arX

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v1 [

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2012

Temporal disorder in up-down symmetric systems

Ricardo Martınez-Garcıa,1 Federico Vazquez,2 Cristobal Lopez,1 and Miguel A. Munoz3

1IFISC, Instituto de Fısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, Spain2Max-Planck-Institut fur Physik Komplexer Systeme, Nothnitzer Str. 38, 01187 Dresden, Germany

3Instituto Carlos I de Fısica Teorica y Computacional, Universidad de Granada, 18071 Granada, Spain(Dated: February 27, 2012)

The effect of temporal disorder on systems with up-down Z2 symmetry is studied. In particular, weanalyze two well-known families of phase transitions: the Ising and the generalized voter universalityclasses, and scrutinize the consequences of placing them under fluctuating global conditions. Weobserve that variability of the control parameter induces in both classes “Temporal Griffiths Phases”(TGP). These recently-uncovered phases are analogous to standard Griffiths Phases appearing insystems with quenched spatial disorder, but where the roles of space and time are exchanged. TGPsare characterized by broad regions in parameter space in which (i) mean first-passage times scalealgebraically with system size, and (ii) the system response (e.g. susceptibility) diverges. Our resultsconfirm that TGPs are quite robust and ubiquitous in the presence of temporal disorder. Possibleapplications of our results to examples in ecology are discussed.

PACS numbers: 64.60.Ht, 05.40.Ca, 05.50.+q, 05.70.Jk

I. INTRODUCTION

Systems with up-down Z2 symmetry –such as the Isingmodel– are paradigmatic in Statistical Mechanics. Someof them –such as the voter model– exhibit absorbingstates, a distinctive feature of non-equilibrium dynamics[1–3]. Absorbing states are configurations of the systemcharacterized by the lack of fluctuations, where the dy-namics becomes frozen and the system remains trapped.In the last years, a great interest has been given to thisclass of models with two symmetric absorbing states [1–9], which are of high relevance in diverse problems in theecological, biological, and social sciences, such as speciescompetition [10], neutral theories of biodiversity [11], al-lele frequency in genetics [12], opinion formation [13],epidemics propagation [14], or language spreading [15].

Phase transitions into absorbing states are quite uni-versal. Systems exhibiting one absorbing state belonggenerically to the very robust Directed Percolation (DP)universality class and share the same set of critical expo-nents, amplitude ratios, and scaling functions. When thisgeneral rule is broken it is because some extra symmetryor conservation law is present in the system [1–3]. Thisis the case of the class of systems with Z2-symmetricabsorbing states, which may exhibit a phase transitionwith critical scaling differing from DP, usually referredas Generalized voter (GV), also called “parity conserv-ing”, “DP2” or “directed Ising”, universality class (see[1, 2, 5] and references therein). More interestingly, ana-lytical and numerical studies [4, 5, 7–9] have shown that,depending on some details, Z2-symmetric models mayundergo either a unique GV-like phase transition separat-ing an active/symmetric phase from an absorbing one or,alternatively, such a transition can split into two separateones: a Ising-like transition in which the Z2 symmetry isbroken, and a second DP-like transition below which thebroken-symmetry phase collapses into the correspondingabsorbing state. In particular, a general Langevin equa-

tion, aimed at capturing the phenomenology of these sys-tems, was proposed in [4]; depending on general featuresit may exhibit a DP, an Ising, or a GV transition.

In many situations, Z2 symmetric systems are not iso-lated but, instead, affected by external conditions or byenvironmental fluctuations. The question of how exter-nal variability affects diversity, robustness, and evolutionof complex systems, is of outmost relevance in differentcontexts. Take, for instance, the example of the neutraltheory of biodiversity: if there are two Z2-symmetric (orneutral) species competing, what happens if dependingon environmental conditions one of the two species is fa-vored at each time step in a symmetric way? Does suchenvironmental variability enhance species coexistence ordoes it hinder it? [16–20].

Motivated by these questions, we study how basicproperties of Z2 symmetric systems, such as responsefunctions and first-passage times, are affected by thepresence of temporal disorder.

Some previous works have explored the effects of fluc-tuating global conditions in simple models exhibitingphase transitions [21–23]. Temporal disorder has beenshown to be a highly relevant perturbation around DPphase transitions in all dimensions (in apparent contra-diction with the Harris criterion for the relevance of dis-order [21]), while temporal disorder has been shown tobe relevant at Ising transition only at and above three di-mensions. More recently, a modified version of the sim-plest representative of the DP class –i.e. the Contact-Process– equipped with temporal disorder was studiedin [24]. In this model, the control parameter (birth prob-ability) was taken to be a random variable, varying ateach time unit. As the control parameter is allowed totake values above and below the transition point of thepure contact process, the system alternates between thetendencies to be active or absorbing. As shown in [24]this dynamical frustration induces a logarithmic typeof finite-size scaling at the transition point and gener-

