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Miguel − 2 and dIsiuoCro eF´ısica Computacional, Te´orica de y I Carlos Instituto nd hs rniin eoescn-re/otnosones second-order/continuous first-order/discontinuous become i.e., transitions point; may phase critical by rounding a the sharp) into remarkably, less result more and even (made low-dimensional and, Aizenman rounded disorder, in by are systems transitions shown equilibrium phase As first-order a Wehr, two coexist. which as at phases value abruptly threshold distinct a change crosses etc., parameter control mag- density, the energy, as such netization, properties system which for transitions, fields. random symmetry of broken presence of the absorbing in states of exist consequence, can presence in the There- and, to states, [11–14]. owing energy systems free equilibrium as thing genuine a such fore, in not which 10] for is [9, states there absorbing random cases with of systems one-dimensional non-equilibrium introduction in the even phase to fields second-order survive that may shows transitions [8], Vojta and Barghathi phase quantum to 7]. extended [6, nu- been transitions and has experimentally (iv) both and merically, examples countless in ified es(hr h reeeg swl defined), well is energy free the (where tems etal ra h ruetaoe[5]), above argument the break tentially hc rv htn ymtybekn cusee at even occurs calculations, breaking case symmetry marginal group the no renormalization that prove rigorous which more by e snwsittedsuso ofis-re phase first-order to discussion the shift now us Let by work recent case, equilibrium the with contrast In h myM argument Imry-Ma The yadfidte ob optbewt the with compatible be to them find and ty rniin n hi ocmtn phase concomitant their and transitions s h myM ruetde o pl oteenon- these to apply not does argument Imry-Ma the ddsre.Tu,w ocueta first that conclude we Thus, disorder. ed uu rniin ntodmninlsys- two-dimensional in transitions nuous saeas rcue nnon-equilibrium in precluded also are ns dsses o vni eunl non- genuinely in even not systems, ed rsneo admfils nta,dis- Instead, fields. random of presence rniin.TeImry-Ma-Aizenman- The transitions. e fqece iodr eas study also We disorder. quenched of e c iodri nrdcd eewe Here introduced. is disorder nce d weeruhitrae ol po- could interfaces rough (where 2 = ,Spain a, ,Spain a, i) n-equilibrium od o qiiru sys- equilibrium for holds iii) a enver- been has ii) sbacked is 2 upon introducing (random-field) disorder [5]. A simi- lar conclusion applies to the case of random interactions System with 2nd order 1st order [5, 15, 16]; indeed, a random distribution of interactions Random Fields (spontaneous (phase (e.g., bonds) locally favors one of the two phases, and d ≤ 2 sym. breaking) coexistence) thus, it has the same effects as random fields. Different Monte Carlo results support this conclusion; furthermore they suggest that the disorder-induced continuous transi- Equilibrium NO [4] NO [4, 5, 15, 16] tion exhibits critical exponents which are consistent with those of the corresponding pure model. An argument ex- plaining these findings was put forward by Kardar et al. Non-equilibrium YES [8] [17]. (abs. states) ? In close analogy with the argument above for the ab- sence of symmetry-breaking, in the case of phase coex- TABLE I: Random fields in low-dimensional disor- istence as well, regions (or “islands”) of arbitrary size dered systems. Summary of the effects of quenched of one of the phases appear in a stable way within the random fields on the existence of continuous/second-order other. Therefore, islands exist within islands in any of transitions (with spontaneously symmetry breaking), and the two phases in a nested way, leading always to hybrid discontinuous/first-order (with associated phase coexistence) states. Hence, two distinct phases cannot possibly coex- phase transitions in d ≤ 2 systems. Both, the equilibrium and ist and first-order transitions are precluded in disordered non-equilibrium cases are considered, the latter including the low-dimensional equilibrium systems. possibility of one or more absorbing states. Thus, the question arises as to whether shifting to the non-equilibrium realm entails the shattering of a fun- damental cornerstone of equilibrium statistical mechan- with absorbing states [20–24] or, instead, belong to a ics as it happens for continuous phase transitions (see new universality class defined by this disorder-induced table 1 for a synthetic summary); do first-order phase criticality. transitions, and, hence, phase coexistence, exist in low- If no novel universal behavior emerges, then it is ex- dimensional non-equilibrium disordered systems? pected for the model to behave as a standard two- Aimed at shedding some light on this issue, we study dimensional contact process (or directed percolation) non-equilibrium models with absorbing states in the pres- with quenched disorder with the following main features ence of disorder. More specifically, we study a variant of [20, 