Branching annihilating random walk with long-range repulsion: logarithmic scaling, reentrant phase transitions, and crossover behaviors
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Branching annihilating random walk with long-range repulsion: logarithmic scaling,reentrant phase transitions, and crossover behaviors
Su-Chan Park ( 박 수 ᄎ ᅡ ᆫ ) Department of Physics, The Catholic University of Korea, Bucheon 14662, Republic of Korea (Dated: September 8, 2020)We study absorbing phase transitions in the one-dimensional branching annihilating random walkwith long-range repulsion. The repulsion is implemented as hopping bias in such a way that a particleis more likely to hop away from its closest particle. The bias strength due to long-range interactionhas the form εx − σ , where x is the distance from a particle to its closest particle, 0 ≤ σ ≤
1, andthe sign of ε determines whether the interaction is repulsive (positive ε ) or attractive (negative ε ).A state without particles is the absorbing state. We find a threshold ε s such that the absorbingstate is dynamically stable for small branching rate q if ε < ε s . The threshold differs significantly,depending on parity of the number ℓ of offspring. When ε > ε s , the system with odd ℓ can exhibitreentrant phase transitions from the active phase with nonzero steady-state density to the absorbingphase, and back to the active phase. On the other hand, the system with even ℓ is in the activephase for nonzero q if ε > ε s . Still, there are reentrant phase transitions for ℓ = 2. Unlike the caseof odd ℓ , however, the reentrant phase transitions can occur only for σ = 1 and 0 < ε < ε s . We alsostudy the crossover behavior for ℓ = 2 when the interaction is attractive (negative ε ), to find thecrossover exponent φ = 1 . σ = 0. I. INTRODUCTION
The branching annihilating random walk (BAW) [1, 2]is a nonequilibrium reaction-diffusion system with pairannihilation, A + A → ∅ , and branching, A → ( ℓ + 1) A .Throughout this paper, we reserve ℓ to denote the num-ber of offspring. In a typical setting, particles diffusesymmetrically on a d -dimensional lattice and pair anni-hilation occurs when two particles happen to occupy thesame site. The vacuum state without particles is an ab-sorbing state in that probability current from other statesto the vacuum state is nonzero, while transition from thevacuum state to any other state is prohibited.If branching rarely occurs, then the steady-state par-ticle density (of an infinite system) is zero, which is akey characteristic of an absorbing phase. For sufficientlylarge branching rate, the system can be in an active phasewith nonzero steady-state particle density. The absorb-ing phase transition from the absorbing phase to the ac-tive phase occurs at a certain critical branching rate. Inthis paper, we are interested in critical behaviors of one-dimensional systems and in the following the dimensionis always assumed 1.The universality class to which the BAW belongs dif-fers according to parity of ℓ [3–6]. The BAW with odd ℓ belongs to the directed percolation (DP) universalityclass, as is consistent with the DP conjecture [7, 8]. TheBAW with even ℓ can be mapped to a kinetic Ising modelwith two symmetric absorbing states [9] (see Sec. V Afor a mapping) and belongs to the directed Ising (DI)universality class [10]. For a review of these two uni-versality classes, see, e.g., Ref. [11–13]. In the literature,the DI class is also called the parity-conserving universal-ity class [14–16], DP2 class [11], or the generalized voterclass [17], depending on which property of this universal-ity class is emphasized.Recently, a modified version of the BAW was suggested by introducing hopping bias in such a way that a particleis attracted by its closest particle [18, 19]. The strengthof the bias depends on the distance x between a particleand its closest particle in a power-law fashion x − σ . Forodd ℓ , this bias does not alter the critical behavior [18].On the other hand, for even ℓ , the modified model inone dimension does not belong to the DI class if σ <