2

ates a subregion in the active phase characterized by ageneric algebraic scaling (rather than the usual exponen-tial, Kramers-like, behavior) of the extinction times withsystem size. More strikingly, this subregion is also char-acterized by generic divergences in the system suscepti-bility, a property which is reserved for critical points inpure systems. This phenomenology is akin to the onein systems with quenched “spatial” disorder [26], whichshow algebraic relaxation of the order parameter, and sin-gularities in thermodynamic potentials in broad regionsof parameter space: the so-called, Griffiths Phases [25].The remarkable peculiarities of standard Griffiths phasesstem from the existence of (exponentially) rare –locallyordered– regions which take a (exponentially) long timeto decay, inducing an anomalously slow decay in the dis-ordered phase.In the case of temporal disorder, an analogy with Grif-

fiths phases can be made, in the sense that very long(exponentially rare) time intervals (corresponding to anabsorbing phase of the pure model) of the control pa-rameter have a large influence on the system dynamicseven when the overall system is in its active phase. Thesephenomenological similarities between systems with spa-tial and temporal disorder led to introduce the conceptof “Temporal Griffiths Phases” (TGP) [24].In order to investigate whether the anomalous behavior

that leads to TGPs around absorbing state (DP) phasetransitions is a universal property of systems in otheruniversality classes –and in particular, in up-down sym-metric systems– we study the possibility of having TGPsaround Ising and GV transitions. We scrutinize simplemodels in these two classes and assume that the cor-responding control parameter changes randomly in time,fluctuating around the transition point of the correspond-ing pure model, and study the susceptibility as well asmean-first passage times. We mainly focus on the mean-field (high dimensional) limit, since it allows for analyti-cal treatment via a Langevin approach, but we also pro-vide numerical results and some theoretical considera-tions for low dimensional systems.The paper is organized as follows. In section II, we

develop a general mean-field description of models withvarying control parameters in terms of collective vari-ables. In section III and IV, we show analytical and nu-merical results for the Ising and GV transitions, respec-tively. In section V, a short summary and conclusionsare presented.

II. MEAN-FIELD THEORY OF Z2-SYMMETRIC

MODELS WITH TEMPORAL DISORDER

Interacting particle models evolve stochastically overtime. A useful technique to study such systems is themean-field (MF) approach, which implicitly assumes a“well-mixed” situation, where each particle can interactwith any other, providing a sound approximation in highdimensional systems. One way in which the mean-field

limit can be seen at work is by analyzing a fully con-nected network (FCN), where each node (particle) is di-rectly connected to any other else, mimicking an infinitedimensional system.In the models we study here, states can be labeled

with occupation-number variables ρi taking a value 1 ifnode i is occupied or 0 if it is empty, or alternatively byspin variables Si = 2ρi − 1, with Si = ±1. Using theselatter, the natural order parameter is the magnetizationper spin, defined as

m =1

N

N∑

i=1

Si, (1)

where N is the total number of particles in the system.The master equation for the probability P (m, t) of havingmagnetization m at a given time t, is

P (m, t+ 1/N) = ω+ (m− 2/N, b)P (m− 2/N, t) (2)

+ ω− (m+ 2/N, b)P (m+ 2/N, t)

+ [1− ω−(m, b)− ω+(m, b)]P (m, t),

where ω±(m, b) are the transition probabilities from astate with magnetization m to a state with magneti-zation m ± 2/N . This describes a process in which a“spin” is randomly selected at every time-step (of lengthdt = 1/N), and inverted with a probability that dependson m and the control parameter b. The allowed magneti-zation changes in an individual update, ∆m = ±2/N , areinfinitesimally small in the N → ∞ limit. In this limit,one can perform a standard Kramers-Moyal expansion[29, 30] leading to the Fokker-Planck equation

∂P (m, t)

∂t= − ∂

∂m[f(m, b)P (m, t)]+

1

2

∂2

∂m2[g(m, b)P (m, t)] ,

(3)with drift and diffusion terms given, respectively, by

f(m, b) = 2 [ω+(m, b)− ω−(m, b)] , (4)

g(m, b) =4 [ω+(m, b) + ω−(m, b)]

N. (5)

From Eq. (3), and working in the Ito scheme (as justifiedby the fact that it comes from a discrete in time equation[28]), its equivalent Langevin equation is [30]

m = f(m, b) +√

g(m, b) η(t), (6)

where the dot stands for time derivative, and η(t) isa Gaussian white noise of zero-mean and correlations〈η(t)η(t′)〉 = δ(t − t′). The diffusion term is propor-

tional to 1/√N , and therefore, it vanishes in the ther-

modynamic limit (N → ∞), leading to a deterministicequation for m.The drift and diffusion coefficients in Eq. (6) depend

not only on the magnetization, but also on the parameterb. To analyze the behavior of the system when b changes

3

0 1 2 3 4 5 6 7t/τ

-1

-0.5

0

0.5

1ξ

0 1 2 3 4 5 6 7t/τ

0.5

1

1.5b

b0 + σ

b0 - σ

(a)

(b)

b0

FIG. 1: (a) Typical realization of the colored noise ξ(t), astep like function that takes values between +1 and −1. (b)Stochastic control parameter b(t) = b0 + σξ(t) according tothe values of the noise in (a), b0 = 1.03 and σ = 0.4.

randomly over time, and following previous works [19,24], we allow b to take a new random value, extractedfrom a uniform distribution, in the interval (b0−σ, b0+σ)at each MC step, i.e., every time interval τ = 1. Thus,we assume that the dynamics of b(t) obeys an Ornstein-Uhlenbeck process

b(t) = b0 + σ ξ(t), (7)

where ξ(t) is a step-like function that randomly fluctuatesbetween −1 and 1, as depicted in Fig. 1a. Its averagecorrelation is

< ξ(t)ξ(t+∆t) > =

13 (1− |∆t|/τ) for |∆t| < τ

0 for |∆t| > τ ,(8)

where the bar stands for time averaging. Parameters b0and σ are chosen with the requirement that b takes val-ues at both sides of the transition point of the pure model