22–24]: the well-known “contact process”, sometimes called the • there should be a critical point separating the ac- “quadratic contact process”, in which two particles are tive from the absorbing phase, needed to generate an offspring while isolated particles can spontaneously disappear [11–13, 18, 19]. As a first • at criticality, a logarithmic or activated type of scal- step, we verify that the pure version of the model exhibits ing (rather than algebraic) should be observed. For a first-order transition separating an active phase from an instance, for quantities related to activity spreading absorbing one. Then we introduce disorder in the form of such as the survival probability, averaged number a site-dependent transition rates and investigate whether of particles, and radius from a localized initial seed, −δ¯ θ¯ the discontinuous character of the transition survives. we expect Ps(t) ∼ [ln(t/t0)] , N(t) ∼ [ln(t/t0)] , 1/Ψ and R(t) ∼ [ln(t/t0)] , respectively; t0 is some crossover time, and δ¯, θ¯, and Ψ should take the values already reported in the literature [23].
II. TWO POSSIBLE SCENARIOS • there should be a sub-region of the absorbing phase, right below the critical point, exhibiting generic Two alternative scenarios might be expected a priori algebraic scaling with continuously varying expo- for the impure/disordered model: nents, i.e. a Griffiths phase [25]. Griffiths phases stem from the existence of rare regions where the 1. the Imry-Ma argument breaks down in this non- disorder takes values significantly different from its equilibrium case and a first order phase transition average [24]. is observed, or 2. the Imry-Ma prediction holds even if the system These features follow from a strong-disorder renormal- ization group approach for the disordered contact pro- is a non-equilibrium one, and a disorder-induced second-order phase transition emerges. cess, which concludes that this anomalous critical behav- ior can be related to the random transverse-field Ising If the latter were true, we could then ask what uni- model for sufficiently strong disorder [20], and have been versality class such a continuous transition belongs to. A confirmed in computational studies which suggested that priori, it could share universality class with other already- this behavior is universal regardless of disorder strength known critical phase transitions in disordered systems [23, 24, 26]. 3
III. MODEL AND RESULTS As customarily done, we perform two types of exper- iments [11–14], considering as initial condition either a We study the simplest non-equilibrium model with homogeneous state, i.e., a fully occupied lattice of linear absorbing states exhibiting a first-order/discontinuous size L, or a localized seed, consisting in this case of a few, transition. Given that, owing to different reasons, one- at least a couple, neighboring particles in an otherwise dimensional systems with absorbing states rarely ex- empty lattice. hibit first-order phase transitions (even in pure systems) Homogeneous initial conditions– Figure 1 shows re- [27, 28], here we focus on the physically more relevant sults of computer simulations for the temporal decay two-dimensional case. In this two-dimensional reaction- of the particle density from ρ(t = 0) = 1. The upper diffusion contact-process-like model [11–14], individual panel shows an abrupt change of behavior at a thresh- particles disappear at a fixed rate, µ, while a pair of old value pdthr ≈ 0.0747; activity survives indefinitely nearest-neighbor particles is required to create an off- for pd < pdthr (at least up to the considered maximum spring at some rate λ time) and the particle density converges to relatively large steady state values (ρ ≈ 0.6), while activity dies µ λ A →∅ , 2A → 3A, (1) off exponentially for any pd > pdthr . This behavior is compatible with a first order phase transition, but the with the additional (“hard-core” or “Fermionic”) con- location of the threshold value has to be considered as a straint preventing sites from housing more than one par- rough estimate. ticle. This restriction can be relaxed at the cost of intro- To better locate the transition point, we study the ducing a reaction mean survival time (MST) as a function of system size. Figure 1b shows a non-standard non-monotonous depen- λ 2 3A → 2A, (2) dence of the MST as a function of size N = L . As we see, there are two regimes: i) for L
0 a) 100 10
10-2 0 0747 ρ ρ
10-4
0 2 4 6 100 102 104 106 10 10 10 10 t b FIG. 2: (Color online) Bistability at the transition point of the pure model. Log-log plot of the averaged particle density as a function of time for different initial con- ditions in the neighborhood of the transition point (results here are for pd = 0.07315). Depending on the initial density, the system stabilizes in the active or in the absorbing phase. The selected initial densities are equi-spaced in the interval [0.005, 1] with constant increments 0.05; system size N = 2562 and averages performed over up to 106 realizations.