1. Rather, the critical exponents vary with σ [19]. InRef. [20], the different critical behavior for σ < ℓ is even. However, a carefulanalysis to be presented in Sec. III shows that the criticalbranching rate can be nonzero for odd ℓ . Rather unex-pectedly, the model exhibits intriguing reentrant phasetransitions, which will be analyzed in Sec. IV for ℓ = 1and in Sec. V for ℓ = 2. The results are summarized inSec. VI. II. MODEL
This section defines the model to be studied and sum-marizes established results that are relevant to later dis-cussion. We consider a one-dimensional lattice of size L with periodic boundary conditions. Each site is eitheroccupied by at most one particle or vacant. A particlebranches with rate q or hops to one of its nearest neigh-bors with rate 1 − q (0 ≤ q ≤ i branches, its ℓ offspring are placed at consecutivesites i + ξ, i + 2 ξ, . . . , i + ℓξ , where ξ is a random variablethat takes 1 or − i hops, it moves to site i ± ± B i ) /
2, where B i = ε sgn( R i − L i ) (min { R i , L i } + µ ) − σ , (1) R i ≡ min { x | s i + x = 1 , ≤ x ≤ L } ,L i ≡ min { x | s i − x = 1 , ≤ x ≤ L } . In the above equation, sgn( x ) is the sign of x withsgn(0) = 0, σ is nonnegative, and ε , µ are constantswith restriction | ε | < (1 + µ ) σ to ensure − < B i <
1. Iftwo particles happen to occupy a same site by any tran-sition event, then pair annihilation ( A + A → ∅ ) of thetwo particles occurs immediately.The stochastic dynamics of the model can be repre-sented as1 i ℓ Y k =1 c i + k → i ℓ Y k =1 ¯ c i + k with rate q , i ℓ Y k =1 c i − k → i ℓ Y k =1 ¯ c i − k with rate q , i c i +1 → i ¯ c i +1 with rate (1 − q ) 1 + B i , i c i − → i ¯ c i − with rate (1 − q ) 1 − B i , (2)where c j stands for the occupation number at site j with¯ c j ≡ − c j and 1 i (0 i ) means that site i is occupied(vacant). Monte Carlo simulations for the rule (2) wereperformed in the following way. Assume that there are M particles at time t . Choose one among M particles atrandom with equal probability. Assume that a particle atsite i is chosen. After the choice, we generate a randomnumber θ that is uniformly distributed in 0 < θ <
1. If θ < q/
2, then ℓ offspring are placed on the right-handside of i . Else if θ < q , then ℓ offspring are placed onthe left-hand side of i . Else if θ < q + (1 − q )(1 + B i ) / /M .The model will be called the BAW with long-rangeinteraction (BAWL). For positive (negative) ε , a parti-cle is in a sense repulsed (attracted) by its closest par-ticle located at min { R i , L i } . Thus, we will refer tothe model with positive (negative) ε as the BAW withlong-range repulsion ( attraction ) to be abbreviated asBAWLR (BAWLA).We define the particle density ρ ( t ) as ρ ( t ) = lim L →∞ L L X i =1 h c i i , (3) where c i is the occupation number at site i at time t and h· · · i stands for average over ensemble. In actualsimulations, the system size L is so large that a finite-size effect is negligible up to the observation time and thesystem evolves from the fully-occupied initial conditionwith ρ (0) = 1.The BAWL with q = 0 is the annihilating random walkwith long-range interaction (AWL), extensively studiedin Ref. [21] (see also Refs. [19, 22] for the AWL withattractive interaction). We summarize some results ofRef. [21] that are relevant to later discussion. For thetwo-particle initial condition in which there are only twoparticles in a row at t = 0, the probability P s that thetwo particles are never annihilated is P − s = 1 + ∞ X i =1 i Y k =1 − ε ( k + µ ) − σ ε ( k + µ ) − σ . (4)For σ = 0 and σ = 1, P s becomes P s ( σ = 0) = 2 ε ε Θ( ε ) , (5) P s ( σ = 1) = 2 ε − µ + ε Θ(2 ε − , (6)where Θ( · ) is the Heaviside step function. For 0 < σ < P s is nonzero for ε > ε ≤