(see Fig. 1b), that is, the model with constant b. Thus,the system randomly shifts between the tendencies to bein one phase or the other (see Fig. 2). The model presentsboth intrinsic and extrinsic fluctuations, as representedby the white noise η(t) and the colored noise ξ(t), re-spectively. Plugging the expression Eq. (7) for b(t) intoEq. (6), and retaining only linear terms in the noise onereadily obtains

m = f0(m) +√

g0(m) η(t) + j0(m) ξ(t), (9)

where f0(m) ≡ f(m, b0), g0(m) ≡ g(m, b0) and j0(m) isa function determined by the functional form of f(m, b),that might also depend on b0. To simplify the analysis,we assume that relaxation times are much longer than theautocorrelation time τ , and thus take the limit τ → 0in the correlation function Eq. (8), and transform theexternal colored noise ξ into a Gaussian white noise with

effective amplitude K ≡∫ +∞

−∞ < ξ(t)ξ(t+∆t) >d∆t =

τ/3. Then, we combine the two white noises into aneffective Gaussian white noise, whose square amplitudeis the sum of the squared amplitudes of both noises [30],and finally arrive at

m = f0(m) +√

g0(m) +Kj20(m) γ(t), (10)

where 〈γ(t)〉 = 0 and 〈γ(t)γ(t′)〉 = δ(t− t′).In the next two sections, we analyze the dynamics

of the kinetic Ising model with Glauber dynamics anda variation of the voter model (the, so-called, q-votermodel) –which are representative of the Ising and GVtransitions respectively– in the presence of external noise.For that we follow the strategy developed in this sectionto derive mean-field Langevin equations and present alsoresults of numerical simulations (for both finite and infi-nite dimensional systems), as well as analytical calcula-tions.

III. ISING TRANSITION WITH TEMPORAL

DISORDER

We consider the kinetic Ising model with Glauber dy-namics [34], as defined by the following transition rates

Ωi(Si → −Si) =1

2

1− Si tanh

b

2d

∑

j∈〈i〉

Sj

. (11)

The sum extends over the 2d nearest neighbors of a givenspin i on a d-dimensional hypercubic lattice, and b =Jβ is the control parameter. J is the coupling constantbetween spins, which we set to 1 from now on, and β =(kBT )

−1. Note that b in this case is proportional to theinverse temperature.

A. The Langevin equation

In the mean-field case, the cubic lattice is replaced bya fully-connected network in which the number of neigh-bors 2d of a given site is simply N − 1. Then, the tran-sition rates of Eq. (11) can be expressed as

Ω±(m, b) ≡ Ω(∓ → ±) =1

2[1± tanh (bm)] . (12)

which implies ω±(m, b) =1∓m2 Ω±(m, b) for jumps in the

magnetization. Following the steps in the previous sec-tion, and expanding Ω± to third order in m, we obtain

m = a0m−c0m3+

√

1− b0m2

N+Kσ2m2(1− b20m

2)2 γ(t),

(13)where b0 is the mean value of the stochastic control pa-rameter, a0 ≡ b0 − 1, and c0 ≡ b30/3.The potential V (m) = −a0

2 m2 + c0

4 m4 associated with

the deterministic term of Eq. (13) has the standard shape

4

Critical point

Control parameter (b)

Order

param

eter

(m)

Tim

e

bmaxbmin

FIG. 2: Schematic representation of the fluctuating controlparameter in the Ising model with Glauber dynamics. Itmakes the system shift between the ordered phase to the dis-ordered one.

of the Ising class, that is, of systems exhibiting a sponta-neous breaking of the Z2 symmetry. A single minimumat m = 0 exists in the disordered phase, while two sym-metric ones, at ±

√

a/c exist below the critical point.

B. Numerical Results

By performing simulations of the Langevin equationEq. (13) or, equivalently, by performing Monte Carlosimulations of the particle system on a fully connectednetwork, we observe that the critical point is located atthe value b0,c = 1 (identical to that of the pure case,bc,pure = 1; see below for details). Having located thetransition point, we study two magnitudes that are rel-evant in systems with temporal disorder [24], the meancrossing time (or mean-first passage time) and the sus-ceptibility. The crossing time is the time employed by thesystem to reach the disordered zero-magnetization statefor the first time, starting from a fully ordered state with|m| = 1 (see Fig. 3). Crossing times were calculated bynumerically integrating Eq. (13) for different realizationsof the noise γ and averaging over many independent re-alizations. Results are shown in the Fig. 4.

As it is characteristic of TGPs [24], at the criticalpoint, the scaling is of the form T ∼ [lnN ]α with a nu-merically determined exponent value α = 2.81 for σ = 0.4(see inset of Fig. 4). This is to be compared with the stan-dard power-law scaling T ∼ Nβ characteristic of puresystems, i.e. for σ = 0. Moreover, a broad region show-ing algebraic scaling T ∼ N δ with a continuously varyingexponent δ(b0) (δ → 0 as b0 → b+0,c) appears in the or-dered phase b0 > b0,c. Both α and δ are not universaland depend on the noise strength σ. Finally, in the disor-dered phase the scaling of T is observed to be logarithmic,T ∼ lnN .