marks a change of tendency, signaling the frontier be- tween the absorbing and active phases. In the active
phase (pd < pdthr ) and for large values of t, both Ns(t) and R2 grow approximately as t2 (as expected for ballis- tic expansion), while Ps(t) converges to a constant (i.e. some runs do survive indefinitely). On the other hand, in FIG. 1: (Color online) Decay from a homogeneous ini- the absorbing phase all three quantities curve downwards tial condition in the pure model . (a) Time evolution of the total averaged particle density for N = L2 = 2562; a first indicating exponential extinction. order phase transition can be observed near p ≈ 0.0747. Thus, the pure model exhibits a discontinuous transi- dthr ≈ (b) Mean survival time, tF , required to reach an arbitrar- tion at some value of pdthr 0.073, which separates a ily small density value here fixed to 0.01 (results are robust phase of high activity from an absorbing one. Observe, against variations of this choice) as a function of system size. that the estimation of the transition point is compatible Up to 103 realizations have been used to average these re- with the interval obtained above. sults. From this finite size analysis, the threshold point can be bounded to lie in the interval [0.070, 0.075]. B. Disordered model point but for finite system sizes the coexistence region has In the disordered version of the model, each lattice site some non-vanishing thickness. The existence of bistabil- has a random uncorrelated (death) probability. In partic- ity makes a strong case for the discontinuous character ular, we take pd(x)= pdr where pd is a constant and r is of the transition. a homogeneously-distributed random number r ∈ [0, 2] Spreading experiments from a localized seed– We con- (and, thus, the mean value is pd). Spatial disorder is sider a few (at least 2) neighboring particles at the center refreshed for each run, to ensure that averages are inde- of an otherwise empty lattice, and monitor how activity pendent of any specific realization of the disorder. spreads from that seed. Each simulation run ends when- Homogeneous initial conditions– We have computed ever the absorbing state is reached or when activity first time series for i) the mean particle density averaged over touches the boundary of the system. We monitor the all runs and ii) the mean particle density for surviving averaged squared radius from the origin R2(t), the aver- runs (i.e. those which have not reached the absorbing 4 aged number of particles over surviving trials, Ns(t), and state). Figure 4 shows time evolution after up to 2 × 10 the survival probability, Ps(t) [11]. Figure 3 shows log- realizations. Results are strikingly different from those log plots of these three quantities as a function of time. of the pure-model.
