Universality class of nonequilibrium phase transitions with infinitely many absorbing states

Non-equilibrium dynamic hyperuniform states

Disordered hyperuniform structures are an exotic state of matter having suppressed density fluctuations at large length-scale similar to perfect crystals and quasicrystals but without any long range orientational order. In the past decade, an increasing number of non-equilibrium systems were found to have dynamic hyperuniform states, which have emerged as a new research direction coupling both non-equilibrium physics and hyperuniformity. Here we review the recent progress in understanding dynamic hyperuniform states found in various non-equilibrium systems, including the critical hyperuniformity in absorbing phase transitions, non-equilibrium hyperuniform fluids and the hyperuniform structures in phase separating systems via spinodal decomposition.

Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Hyperuniformity emerges generically in the coarsening regime of phase-separating fluids. Numerical studies of active and passive systems have shown that the structure factorS(q) behaves asqςforq → 0, with hyperuniformity exponentς = 4. For passive systems, this result was explained in 1991 by a qualitative scaling analysis of Tomita, exploiting isotropy at scales much larger than the coarsening length. Here we reconsider and extend Tomita's argument to address cases of active phase separation and of non-constant mobility, again findingς = 4. We further show that dynamical noise of varianceDcreates a transientς = 2 regime forq^≪q^∗∼Dt[1-(d+2)ν]/2, crossing over toς = 4 at largerq^. Here,νis the coarsening exponent for the domain sizeℓ, such thatℓ(t)∼tν, andq^∝qℓis the rescaled wavenumber. In diffusive coarseningν=1/3, so the rescaled crossover wavevectorq^∗vanishes at large times whend⩾2. The slowness of this decay suggests a natural explanation for experiments that observe a long-livedς = 2 scaling in phase-separatingactivefluids (where noise is typically large). Conversely, ind = 1, we demonstrate that with noise theς = 2 regime survives ast→∞, withq^∗∼D5/6. (The structure factor is not then determined by the zero-temperature fixed point.) We confirm our analytical predictions by numerical simulations of continuum theories for active and passive phase separation in the deterministic case and of Model B for the stochastic case. We also compare them with related findings for a system near an absorbing-state transition rather than undergoing phase separation. A central role is played throughout by the presence or absence of a conservation law for the centre of mass positionRof the order parameter field.

Yielding Is an Absorbing Phase Transition with Vanishing Critical Fluctuations

The yielding transition in athermal complex fluids can be interpreted as an absorbing phase transition between an elastic, absorbing state with high mesoscopic degeneracy and a flowing, active state. We characterize quantitatively this phase transition in an elastoplastic model under fixed applied shear stress, using a finite-size scaling analysis. We find vanishing critical fluctuations of the order parameter (i.e., the shear rate), and relate this property to the convex character of the phase transition (β>1). We locate yielding within a family of models akin to fixed-energy sandpile (FES) models, only with long-range redistribution kernels with zero modes that result from mechanical equilibrium. For redistribution kernels with sufficiently fast decay, this family of models belongs to a short-range universality class distinct from the conserved directed percolation class of usual FES, which is induced by zero modes.

Absorbing phase transitions in systems with mediated interactions

Experiments of periodically sheared colloidal suspensions or soft amorphous solids display a transition from reversible to irreversible particle motion that, when analyzed stroboscopically in time, is interpreted as an absorbing phase transition with infinitely many absorbing states. In these systems, interactions mediated by hydrodynamics or elasticity are present, causing passive regions to be affected by nearby active ones. We show that mediated interactions induce a universality class of absorbing phase transitions distinct from conserved directed percolation, and we obtain the corresponding critical exponents. We do so with large-scale numerical simulations of a minimal model for the stroboscopic dynamics of sheared soft materials and we derive the minimal field theoretical description.

Nonequilibrium phase coexistence and criticality near the second explosion limit of hydrogen combustion

While hydrogen is a promising source of clean energy, the safety and optimization of hydrogen technologies rely on controlling ignition through explosion limits: pressure-temperature boundaries separating explosive behavior from comparatively slow burning. Here, we show that the emergent nonequilibrium chemistry of combustible mixtures can exhibit the quantitative features of a phase transition. With stochastic simulations of the chemical kinetics for a model mechanism of hydrogen combustion, we show that the boundaries marking explosive domains of kinetic behavior are nonequilibrium critical points. Near the pressure of the second explosion limit, these critical points terminate the transient coexistence of dynamical phases-one that autoignites and another that progresses slowly. Below the critical point temperature, the chemistry of these phases is indistinguishable. In the large system limit, the pseudo-critical temperature converges to the temperature of the second explosion limit derived from mass-action kinetics.