We have also performed Monte Carlo simulations ofthe time-disordered Glauber model on two- and three-dimensional cubic lattices with nearest neighbor inter-actions. In d = 2, a shift in the critical point was

0 20 40 60 80 100 120t

0

0.2

0.4

0.6

0.8

1

1.2

m

TCrossing time

∆m

=2/N

Jump L

ength

Random walk

mapping

FIG. 3: Single realization of the stochastic process. The sys-tem starts with all the spins in the same state (m = 1) andthe dynamics is stopped when it crosses m = 0, which de-fines the crossing time in the Ising model. We take σ = 0.4,b0 = 0.98 and system size N = 106. On the right margin wesketch the mapping of the problem to a Random Walk withjump length |∆m| = 2/N .

104

106

108N

102

104

106

108

T

b0 = 1.10

b0 = 1.09

b0 = 1.08

b0 = 1.07

b0 = 1.06

b0 = 1.05

b0 = 1.04

b0 = 1.03

b0 = 1.02

b0 = 1.01

b0,c

= 1.00

b0 = 0.99

b0 = 0.98

MC

101ln(N)

102

104

T

FIG. 4: Main: Log-log plot of the crossing time T (N) forthe Ising Model with Glauber dynamics in mean field. MonteCarlo simulations on a FCN (circles) and numerical integra-tion of the Langevin equation Eq. (13) with σ = 0.4 (solidlines). There is a region with generic algebraic scaling ofT (N) and continuously varying exponents. Inset: log-log plotof T (N) vs. lnN . It is estimated at criticality T ∼ (lnN)2.81.

found: from bc,pure = 0.441(1) in the pure model tob0,c = 0.605(1) for σ = 0.4. However, the scaling be-havior of T with N resembles that of the pure model,with T ∼ Nβ at criticality (with an exponent numericallyclose to that of the pure model [3]), and an exponentialgrowth T ∼ exp(cN), where c is a positive constant, inthe ordered phase (Arrhenius law). Thus, no region ofgeneric algebraic scaling appears in this low-dimensionalsystem. On the contrary, in d = 3, results qualitativelysimilar to mean-field ones are recovered (see Fig. 5). Thecritical point is shifted from bc,pure = 0.222(1) (calcu-lated in [31]) to b0,c = 0.413(2), with a critical exponentα(d = 3) = 5.29 for σ = 0.4, and generic algebraic scal-ing in the ordered phase. In conclusion, our numerical

5

102

104

106N

102

104

106

108

T

b0 = 0.46

b0 = 0.45

b0 = 0.44

b0 = 0.43

b0 = 0.42

b0 = 0.41

b0 = 0.40

101ln(N)

102

104

T

b0 = 0.413

FIG. 5: Main: Log-log plot of the crossing time T (N) for theIsing Model with Glauber dynamics in d = 3. Monte Carlosimulations on a regular cubic lattice with σ = 0.4. There is aregion b ∈ [0.42, 0.46] with generic algebraic scaling of T (N)and continuously varying exponents. Inset: log-log plot ofT (N) vs. ln(N). It is estimated at criticality T ∼ (lnN)5.29.

studies suggest that the lower critical dimension for theTGPs in the Ising transition is dc = 3. This is in agree-ment with the analytical finding in [22], establishing thattemporal disorder is irrelevant in Ising-like systems be-low three dimensions. This result is to be compared withdc = 2 numerically reported for the existence of TGPsin DP-like transitions [24] (observe, however, that tem-poral disorder, in this case, affects the value of criticalexponents at criticality in all spatial dimensions). Fur-ther studies are needed to clarify the relation betweendisorder-relevance at criticality and the existence or notof TGPs.We have also measured the susceptibility χ, defined

as the response function to an external field h in thevanishing field limit

χ = limh→0

∂m

∂h, (14)

where m denotes the stationary magnetization averagedover many independent realizations. In the presence ofan external field, the transition rates become Ω±(m, b) =12 [1± tanh (bm+ h)]. Expanding the hyperbolic tan-gent up to third order in m and to first order in h, weobtain the following Langevin equation

m = a0m−c0m3+h(

1− b2m2)

+√Kσm(1−b20m2) γ(t),

(15)where we have considered the N → ∞ limit (g0 = 0).The average magnetization m for a given field h was

calculated by integrating the Langevin equation and thentaking averages over noise realizations. The susceptibil-ity can be computed, for different values of b0, as thederivative of m with respect to h. Generic divergences ofthe form χ ∼ hυ+Constant (with υ < 0 as h→ 0) appearin a broad region b0 ∈ [b0,c − σ2/2, b0,c + σ2/2], centeredaround b0,c, with symmetric exponents around the criti-cal point (see Fig. 6). These results agree with those ob-tained through Monte Carlo simulations on a FCN (not

10-10

10-8

10-6

10-4

h

102

104

106

108

χ

b0 = 0.996

b0 = 0.997

b0 = 0.998

b0 = 0.999

b0,c

= 1.000

b0 = 1.001

b0 = 1.002

b0 = 1.003

b0 = 1.004

FIG. 6: Main: Log-log plot of the susceptibility as a func-tion of the external field for different values of b0 ∈ [b0,c −σ2/2, b0,c + σ2/2] with σ = 0.1, obtained by integratingEq. (15). Generic divergences with symmetric exponentsaround the critical value b0.c = 1 are observed.

shown). In finite dimensions, given the required large sys-tems sizes and small fields, we could not conclude aboutthe existence or not of generic divergences.

