins), symmetric (tie of opposite sides), and absorbing neutral (indecision wins). We find the critical points and exponents that characterize the phase transitions between them. The symmetry breaking transition belongs to the usual Ising mean-field universality class, but the absorbing-phase transitions, with β=0.5, are out of the paradigmatic directed percolation class. Moreover, ordered phases can emerge by increasing social temperature.

Multicritical absorbing phase transition in a class of exactly solvable models.

We study diffusion of hard-core particles on a one-dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value n+1; an initial separation larger than n+1 can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls below a critical threshold ρ_{c}=1/(n+1). We find that the ϕ_{k}, the density of 0-clusters (0 representing vacancies) of size 0≤k<n, vanish at the transition point along with activity density ρ_{a}. The steady state of these models can be written in matrix product form to obtain analytically the static exponents β_{k}=n-k and ν=1=η corresponding to each ϕ_{k}. We also show from numerical simulations that, starting from a natural condition, ϕ_{k}(t)s decay as t^{-α_{k}} with α_{k}=(n-k)/2 even though other dynamic exponents ν_{t}=2=z are independent of k; this ensures the validity of scaling laws β=αν_{t} and ν_{t}=zν.

Temporal disorder does not forbid discontinuous absorbing phase transitions in low-dimensional systems.

Recent papers have shown that spatial (quenched) disorder can suppress discontinuous absorbing phase transitions. Conversely, the scenario for temporal disorder is still unknown. To shed some light in this direction, we investigate its effect in three different two-dimensional models which are known to exhibit discontinuous absorbing phase transitions. The temporal disorder is introduced by allowing the control parameter to be time dependent p→p(t), either varying as a uniform distribution with mean p[over ¯] and variance σ or as a bimodal distribution, fluctuating between a value p and a value p_{l}≪p. In contrast to spatial disorder, our numerical results strongly suggest that such uncorrelated temporal disorder does not forbid the existence of a discontinuous absorbing phase transition. We find that all cases are characterized by behaviors similar to their pure (without disorder) counterparts, including bistability around the coexistence point and common finite-size scaling behavior with the inverse of the system volume, as recently proposed [M. M. de Oliveira et al., Phys. Rev. E 92, 062126 (2015)PLEEE81539-375510.1103/PhysRevE.92.062126]. We also observe that temporal disorder does not induce temporal Griffiths phases around discontinuous phase transitions, at least not for d=2.

Comparing the influence of distinct kinds of temporal disorder in a low-dimensional absorbing transition model.

Recently it was stated that temporal disorder constitutes a relevant perturbation in absorbing phase transitions for all dimensions. However, its effect on systems other than the standard contact process (CP), its competition with other ingredients (e.g., particle diffusion), and other kinds of disorder (besides the standard types) are unknown. In order to shed some light on the above-mentioned points, we investigate a variant of the usual CP, namely, the triplet annihilation model, in which the competition between triplet annihilation and single particle diffusion leads to an unusual phase diagram behavior, with reentrant shape and endless activity for sufficiently large diffusion rates. Two kinds of time-dependent disorder have been considered. In the former, it is introduced in the creation-annihilation parameters (as commonly considered in recent studies), whereas in the latter, the diffusion rate D is allowed to be time dependent. In all cases, the disorder follows a uniform distribution with fixed mean and width σ. Two values of σ have been considered in order to exemplify the regime of "weaker" and "stronger" temporal disorder strengths. Our results show that in the former approach, the disorder suppresses the reentrant phase diagram with a critical behavior deviating from the directed percolation (DP) universality class in the regime of low diffusion rates, while they strongly suggest that the DP class is recovered for larger hopping rates. An opposite scenario is found in the latter disorder approach, with a substantial increase of reentrant shape and the maximum diffusion, in which the reentrant shape also displays a critical behavior consistent with the DP universality class (in similarity with the pure model). In order to compare with very recent claims, the results from taking a bimodal distribution and critical behavior in the limit of strong disorder are presented. Also, the results derived from the mean-field theory are performed, presenting partial agreement with numerical results. Lastly, a comparison with the diffusive disordered CP is undertaken.

Critical behavior for random initial conditions in the one-dimensional fixed-energy Manna sandpile model.

A fixed-energy Manna sandpile model undergoes an absorbing phase transition at a critical ρ_{c}, where an order parameter ϕ(t) decays as t^{-α} in time t. As the prototype of the Manna class, the model has been extensively studied in one dimension. However, the previous estimates of ρ_{c} and some critical exponents are different, depending on the types of initial conditions; random, natural, and regular conditions. The estimates of ρ_{c} for the random and the regular conditions are the lower and the upper bound among currently known estimates, respectively. In this work, for the random conditions, ρ_{c} and α are measured by taking into account finite-size (FS) effects. At the previous estimate of ρ_{c}, simulation results show that the temporal decay of ϕ(t) is strongly affected by the FS effects up to much larger system size (∼10^{6}). For the sizes for which ϕ(t) is independent up to t=2×10^{7}, we estimate ρ_{c}=0.8925(1) and α=0.110(5), which clearly differ from the previous results for the random conditions, ρ_{c}=0.89199(5) and α=0.141(24). Instead, the present ρ_{c} agrees with ρ_{c}=0.89255(2) of the regular conditions. In addition, the present α is substantially distinguishable from the results of the other types of initial conditions, α=0.159(3) and 0.146(2) for the natural and the regular conditions, respectively, which supports the claim of the initial condition dependence of dynamical exponents in the Manna class.

Noise-induced absorbing phase transition in a model of opinion formation

In this work we study a 3-state (+1, −1, 0) opinion model in the presence of noise and disorder. We consider pairwise competitive interactions, with a fraction p of those interactions being negative (disorder). Moreover, there is a noise q that represents the probability of an individual spontaneously change his opinion to the neutral state. Our aim is to study how the increase/decrease of the fraction of neutral agents affects the critical behavior of the system and the evolution of opinions. We derive analytical expressions for the order parameter of the model, as well as for the stationary fraction of each opinion, and we show that there are distinct phase transitions. One is the usual ferro–paramagnetic transition, that is in the Ising universality class. In addition, there are para-absorbing and ferro-absorbing transitions, presenting the directed percolation universality class. Our results are complemented by numerical simulations.

Influence of competition in minimal systems with discontinuous absorbing phase transitions

Contact processes (CP’s) with particle creation requiring a minimal neighborhood (restrictive or threshold CP’s) present a novel sort of discontinuous absorbing transitions, that revealed itself robust under the inclusion of different ingredients, such as distinct lattice topologies, particle annihilations and diffusion. Here, we tackle on the influence of competition between restrictive and standard dynamics (that describes the usual CP and a continuous DP transition is presented). Systems have been studied via mean-field theory (MFT) and numerical simulations. Results show partial contrast between MFT and numerical results. While the former predicts that considerable competition rates are required to shift the phase transition, the latter reveals the change occurs for rather limited (small) fractions. Thus, unlike previous ingredients (such as diffusion and others), limited competitive rates suppress the phase coexistence.