0. When necessary,one can numerically calculate P s for 0 < σ < i − r , i , i + r and particles at i ± r are assumed immobile, the time T ( r ) taken for the particle in the middle to encounterone of the two particles at i ± r behaves for large r as T ∼ r σ exp (cid:0) C σ r − σ (cid:1) , σ < ,r ε , σ = 1 & 2 ε > ,r ln r, σ = 1 & 2 ε = 1 ,r , otherwise. (7) C σ ≡ ( ln[(1 + ε ) / (1 − ε )] , σ = 0 , ε/ (1 − σ ) , < σ < , (8)for positive ε and T ∼ ( r σ , σ ≤ ,r , σ > , (9)for negative ε .Interpreting 2 /T with r = 1 /ρ ( t ) as the rate of thedensity decrease per particle at time t , it was found [21]that ρ ( t ) ∼ (ln t ) − / (1 − σ ) , σ < ε > ,t − / (1+2 ε ) , σ = 1 & 2 ε > ,t − / , otherwise, (10)for ε > ρ ( t ) ∼ ( t − / (1+ σ ) , σ ≤ ,t − / , σ ≥ , (11)for ε < σ = 0 and positive ε , whichclaims that ρ ( t ) behaves as (ln t ) − b with b to vary contin-uously, depending on ε . We think, however, the neglectof ln ln t correction [21] errorneously gives this conclusion. III. STABILITY OF THE ABSORBING STATE
Since P s for σ < ε > σ < q is nonzero.To check the validity of this claim, we investigate thestability of the absorbing state for small q .To this end, we consider an initial condition that eachsite is occupied with probability ρ . Assuming both ρ and q are small (0 < ρ ≪ , q ≪ ℓ = 1 and general cases will be discussedlater.The rate of density decrease due to pair annihilationis roughly 2 /T ( r ) with r = 1 /ρ [21]. To estimate therate of density increase by branching, we assume thata particle at site i , to be called the parent, begets itsoffspring at t = 0. With probability P s the offspringwill escape from the parent and the number of particlesincreases by 1, while with probability 1 − P s the parentand the offspring undergo pair annihilation, which makesthe number of particles decrease by 1. Hence, the rateof density increase by branching is q [ P s − (1 − P s )] = q (2 P s − d ln ρdt ≈ q (2 P s − − T (1 /ρ ) . (12)If P s < /
2, then the branching triggers even faster(exponential) density decay, which makes the absorbingstate stable. On the other hand, if P s > /
2, then theabsorbing state is unstable, because 1 /T → ρ →
0; see Eq. (7). Hence the condition for the absorbingstate to be stable is P s < /
2. As we will see soon, thecondition will differ significantly for ℓ = 2.Unlike the claim at the beginning of this section, thereshould be a positive threshold ε s determined by P s ( ε s ) = such that for 0 < ε < ε s , the steady-state density forsmall q is zero and, in turn, the critical point q c shouldbe nonzero. From Eq. (4), ε s is found as ε s = / , σ = 0 , .
438 213 752 , σ = 0 . , µ = 1 , ( µ + 2) / , σ = 1 , (13) . . . . ρ t δ D P t . . . FIG. 1. Semilogarithmic plots of ρt − δ DP vs. t for σ = 0 . ε = 0 . µ = 0, and ℓ = 1 with q = 0 . δ DP ≈ .
159 46 is the critical decayexponent of the DP class. The curve for q = 0 . q c = 0 . where the numerical value of ε s for σ = 0 . P s < /
2, Eq. (12) suggests that the densitydecreases exponentially for small but nonzero q , whichis a typical feature of the absorbing phase of systemsbelonging to the DP class. This is usually interpreted asthe generation of spontaneous annihilation by the chainof reaction A → A →
0. Hence, we anticipate that theBAWL with ℓ = 1 belongs to the DP class if P s < / σ = 0 . ε = 0 .
5, and µ = 0 with L = 2 . P s for this caseis about 0.445. If this case does indeed belong to the DPclass, then the density at the critical point should decayas ρ ( t ) ∼ t − δ DP , (14)with the critical decay exponent δ DP = 0 .