C. Analytical results

Let us consider the Langevin equation Eq. (13) in theThermodynamic limit (g0(m) = 0). Given that the re-maining intrinsic noise comes from a transformation of acolored noise into a white noise, the Stratonovich inter-pretation is to be used to obtain its associated Fokker-Planck equation (see e.g. [28])

∂P (m, t)

∂t= − ∂

∂m

[

f0(m) +K

2j0(m)j′0(m)

]

P (m, t)

+1

2

∂2

∂m2

Kj20(m)P (m, t)

. (16)

Imposing the detailed balance (fluxless) condition, it isstraightforward to obtain the steady state solution

Pst(m) ∝ exp

(

−c0m2

σ2

)

|m|2a0/σ2−1, (17)

with a power-law singularity at the origin, which is a dis-tinctive trait of a Langevin equation with linear multi-plicative noise [32, 33]. By performing a calculation anal-ogous to that in [24], we have analytically computed thesystem susceptibility and found that χ ∼ hυ +Constant,as mentioned earlier, and in agreement with previous re-sults found in [24, 32, 33], with

υ =2(b0 − 1)

σ2− 1. (18)

This, in particular, implies that the susceptibility di-verges when υ < 0 as h → 0 or, in terms of the control

6

parameter b0 = 1+a0, when b0 takes a value in the region1 − σ2/2 < b0 < 1 + σ2/2 centered at the critical pointb0,c = 1. The values of the exponent υ agree well withthose of Fig. 6 at some distance from the critical point.For instance, an analytical value υ = −0.40 for b0 = 1.003corresponds to a numerical value υnum = −0.39, andυ = −0.60 for b0 = −1.002 to a value υnum = −0.59.However, the analytical exponent υ = −1 at the criticalpoint is not in good agreement with the numerical resultυnum = −0.88, indicating that the asymptotic regime hasnot been numerically reached.We next provide analytical results for the crossing

time. Starting from the N-independent Fokker-Planckequation Eq. (16), an effective dependence on N is im-plemented by calculating the first-passage time to thestate m = |2/N | rather than m = 0. This is equivalentto the assumption that the system reaches the zero mag-netization state with an equal number N+ = N− = N/2of up and down spins when |m| < 2/N , that is, whenN/2− 1 < N+ < N/2 + 1. The mean-first passage timeT associated with the Fokker-Planck equation Eq. (16)obeys the differential equation [30]

K

2j20 (m)T ′′(m) +

[

f0(m) +K

2j0(m)j′0(m)

]

T ′(m) = −1,

(19)with absorbing and reflecting boundaries at |m| = 2/Nand |m| = 1, respectively. The solution, starting at timet = 0 from m = 1 is given by

T (m = 1) = 2

∫ 1

2/N

dy

ψ(y)

∫ 1

y

ψ(z)

Kj20(z)dz, (20)

where

ψ(x) = exp

∫ x

2/N

2f0(x′) +Kj0(x

′)j′0(x′)

Kj20(x′)

dx′

. (21)

Computing these integrals (see Appendix A) we obtain

T ∼

lnN/(b0 − 1) for b0 < 13(lnN)2/σ2 for b0 = 1

N6(b0−1)

σ2 for b0 > 1.

(22)

These expressions qualitatively agree with the numericalresults of Fig. 4, showing that T grows logarithmicallywith N in the absorbing phase b0 < 1, as a power law inthe active phase b0 > 1, and as a power of lnN (i.e.poly-logarithmically) at the transition point b0,c = 1.The critical exponent α = 2, as well as the exponentsδ = 6(b0 − 1)/σ2 do not agree well with the numer-ically determined exponents. This is probably due toto the fact that we have neglected the 1/

√N term by

taking g0 = 0, which becomes of the same magnitudeas the j0 term when |m| approaches 2/N . Indeed, thiswas confirmed (not shown) by testing that analytical ex-pressions Eq. (22) agree very well with numerical inte-grations of Eq. (13) performed for g0 = 0, and setting

the crossing point at m = 2/N . In summary, this an-alytical approach reproduces qualitatively –and in somecases quantitatively– the above reported non-trivial phe-nomenology.

IV. GENERALIZED VOTER TRANSITION

WITH TEMPORAL DISORDER

We study in this section the GV transition [5], whichappears when a Z2-symmetry system simultaneouslybreaks the symmetry and reaches one of the two absorb-ing states. A model presenting this type of transition isthe nonlinear q-voter model, introduced in [35]. The mi-croscopic dynamics of this nonlinear version of the votermodel consists in randomly picking a spin Si and flip-ping it with a probability that depends on the state ofq randomly chosen neighbors of Si (with possible rep-etitions). If all neighbors are at the same state, thenSi adopts it with probability 1 (which implies, in par-ticular, that the two completely ordered configurationsare absorbing). Otherwise, Si flips with probability b.Therefore, the probability that Si flips at a given time, isa function of the fraction x of disagreeing (antiparallel)neighbors

f(x, b) = xq + b[1− xq − (1− x)q ], (23)

where b is taken as the control parameter. Three typesof transitions, Ising, DP and GV can be observed in thismodel depending on the value of q [35]. Here, we focuson the q = 3 case, for which a unique GV transition atbc = 1/3 has been reported [35].