In all cases, we find a threshold value pdthr ≈ 0.073 that For values below threshold, pd 0 108 a) 10 2 , 0 0.0730 -2 s 10 10 0.0731 , 0.0732 0 0755 ρ s PNR -8 -4 10 10 3 10 105 t FIG. 3: (Color online) Spreading experiments for the b) pure model. Double-logarithmic plot of the (from bottom to top), (i) the survival probability Ps(t), (ii) the averaged number of particles Ns(t) (averaged over surviving runs), and (iii) the averaged squared radius (averaged over of all runs), as a function of time, for N = 10242 using up to 4 × 1010 experiments. Curves for R2(t) have been shifted upwards for clarity. In spite of the large number of runs used to average, ρ curves are still noisy. This is due to the fact that, being very close to the transition point, a large fluctuation is needed for the system to ”jump” to the active phase from the vicinity of the absorbing one and only a few runs reach large times. ticle density converges to a constant value for asymptot- ically large times, while for pdc > 0.077 curves decay as power laws (a much more precise estimation of the criti- cal point will be computed below). The generic algebraic FIG. 4: (Color online) Density decay from a homoge- decay is observed for a wide range of pd; however, the transient before the power-law regime increases with p , neous initial conditions in the disordered model. (a) d Particle density averaged over all trials in a lattice of size 2562 which makes it difficult to determine the exact bound- 4 and up to 2 × 10 realizations (curves in the active phase are aries of the mentioned range. The presence of generic plotted with dashed lines). Observe the presence of a broad algebraic scaling in an extended region is the trademark region with generic power-law behavior, i.e. a Griffiths phase of Griffiths phases. which starts roughly at pd = 0.0775. (b) As (a) but averaging Plotting the activity over the surviving trials [Fig. 4b], only over surviving trials. Note the non-monotonic behavior we observe that the evolution is non-monotonous in the in the Griffiths phase (see main text for details). absorbing (Griffiths) phase: the curves decrease up to a minimum value and then increase. This stems from the fact that realizations with large rare active regions re- main active for longer times than those with smaller ones; tion of time; for all of them, we clearly observe generic as realizations with only relatively small rare-regions pro- asymptotic power laws with continuously varying expo- gressively die out, those with larger and larger rare- nents. These spreading quantities also allow us to scru- regions are filtered through and, thus, the overall average tinize the behavior at the critical point. As discussed density grows as a function of time, being limited only in the Introduction, in a disordered system as the one by system size. under study, we expect logarithmic (activated scaling) In addition, we observe that, contrarily to the pure at criticality. Indeed, Figure 7 shows results for the case, there is no bistability around the transition point usual spreading quantities represented in a double log- (figure 5). Indeed, very near to the transition point arithmic plot of the different quantities as a function of (pd =0.07650), all curves regardless of their initial value ln(t/t0). The value of t0 is in principle unknown and converge to a unique well-defined stationary density close constitutes a significant error source [23]. We fix it as to zero, as appropriate for a continuous transition to an the value of t such that it gives the best straight lines at absorbing state. the transition point for all three quantities [23]). Right Spreading experiments from a localized seed– Figure 6 at the critical point (pc ≈ 0.07652 to be obtained with shows results for three spreading observables as a func- more accuracy below) a straight asymptotic behavior in- 6 0 10 £¤ 104 10-2 N ρ 100 10-4 0 102 104 106 108 b¥ 10 t 10-2 FIG. 5: (Color online) Absence of bistability in the disordered model. Double-logarithmic plot of the averaged s P -4 particle density as a function of time for pd = 0.0765, with 10 N = 2562, and up to 106 realizations. Initial densities are ρ0 = 0.00006, 0.01, 0.2, 0.3, 0.4, 0.7, 1. Regardless of the initial condition, the system stabilizes to a constant small value of -¢ the density, as expected for a second-order phase transition. 