159 46 [24]of the DP class. In Fig. 1, we depict ρt δ DP versus t on a semilogarithmic scale for q = 0 . , . q = 0 . q = 0 . q c = 0 . P s > /
2, thesteady state should have nonzero density for small butnonzero q . Setting the time derivative of ρ in Eq. (12) tobe 0, the steady-state density ρ s can be estimated by T (1 /ρ s ) ∼ /q, (15) − − ε = 0 . ℓ = 1 ε = 0 . ℓ = 2 ε = 0 . ℓ = 2 q − / ( + ε ) ρ qt FIG. 2. Data collapse plots of q − / (1+2 ε ) ρ vs. qt for ε = 0 . ℓ = 2 (top), ε = 0 . ℓ = 2 (middle), and ε = 0 . ℓ = 1(bottom) on a double logarithmic scale. Here, σ is set 1. q for ε = 0 . − to 5 × − and q for ε = 0 . ℓ = 1 and ℓ = 2) ranges from 10 − to 5 × − . Todistinguish collapse plots from each other, we multiply ρ byan arbitrary constant for each parameters set. which, along with Eq. (7), gives ρ s ∼ ( ( − ln q ) − / (1 − σ ) , σ < ,q / (1+2 ε ) , σ = 1 , (16)for small q and ε > ε s . Hence, there is a continuoustransition with q c = 0.Now, we make an ansatz for a scaling function thatdescribes the absorbing phase transition with q c = 0.From the above discussion, 1 /q should be a characteristictime scale of the system. For σ = 1, the density wouldbe described by a scaling function of the form ρ ( t ; q ) = q / (1+2 ε ) G ( qt ) . (17)To ensure that ρ ( t ; q ) has the desired behavior as t → ∞ ,the scaling function G should have the following asymp-totic behavior: G ( x ) ∼ x → ∞ and G ( x ) ∼ x − / (1+2 ε ) as x → x → t → ∞ and qt → σ <
1, the logarithmic behavior should be incorpo-rated properly into the scaling function. Since the char-acteristic time scale is 1 /q and there is a logarithmic den-sity decay at q = 0, a proper scaling parameter shouldbe − ln t/ ln q . Thus, we write ρ ( t ; q ) = ( − ln q ) − / (1 − σ ) G σ ( − ln t/ ln q ) , (18)for σ <
1. The desired asymptotic behaviors for large t are reproduced if G σ ( x ) ∼ x − / (1 − σ ) for small x and G σ ( x ) ∼ x . Notice that if we choose qt asa scaling parameter as in Eq. (17), we cannot reproducethe desired asymptotic behavior in the small qt limit.Before confirming the validity of the scaling ansatz bynumerical simulations, we will discuss the stability of theabsorbing state for ℓ >
048 0 0 . . . . (a)(b) ( − l n q ) ρ − ln t/ ln q − − − ( − l n q ) ρ − ln t/ ln q − − − FIG. 3. Plots of ( − ln q ) / (1 − σ ) ρ vs − ln t/ ln q for (a) σ = 0, ℓ = 1 and (b) σ = 0 . ℓ = 2. Here, we set µ = 0 and ε = 0 . t , all data are well collapsed onto a single curve. Right after a particle branches, it and its offspring forma cluster of size ℓ +1. Particles in the middle of the clusterare likely to be annihilated in short time. If ℓ is odd, onlytwo particles in the cluster are likely to remain in shorttime and the fate of the cluster is almost identical to thecase of ℓ = 1. Hence the scenario for ℓ = 1 is applicableto the BAWL with odd ℓ , though the value of ε s candiffer depending on ℓ .If ℓ is even, then the size of the cluster shrinks to 3 inshort time. By ˜ P s , we denote the probability that no pairannihilation happens in the AWL if the system evolvesfrom a configuration only with three particles in a row.That is, with probability ˜ P s all the three particles surviveand with probability 1 − ˜ P s only one particle survives.Since at least one particle always survives, the rate ofthe density increase by branching is 2 ˜ P s , which gives d ln ρdt ≈ q ˜ P s − T . (19)Now we discuss the relation between P s and ˜ P s . Sincethe particle in the middle is more likely to be annihi-lated than particles in the two-particle initial condition,˜ P s cannot be larger than P s . In the mean time, if P s > P s > P s >
0. Therefore, ˜ P s is positive if andonly if P s is positive. Accordingly, if P s is positive and ℓ is even, then the absorbing state is unstable and thescaling functions (17) and (18) are expected to describethe critical phenomena. If P s = ˜ P s = 0 and q is small,then Eq. (19) shows that only pair annihilation governsthe long-time behavior of the particle density. . . . . ρ t δ D I t FIG. 4. Semilogarithmic plots of ρt δ DI vs. t for q = 0 . . . ℓ = 2, σ = 1, µ = 0 .