A. The Langevin equation

In the MF limit (FCN) [36], the fractions of antiparallelneighbors of the two types of spins Si = 1 and Si = −1are x = (1−m)/2 and x = (1+m)/2, respectively. Thus,the transition probabilities are

ω±(m, b) =1∓m

2f

(

1±m

2, b

)

. (24)

Following the same steps as in the previous section, weobtain the Langevin equation

m =1− 3b0

2m(1−m2) + (25)

√

(1−m2) (1 + 6b0 +m2)

N+

9K

4σ2m2(1−m2)2 γ(t).

Let us remark that the potential in the nonlinear votermodel (Fig. 7) differs from that for the Ising model. Ow-ing to the fact that the coefficients of the linear and cubicterm in the deterministic part of Eq. (25) coincide (ex-cept for their sign), the system exhibits a discontinuousjump at the transition point, where the potential min-imum changes directly from m = 0 in the disordered

7

-1 -0.5 0 0.5 1m

-0.02

-0.01

0

0.01

0.02

V

FIG. 7: Potential for the GV transition in a mean field ap-proach. The dashed, solid and dot-dashed lines correspond tothe paramagnetic phase, critical point, and the ferromagneticphase, respectively.

phase to m = ±1 in the ordered one. Furthermore, thepotential vanishes at the critical point [4].

B. Numerical Results

The ordering time, defined as the averaged time re-quired to reach a completely ordered configuration (ab-sorbing state) starting from a disordered configuration,is the equivalent of the crossing time above. We havemeasured the mean ordering time T by both, integratingthe Langevin equation Eq. (25) and running Monte Carlosimulations of the microscopic dynamics on FCNs and fi-nite dimensions. In Fig. 8 we show the MF results. Weobserve that T has a similar behavior to the one foundfor the mean crossing time in the Ising model, and forthe mean extinction time for the contact process [24].That is, a critical scaling T ∼ [lnN ]α at the transitionpoint b0,c = 1/3, with a critical exponent α = 3.68 forσ = 0.3, a logarithmic scaling T ∼ lnN in the absorbingphase b0 < b0,c, and a power law scaling T ∼ N δ withcontinuously varying exponent δ(b0) in the active phaseb0 > b0,c.

Monte Carlo simulations on regular lattices of dimen-sions d = 2 and d = 3 revealed that there is no signifi-cant change in the scaling behavior respect to the puremodel (not shown). The critical point shifts in d = 2and remains very close to its mean-field value in d = 3,but results are compatible with the usual critical (pure)voter scaling T2d ∼ N lnN and T3d ∼ N . In the absorb-ing phase T grows logarithmically with N , while in theactive phase T grows exponentially fast with N , as in thepure-model case. Therefore, in these finite dimensionalsystems we do not find any TGP nor other anomalous ef-fects induced by temporal disorder, although we cannotnumericaly exclude their existence in d = 3. Such effectsshould be observable, only in higher dimensional systems(closer to the mean-field limit).

100

102

104

106

108N

102

104

106

108

Tb

0 = 0.370

b0 = 0.365

b0 = 0.360

b0 = 0.355

b0 = 0.350

b0= 0.345

b0= 0.340

b0,c

= 1/3

b0 = 0.330

MC

10ln(N)

102

104

T

FIG. 8: Main: Log-log plot of the ordering time as a functionof the system size N in the MF q-voter model. Monte Carlosimulations on a FCN (dots) and numerical integration of theLangevin equation Eq. (25) for values of b going from 0.330(bottom) to 0.370 (top), and σ = 0.3 (lines). In the activephase a finite region with power law scaling is observed. Inset:log-log plot of T as a function of lnN . At the critical pointis T ∼ [lnN ]3.68.

C. Analytical results

The ordering time T can be estimated by assumingthat the dynamics is described by the Langevin equa-tion Eq. (25), and calculating the mean first-passage timefromm = 0 to any of the two barriers located at |m| = 1.It turns out useful to consider the density of up spinsrather than the magnetization

ρ ≡ 1 +m

2. (26)

T is the mean first-passage time to ρ = 0 starting fromρ = 1/2. The Langevin equation for ρ is obtained from

Eq. (25), by neglecting the 1/√N term and applying the

ordinary transformation of variables (which is done em-ploying standard algebra, given that Eq. (25) is inter-preted in the Stratonovich sense) is

ρ = A(ρ) +√KC(ρ)γ(t), (27)

with

A(ρ) = a0ρ(2ρ− 1)(1− ρ),

C(ρ) = 3σρ(2ρ− 1)(1− ρ), (28)

where a0 = 1− 3b0.Now, we can follow the same steps as in section III Cfor the Ising model, and find the equation for the meanfirst-passage time T (ρ) by means of the Fokker-Planckequation. The solution is given by (see Appendix B)

T ∼

lnN/(3b0 − 1) for b0 < 1/3(lnN)2/3σ2 for b0 = 1/3

N2(b0−1/3)

σ2 for b0 > 1/3.

(29)

8

These scalings, which qualitatively agree with the nu-merical results of Fig. 8 for the q-voter, show that thebehavior of T is analogous to the one observed in theIsing transition of section III and in the DP transitionfound in [24]. Therefore, we conclude that TGPs appeararound GV transitions in the presence of external varyingparameters in high dimensional systems.For the GV universality class the renormalization

group fixed point is a non-perturbative one [37], becom-ing relevant in a dimension between one and two. A fieldtheoretical implementation of temporal disorder in thistheory is still missing, hence, theoretical predictions andsound criteria for disorder relevance are not available.