10 c¦ dicates that results are compatible with logarithmic (i.e. activated) type of scaling. The best estimates for the 4 (pseudo)-exponents listed in section II are: δ¯ ≈ 1.90, 10 θ¯ ≈ 2.09, and Ψ ≈ 0.43, which are compatible with the values reported in the literature for the universal- 2 ¡ ity class of directed percolation with quenched disorder a (i.e. δ¯ =1.9(2), θ¯ =2.05(20), Ψ = 0.51(6)). R 100 Similarly, following the work of Vojta and collabora- tors [23], we represent in Fig. 8 one of the spreading quantities as a function of another one, e.g. N(t) as a function of P (t) to eliminate the free variable t from s 0 -4 the plot. This type of plot allows for the identification of 10 2 4 6 power-law dependencies rather than logarithmic ones, i.e. 10 10 10 −θ/¯ δ¯ N(t) ∼ Ps(t) . If the second-order phase transition belongs to the universality class of the directed percola- t tion with quenched disorder (see section II), we should − FIG. 6: (Color online) Spreading experiments in the have N(t) ∼ P (t) 1.08(15), using as a reference the val- s disordered model. Double logarithmic plot of the three ues in the literature [23]. Indeed, as shown in Figure 8 we −1.10(2) usual spreading quantities showing the presence of generic obtain N(t) ∼ Ps(t) , in very good agreement with power-laws with continuously varying exponents all along the expected value [23], and this is the method by which the Griffiths phase (pd . 0.07652). Parameter values (from the critical point location, pd ≈ 0.07652, is obtained with top to bottom) 0.06, 0.07, 0.073, 0.075, 0.07652, 0.077, 0.0775, best accuracy. 0.078, 0.0785, 0.079, 0.0795 (curves in the active phase are plotted with dashed lines), up to 5 × 107 realizations. IV. CONCLUSIONS AND DISCUSSION Results are rather similar to those reported for the In contrast with the pure model, in the disordered case standard contact process with quenched disorder. In- we have found a Griffiths phase and a second-order phase deed, results are fully compatible (up to numerical pre- transition with an activated type of scaling. Therefore, in cision) with the standard strong-disorder fixed point of this non-equilibrium system with one absorbing state the the universality class of the directed percolation with situation remains much as in equilibrium situations: dis- quenched disorder [20, 23, 24]. We believe that our re- order annihilates discontinuous transitions and induces sults are robust upon considering other types of (weaker) criticality. disorders [26]. Thus, two different models with signifi- 7 a) 102 6 0 4 10 !"#$ % 10 1 10 & ')*+, . 2§¨© ln(t/t0) N 2 N 10 102 5x102 100 1 10-2 ( s 1 3 5 0 10 10 10 b) 10 1/Ps FIG. 8: (Color online) Double logarithmic plot of N(t) 10-2 as a function of 1/Ps(t) for spreading experiments at ln(t/t0) criticality in the disordered model [24]. Our best esti- mate for the slope of at the critical point (separatrix of the curves, see main panel) is compatible with the value reported −θ/¯ δ¯≈−1.08 in the literature N(t) ∼ Ps(t) corresponding to the 10-4 universality class of the directed percolation with quenched disorder. The inset shows a zoom around the critical point. Lattice size N = 10242; averages up to 5 × 107 realizations, and same parameters as in Figure 7. c) 4 Griffiths phases and activated scaling at the transition ln(t/t0) 104 point. From a more general perspective, deciding whether 2 novel universal behavior emerges in disorder-induced crit- icality is still an open problem in statistical mechanics. For illustration, let us point out that recent work suggests 2 that disorder-induced second-order phase transitions in 10 an Ising-like system with up-down symmetry does not coincide with Ising transition [35]. Similarly, in Ref. [36] 1 10 2x101 a novel type of critical behavior is found for disorder- induced criticality. In the case studied here, the disorder- ln(t/t0) induced criticality does not seem to lead to novel behav- ior (up to numerical precision); indeed, all evidences sug- FIG. 7: (Color online) Spreading experiments in gest that it behavior coincides with the universality class the disordered model. Double logarithmic plot of of the directed