2, and ε = 0 . δ DI is the criticaldecay exponent of the DI class with numerical value 0 . q = 0 .
167 is flat for more than two logarithmicdecades, while the other curves eventually veer up (0.169:top curve) or down (0.165 : bottom curve), indicating thatthe transition point is q c = 0 . To confirm the scaling ansatz as well as the scenariofor even ℓ , we performed Monte Carlo simulations fortwo cases, ℓ = 1 and ℓ = 2. First, we present simulationsresults for σ = 1 and µ = 0. Figure 2 depicts data-collapse plots using Eq. (17) for ε = 0 . ℓ = 2) and ε = 0 . ℓ = 1 and ℓ = 2). Notice that ε s for ℓ = 1 and µ = 0 is ≈ .
67. The system size in the simulations is L = 2 and number of independent runs ranges from 8to 40. The collapse is almost perfect, which confirms thescaling ansatz (17).Next, we present simulation results for σ = 0 ( ℓ = 1)and σ = 0 . ℓ = 2). For both cases, we set µ = 0 and ε = 0 .
5. The values of q are 10 − , 10 − , and 10 − . Thesystem sizes in the simulations are 2 and 2 for σ = 0and σ = 0 .
5, respectively, The number of independentruns is 8 for all cases. Figure 3 depicts data-collapseplots according to the scaling ansatz (18), which shows anice agreement.Now we discuss absorbing phase transitions of theBAWLR with ℓ = 2 for σ = 1 and 0 < ε < .
5. Inthis regime, P s = 0 and the density decays as t − / forsmall q ; see Eq. (19) along with Eq. (10). Hence, it canbe anticipated that there should be a phase transition atnonzero q c and this case should belong to the DI class.To confirm, we simulated the system of size L = 2 with σ = 1, µ = 0 .
2, and ε = 0 .
35 for q = 0 . . . ρt δ DI as a function of t on a semilogarithmic scale, where δ DI ≈ . q = 0 .
165 (0.169) eventuallyveers down (up), we conclude that the critical point is
TABLE I. Critical points ε c of the BAWL with ℓ = 1 for σ = 0 ( µ = 0; second column), 0 . µ = 1; third column), and1 ( µ = 1; fourth column). Numbers in parentheses indicateuncertainty of the last digits. q ε c σ = 0 σ = 0 . σ = 10 . . . . . . .
452 85(5) 1 . .
001 0 . .
469 75(2) 1 . .
003 0 . . . .
01 0 .
363 40(2) 0 .
526 24(3) 1 . .
03 0 .
403 08(3) 0 .
559 24(2) 1 . . . . . . .
500 05(4) 0 . . . .
516 35(5) 0 .
604 43(3) 1 .
246 15(5)0 . . .
596 05(5) 1 . . .
513 65(5) 0 . . . . . . . . . . . . . .