V. SUMMARY AND CONCLUSIONS

We have investigated the effect of temporal disorderon phase transitions exhibited by Z2 symmetric systems:the (continuous) Ising and (discontinuous) GV transi-tions which appear in many different scenarios. We haveexplored whether temporal disorder induces TemporalGriffiths Phases as it was previously found in standard(DP) systems with one absorbing state. By performingmean-field analyses as well as extensive computer simu-lations (in both fully connected networks and in finite di-mensional lattices) we found that TGPs can exist aroundequilibrium (Ising) transitions (above d = 2) and arounddiscontinuous (GV) non-equilibrium transitions (only inhigh-dimensional systems). Therefore, we confirm thatTGPs may also appear in systems with two symmetricabsorbing states, illustrating the generality of the under-lying mechanism: the appearance of a region, induced bytemporal stochasticity of the control parameter, wherefirst-passage times scale as power laws of the system sizeand where the susceptibility diverges. Temporal disorder,makes the ordered/active phase less stable and makes thesystem highly susceptible to perturbations. This appearsto be a rather general and robust phenomenon.It also seems to be a general property that TGPs do

not appear in low dimensional systems, where standardfluctuations dominate over temporal disorder. In all thecases studied so far, a critical dimension dc –at and belowwhich TGPs do not appear– exist (dc = 1 for DP tran-sitions, dc = 2 for Ising like systems, and dc ≃ 3 for GVones). Calculating analytically such a critical dimensionand comparing it with the standard critical dimension forthe relevance/irrelevance of temporal disorder at the crit-ical point (i.e. at the renormalization group non-trivialfixed point of the corresponding field theory) remains anopen and challenging task.A relevant application of our results is found in models

of ecosystems. In this case, first-passage times are re-lated to typical extinction times, and studying how suchextinction times are affected by system size (e.g. habitatfragmentation) is a problem of outmost relevance. Fu-ture research might be oriented to the effect of temporaldisorder on the formation and dynamics of spatial struc-

tures.

Acknowledgments

R.M-G. is supported by the JAEPredoc programof CSIC. R.M-G. and C.L. acknowledge support fromMICINN (Spain) and FEDER (EU) through Grant No.FIS2007- 60327 FISICOS. MAM acknowledges finan-cial support from the Spanish MICINN-FEDER underproject FIS2009-08451 and from Junta de AndalucıaProyecto de Excelencia P09FQM-4682. We are grate-ful to J.A. Bonachela for useful discussions and a criticalreading of the manuscript.

Appendix A: Analytical calculations of the crossing

time for the mean field Ising model

The mean first passage time to reach an absorbing bar-rier at |m| = 2/N starting from |m| = 1 can be expressedas [30],

T (m = 1) = 2

∫ m=1

2/N

dy

ψ(y)

∫ 1

y

ψ(z)

Kj20(z)dz, (A1)

with

ψ(z) = exp

∫ z

2/N

dz′2f0(z

′) +Kj0(z′)j′0(z

′)

Kj20(z′)

, (A2)

which involves 6th and 4th order polynomial functions.In order to make the integral simpler, functions are ex-panded up to 3rd order,

f0(m) +K

2j0(m)j′0(m) ≈ m(r − sm2)

Kj20(m) ≈ ωm2, (A3)

with ω ≡ σ2/3, r ≡ a0 + ω/2, s ≡ (c0 + 2ωb20). A secondsimplifying assumption is to take 1 as the lower integra-tion limit in Eq. (A2) instead of 2/N (justified becauseψ(z) appears both in the numerator and the denomina-tor of T (m) and the contribution of this limiting value isnegligible). Therefore, Eq. (A2) becomes

ψ(z) = exp

∫ z

1

2z′(r − sz′2)

ωz′2dz′ = zαeβ(1−z2), (A4)

where α ≡ 2r/ω and β ≡ s/ω. The first passage time iswritten as

T = 2

∫ 1

1/N

I(y)

ψ(y)dy, (A5)

where it has been defined

I(y) =

∫ 1

y

ψ(z)

Kj20(z)dz =

eβ

ω

∫ 1

y

zα−2e−βz2

dz, (A6)

which presents a singularity when α = 1 (b0 = 1 ≡ b0,c).This case will be studied separately.

9

1. Case α 6= 1

Integrating by parts Eq. (A6),

I(y) =eβ

ω

[

e−β − e−βy2

yα−1

α− 1+ 2β

∫ 1

y

zαe−βz2

α− 1dz

]

.

(A7)This integral can be solved again integrating by parts,and so on, recursively,

I(y) =1

ω

∞∑

k=0

(2β)k1− e−β(y2−1)yα−1+2k

∏ki=0 α− 1 + 2i

. (A8)

Therefore,

T =2

ω

∞∑

k=0

(2β)k∏k

i=0 α− 1 + 2i[I1(N)− I2(k,N)] , (A9)

where

I1(N) ≡∫ 1

1/N

y−αeβ(y2−1)dy, (A10)

I2(k,N) ≡∫ 1

1/N

y2k−1dy. (A11)

I1(N) is solved by parts. A recursive integration similarto the one in Eq. (A6) has to be performed,

I1(N) =

∞∑

l=0

(−2β)l[1−Nα−1−2leβ(1/N2−1)]

∏lj=0 α− 1 + 2j

. (A12)

On the other hand, I2(k,N) is easily solved

I2(k,N) =

− ln(N−1) = lnN for k = 0,1−N−2k

2k for k ≥ 1.(A13)