Directed percolation with a conserved field and the depinning transition

Conserved directed percolation (C-DP) and the depinning transition of a disordered elastic interface belong to the same universality class, as has been proven very recently by Le Doussal and Wiese [Phys. Rev. Lett. 114, 110601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.110601] through a mapping of the field theory for C-DP onto that of the quenched Edwards-Wilkinson model. Here, we present an alternative derivation of the C-DP field theoretic functional, starting with the coherent-state path integral formulation of the C-DP and then applying the Grassberger transformation, which avoids the disadvantages of the so-called Doi shift. We revisit the aforementioned mapping with focus on a specific term in the field theoretic functional that has been problematic in the past when it came to assessing its relevance. We show that this term is redundant in the sense of the renormalization group.

Exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media

We show that the stochastic field theory for directed percolation in the presence of an additional conservation law [the conserved directed-percolation (C-DP) class] can be mapped exactly to the continuum theory for the depinning of an elastic interface in short-range correlated quenched disorder. Along one line of the parameters commonly studied, this mapping leads to the simplest overdamped dynamics. Away from this line, an additional memory term arises in the interface dynamics; we argue that this does not change the universality class. Since C-DP is believed to describe the Manna class of self-organized criticality, this shows that Manna stochastic sandpiles and disordered elastic interfaces (i.e., the quenched Edwards-Wilkinson model) share the same universal large-scale behavior.

Tuning spreading and avalanche-size exponents in directed percolation with modified activation probabilities

We consider the directed percolation process as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. The model is in a critical state when the activation probability is adjusted at some precise value p(c). Criticality is lost as soon as the probability to activate sites at the first attempt, p(1), is changed. We show here that criticality can be restored by "compensating" the change in p(1) by an appropriate change of the second time activation probability p(2) in the opposite direction. At compensation, we observe that the bulk exponents of the process coincide with those of the normal directed percolation process. However, the spreading exponents are changed and take values that depend continuously on the pair (p(1),p(2)). We interpret this situation by acknowledging that the model with modified initial probabilities has an infinite number of absorbing states.

Coexistence and critical behavior in a lattice model of competing species

In the present paper we study a lattice model of two species competing for the same resources. Monte Carlo simulations for d = 1,2, and 3 show that when resources are easily available both species coexist. However, when the supply of resources is on an intermediate level, the species with slower metabolism becomes extinct. On the other hand, when resources are scarce it is the species with faster metabolism that becomes extinct. The range of coexistence of the two species increases with dimension. We suggest that our model might describe some aspects of the competition between normal and tumor cells. With such an interpretation, examples of tumor remission, recurrence, and different morphologies are presented. In the d = 1 and d = 2 models, we analyze the nature of phase transitions: they are either discontinuous or belong to the directed-percolation universality class, and in some cases they have an active subcritical phase. In the d = 2 case, one of the transitions seems to be characterized by critical exponents that differ from directed-percolation ones, but this transition could be also weakly discontinuous. In the d = 3 version, Monte Carlo simulations are in a good agreement with the solution of the mean-field approximation. This approximation predicts that oscillatory behavior occurs in the present model but only for d ≳ 2. For d ≥ 2, a steady state depends on the initial configuration in some cases.

Cusps, self-organization, and absorbing states

Elastic interfaces embedded in (quenched) random media exhibit metastability and stick-slip dynamics. These nontrivial dynamical features have been shown to be associated with cusp singularities of the coarse-grained disorder correlator. Here we show that annealed systems with many absorbing states and a conservation law but no quenched disorder exhibit identical cusps. On the other hand, similar nonconserved systems in the directed percolation class are also shown to exhibit cusps but of a different type. These results are obtained both by a recent method to explicitly measure disorder correlators and by defining an alternative new protocol inspired by self-organized criticality, which opens the door to easily accessible experimental realizations.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Nontrivial critical crossover between directed percolation models: effect of infinitely many absorbing states

At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.

Absorbing states and elastic interfaces in random media: two equivalent descriptions of self-organized criticality

We elucidate a long-standing puzzle about the nonequilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved nondiffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.

Sticky grains do not change the universality class of isotropic sandpiles

We revisit the sandpile model with "sticky" grains introduced by Mohanty and Dhar [Phys. Rev. Lett. 89, 104303 (2002)] whose scaling properties were claimed to be generically in the universality class of directed percolation for both isotropic and directed models. While for directed models this conclusion is unquestionable, for isotropic models we present strong evidence that the asymptotic scaling in the self-organized regime (in which a stationary critical state exists in the limit of slow driving and vanishing dissipation rate) is, like other stochastic sandpiles, generically in the Manna universality class. This conclusion is drawn from extensive Monte Carlo simulations, and is strengthened by the analysis of the Langevin equations (proposed by the same authors to account for this problem), argued to converge upon coarse-graining to the well-established set of Langevin equations for the Manna class.

Classification of phase transitions in reaction-diffusion models

Equilibrium phase transitions are associated with rearrangements of minima of a (Lagrangian) potential. Treatment of nonequilibrium systems requires doubling of degrees of freedom, which may be often interpreted as a transition from the "coordinate"- to the "phase"-space representation. As a result, one has to deal with the Hamiltonian formulation of the field theory instead of the Lagrangian one. We suggest a classification scheme of phase transitions in reaction-diffusion models based on the topology of the phase portraits of corresponding Hamiltonians. In models with an absorbing state such a topology is fully determined by intersecting curves of zero "energy." We identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations.

Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions

Efficient and accurate integration of stochastic (partial) differential equations with multiplicative noise can be obtained through a split-step scheme, which separates the integration of the deterministic part from that of the stochastic part, the latter being performed by sampling exactly the solution of the associated Fokker-Planck equation. We demonstrate the computational power of this method by applying it to the most absorbing phase transitions for which Langevin equations have been proposed. This provides precise estimates of the associated scaling exponents, clarifying the classification of these nonequilibrium problems, and confirms or refutes some existing theories.

Numerical study of the Langevin theory for fixed-energy sandpiles

The recently proposed Langevin equation, aimed to capture the relevant critical features of stochastic sandpiles and other self-organizing systems, is studied numerically. The equation is similar to the Reggeon field theory, describing generic systems with absorbing states, but it is coupled linearly to a second conserved and static (nondiffusive) field. It has been claimed to represent a different universality class, including different discrete models: the Manna as well as other sandpiles, reaction-diffusion systems, etc. In order to integrate the equation, and surpass the difficulties associated with its singular noise, we follow a numerical technique introduced by Dickman. Our results coincide remarkably well with those of discrete models claimed to belong to this universality class, in one, two, and three dimensions. This provides a strong backing for the Langevin theory of stochastic sandpiles, and to the very existence of this meagerly understood universality class.

Universality class of absorbing transitions with continuously varying critical exponents

The well-established universality classes of absorbing critical phenomena are directed percolation (DP) and directed Ising (DI) classes. Recently, the pair contact process with diffusion (PCPD) has been investigated extensively and claimed to exhibit a different type of critical phenomenon distinct from both DP and DI classes. Noticing that the PCPD possesses a long-term memory effect, we introduce a generalized version of the PCPD (GPCPD) with a parameter controlling the memory strength. The GPCPD connects the DP fixed point to the PCPD point continuously. Monte Carlo simulations strongly suggest that the GPCPD displays, to our knowledge, novel critical phenomena which are characterized by continuously varying critical exponents. The same critical behaviors are also observed in models where two species of particles are coupled cyclically. We present one possible scenario that the long-term memory may serve as a marginal perturbation to the ordinary DP fixed point.

Epidemic processes with immunization

We study a model of directed percolation (DP) with immunization, i.e., with different probabilities for the first infection and subsequent infections. The immunization effect leads to an additional non-Markovian term in the corresponding field theoretical action. We consider immunization as a small perturbation around the DP fixed point in d<6, where the non-Markovian term is relevant. The immunization causes the system to be driven away from the neighborhood of the DP critical point. In order to investigate the dynamical critical behavior of the model, we consider the limits of low and high first-infection rate, while the second-infection rate remains constant at the DP critical value. Scaling arguments are applied to obtain an expression for the survival probability in both limits. The corresponding exponents are written in terms of the critical exponents for ordinary DP and DP with a wall. We find that the survival probability does not obey a power-law behavior, decaying instead as a stretched exponential in the low first-infection probability limit and to a constant in the high first-infection probability limit. The theoretical predictions are confirmed by optimized numerical simulations in 1+1 dimensions.

Universal scaling behavior at the upper critical dimension of nonequilibrium continuous phase transitions

In this work we analyze the universal scaling functions and the critical exponents at the upper critical dimension of a continuous phase transition. The consideration of the universal scaling behavior yields a decisive check of the value of the upper critical dimension. We apply our method to a nonequilibrium continuous phase transition. By focusing on the equation of state of the phase transition it is easy to extend our analysis to all equilibrium and nonequilibrium phase transitions observed numerically or experimentally.

Dynamical properties of the synchronization transition

We use spreading dynamics to study the synchronization transition (ST) of one-dimensional coupled map lattices (CML's). Recently, Baroni et al. [Phys. Rev. E 63, 036226 (2001)] have shown that the ST belongs to the directed percolation (DP) universality class for discontinuous CML's. This was confirmed by accurate numerical simulations for the Bernoulli map by Ahlers and Pikovsky [Phys. Rev. Lett. 88, 254101 (2002)]. Spreading dynamics confirms such an identification only for random synchronized states. For homogeneous synchronized states the spreading exponents eta and delta are different from the DP exponents but their sum equals the corresponding sum of the DP exponents. Such a relation is typical of models with infinitely many absorbing states. Moreover, we calculate the spreading exponents for the tent map for which the ST belongs to the bounded Kardar-Parisi-Zhang (BKPZ) universality class. The estimation of spreading exponents for random synchronized states is consistent with the hyperscaling relation, while it is inconsistent for the homogeneous ones. Finally, we examine the asymmetric tent map. For small asymmetry the ST remains of the BKPZ type. However, for large asymmetry a different critical behavior appears, with exponents being relatively close to those for DP.