percolation with quenched disorder. the three usual spreading quantities as a function of After a careful inspection of the literature in search of ln(t/t0) for different parameter values (from top to bottom 0.07, 0.073, 0.075, 0.07652, 0.077, 0.0775, 0.078 (curves in the discontinuous transitions in disordered non-equilibrium active phase are plotted with dashed lines). Same network low-dimensional systems, we found a very recent work in sizes as in the homogeneous case and up to 106 realizations. which the authors study the popular (two-dimensional) By conveniently choosing t0 = 0.01 (see main text) we observe Ziff-Gulari-Barshad (ZGB) model for catalytic oxidation straight lines at the critical point, pd ≈ 0.07652. From their of carbon monoxide [37] in the presence of catalytic impu- corresponding slopes we measure the associated (pseudo)- rities (a fraction of inert sites) [38]. The pure ZGB model exponents: θ¯ ≈ 2.09, δ¯ ≈ 1, 9, and 2/Ψ ≈ 4.6 (slopes marked is known to exhibit, among many other relevant features, by dashed lines). These values have a large uncertainty, as a discontinuous transition into an absorbing state. How- changes in the value of t0 severely affect them. Estimating ever, after introducing quenched disorder, no matter how these exponents with larger precision is computationally very small its proportion, the discontinuous transition is re- demanding. placed by a continuous one [38], similarly to our findings here. In conclusion, we conjecture that first-order phase cantly different pure versions –i.e. one with a first-order transitions cannot appear in low-dimensional disordered and one with a second-order transition– become very sim- systems with an absorbing state. In other words, the ilar once quenched disorder is introduced. Both exhibit Imry-Ma-Aizenman-Wehr-Berker argument for equilib- 8 rium systems can be extended to non-equilibrium situ- dimensional limit, we consider a fully connected lattice ations including absorbing states. The underlying reason in which each node is nearest neighbor to any other one. for this is that, even if the absorbing phase is fluctuation- n(t) is the total number of particles at a given time less and hence is free from the destabilizing effects the and ρ(t) = n(t)/N the corresponding density. Allowed Imry-Ma argument relies on, the other phase is active changes at any time-step are of the magnitude ±1/N. We and subject to fluctuation effects. Therefore, intrinsic can thus write the overall transition rates as W −(ρ → fluctuations destabilize it as predicted by the Imry-Ma- ρ − 1/N) = µρ and W +(ρ → ρ +1/N) = λρ2(1 − ρ) Aizenman-Wehr-Berker argument, precluding phase co- for creation and annihilation processes, respectively. Ex- existence. panding the associated master equation in power series, Remarkably, in the case studied by Barghathi and Vo- and keeping only the first two leading terms we obtain jta, in which the Imry-Ma argument is violated in fa- the Fokker-Planck equation: vor of a second-order phase transition, the two broken- symmetry states are absorbing ones: once the symmetry 3 2 is broken in any of the two possible ways, the system ∂tP (ρ,t)= −∂ρ[(−λρ + λρ − µρ)P ]+ becomes completely frozen, i.e., free from fluctuation ef- 1 + ∂2[(−λρ3 + λρ2 + µρ)P ], fects, and, consequently, the Imry-Ma argument breaks 2N ρ down. Thus, the only possibility to have first-order phase transitions in low-dimensional disordered systems would equivalent to the (Itˇo) Langevin equation [34]: be to have (in its pure version counterpart) a discontin- uous phase transition between two different fluctuation- less states, and we are not aware of any such transition. 3 2 2 3 µρ − λρ + λρ Therefore, we conclude that quenched disorder forbids ∂tρ = −µρ + λρ − λρ + ξ(t). r N discontinuous phase transitions in low-dimensional non- where ξ(t) is a Gaussian white noise. In the N → ∞ equilibrium systems with absorbing states. (mean-field) limit one recovers the rate equation [Eq.(3)] with its associated discontinuous transitions. Using the Acknowledgments noise term it is possible to derive (using the theory of mean first passage times [32–34]) the scaling of the escape time as a function of the system size. 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