The final expression for the first passage time is

T =2

ω

(

I1(N)− lnN

α− 1

)

+

+2

ω

∞∑

k=1

(2β)k[

I1(N)− (1−N−2k)/2k]

∏ki=0 α− 1 + 2i

,

(A14)

whose asymptotic limit N → ∞ has two different cases.

a. α < 1

In this case, α− 1− 2l < 0 when l ≥ 0 so in I1(N)

1−Nα−1−2leβ(1/N2−1) ∼ 1, (A15)

which leads to

I1(N) =

∞∑

l=0

(−2β)l∏l

j=0 1 + 2j − α≡ C(α, β). (A16)

We have

T ≈2

ω

[

C(α, β)− lnN

α− 1+

∞∑

k=1

(2β)k(

C(α, β) − (2k)−1)

∏l

j=0α− 1 + 2j

]

,

(A17)

and finally,

T ≈ 2

ω(α− 1)lnN. (A18)

b. α > 1.

Considering that Nα−1 ≫ Nα−1−2l, ∀l > 0, only thefirst term is relevant in Eq. (A12) for I1(N). Then

I1(N) ≈ 1− e−βNα−1

1− α≈ e−βNα−1

1− α, (A19)

and in the asymptotic behavior (N ≫ 1) of the meanescape time

T ≈ K(α, β)Nα−1 − 2 lnN

ω(α− 1)∼ Nα−1. (A20)

2. Case α = 1. Critical point

We need to solve

I(y) =

∫ 1

y

ψ(z)

Kj2(z)dz =

eβ

ω

∫ 1

y

y−1e−βz2

dz. (A21)

Expanding the exponential function and integrating, it is

I(y) =eβ

ω

[

− ln y +∞∑

k=1

(−β)k(1− 2y)2k

k!2k

]

. (A22)

Taking Eq. (A22) into Eq. (A5)

T =2eβ

ω

[

I3(N) +

∞∑

k=1

(−β)kk!2k

(I4(N) + I5(k,N))

]

,

(A23)where

I3(N) = −∫ 1

1/N

y−1 ln yeβ(y2−1)dy,

I4(N) =

∫ 1

1/N

y−1eβ(y2−1)dy,

I5(k,N) =

∫ 1

1/N

yk−1eβ(y2−1)dy. (A24)

First of all, let us consider the solution of I3(N) integrat-ing by parts, so that

I3(N) =(lnN)2

2eβ(N

−2−1) + β

∫ 1

1/N

(ln y)2eβ(y2−1)dy,

(A25)

10

and we obtain

I3(N) =(lnN)2

2eβ(N

−2−1) +2β−O(N−1) +O

(

lnN

N

)

,

(A26)which scales in the asymptotic limit as

I3(N) ∼ (lnN)2

2e−β . (A27)

On the other hand, the leading behavior when the size ofthe system is big enough (N ≫ 1) is

I4(N) ∼ e−β lnN + C4(β). (A28)

To solve the last integral,I5(k,N), the exponential func-tion has to be expanded as well. It is

I5(k,N) = e−β∞∑

l=0

βl

l!(k + 2l)

(

1−N−2l−k)

∼ constant,

(A29)when N ≫ 1. It finally leads to an expression for T atcriticality

T ≈2e−β

ω

e−β(lnN)2

2+

∞∑

k=1

(−β)k

k!2k

[

e−β lnN + C′

4(β)]

.

(A30)

In the limit of very large system sizes (N ≫ 1) the meanescape time scales as

T ∼ (lnN)2

ω+

1

ω

∞∑

k=1

(−β)kk!k

lnN +K(β), (A31)

which asymptotically becomes

T ∼ (lnN)2

ω. (A32)

Summing up, the time taken by the system for reachingm = 2/N from an initial condition m = 1 is

T ∼

2ω(α−1) lnN for α < 1,(lnN)2

ω for α = 1,Nα−1 for α > 1.

(A33)

or in terms of the original parameters

T ∼

lnNγ0−1 for b0 < b0,c,3(lnN)2

σ2 for b0 = b0,c,

N6(γ0−1)

σ2 for b0 > b0,c,

(A34)

Appendix B: Analytical calculations of the crossing

time for the mean field nonlinear q-voter model

After performing the change of variables of Eq. (26),the absorbing barrier is placed at ρ = 1/N and the re-flecting one at ρ = 1/2, (which is the initial point). Themean first passage time is given by

T (ρ = 1/2) = 2

∫ ρ=1/2

1/N

dy

ψ(y)

∫ 1/2

y

ψ(z)

Kj20(z)dz, (B1)

with

ψ(z) = exp

∫ z

1/N

dz′2A(z′) +KC(z′)C′(z′)

KC2(z′). (B2)

We expand the polynomials in Eq. (B2) up to secondorder, using Eq. (28), it is

A(ρ) +K

2C(ρ)C′(ρ) ≈ ρ(r − sρ),

KC2(ρ) ≈ ωρ2, (B3)

where we have defined ω ≡ 3σ2, r = w2 − 1 + 3b0 and

s = 3r. These polynomials are similar to the ones ob-tained for the Ising model, but with redefined parame-ters. The integrals are done in a very similar way, and onefinally reaches the following expressions for the crossing(or ordering) time.

T ∼

lnN(3b0−1) for b0 < 1/3(lnN)2

3σ2 for b0 = 1/3

N2(b0−1/3)

σ2 for b0 > 1/3.

(B4)

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