716 65(5)0 . − . − . − . q c = 0 . IV. SINGLE OFFSPRING
In this section, we continue studying absorbing phasetransitions of the BAWLR with ℓ = 1. When it comesto the universal behavior, nothing new beyond the resultin Sec. III is expected. Rather, we are interested in thebehavior of the phase boundary as ε approaches ε s .To this end, we performed Monte Carlo simulations for σ ≤
1. Differently from the numerical studies in Sec. III, q is now fixed and ε is a tuning parameter. The criticalpoint will be denoted by ε c . Since the model belongsto the DP class, the critical point was found using themethod as in Fig. 1 (details not shown here).We simulated the BAWLR for three sets of parameters, { σ = 0 , µ = 0 } , { σ = 0 . , µ = 1 } , and { σ = 1 , µ = 1 } .The threshold values for these parameter sets are givenin Eq. (13). The critical points are tabulated in Table I.It is intriguing that ε c approaches ε s from above. Since ε c becomes smaller than ε s for sufficiently large q , thereshould be a reentrant phase transition as q varies with ε to be fixed slightly larger than ε s .To illustrate the reentrant phase transitions, we depictphase boundaries ( ε c versus q ) in Fig. 5(a). There aretwo salient features of the phase boundaries. First, thereentrant phase transitions indeed occur for ε s < ε < ε b .We roughly estimate ε b as 0.521, 0.605, and 1.26, for σ = 0, 0 .
5, and 1, respectively, from Table I. Second, thebehavior of the phase boundary for small q varies with σ . To be concrete, we analyzed how ε c approaches ε s as q →
0. In Fig. 5(b), ε c − ε s is drawn against q ona double logarithmic scale, along with power-law fitting . . . . . . . . − − − − − − (a)(b) ε c q ε c − ε s q . FIG. 5. (a) Plots of ε c vs. q for σ = 0 (circle), σ = 0 . σ = 1 (square), bottom to top. Error bars arehardly observable in this scale. (b) Log-log plots of ε c − ε s vs. q for σ = 0 (circle), σ = 0 . σ = 1 (square),bottom to top. Line segments with slope 0.18, 0.66, 1.15 (topto bottom) show the results of power-law fitting. results for small q . We find ε c − ε s ∼ q φ (20)with φ = 1 .
15, 0 .
66, and 0 .
18 for σ = 0, 0.5, and 1,respectively. This power-law behavior is reminiscent of acrossover behavior.Now we provide a hand-waving argument as to whythere are reentrant phase transitions. Let us revisit theinitial condition with low density as in Sec. III. Assumethat a particle at site i branches its offspring at site i + 1at time t = 0. If these two particles survive, then nextbranching occurs approximately at time 1 /q . We assumethat the parent is at site j and it branches a secondoffspring at site j + 1. Since the mean distance of thesurviving samples with the two-particle initial conditionincreases as t / (1+ σ ) [21], the average distance betweenthe parent and its first offspring at the time of the sec-ond birth is d ≈ q − / (1+ σ ) . Now there are three particlesat site j , j + 1, and j + d .Let P m ( d ) be the probability that three particles sur-vive forever if the system (with q = 0) evolves from theinitial condition with only three particles at sites j , j + 1and j + d . Obviously, P m ( d ) should be an increasingfunction of d with P m ( d ) → P s as d → ∞ . Accordingly,it is not impossible for P m ( d ) to be smaller than evenif P s is larger than .The rate w of particle-number change due to the twobranching events is w = − q (1 − P s ) + qP s [1 − q (1 − P m ) + qP m ]= q (2 P s − − q P s (1 − P m ) , (21) which modifies Eq. (12) as d ln ρdt ≈ q (2 P s − − q P s (1 − P m ) − T . (22)If P s > / q is extremely small but nonzero, theabsorbing state is unstable as we have seen. Since P m isa decreasing function of q (recall that d is a decreasingfunction of q ), P m may be smaller than from certain q and the stability of the absorbing state can change at,say, q , which can be found approximately as a solutionof q (1 − P m ) = 2 P s − , (23)where P m should be regarded as a function of q . Sincethe system should be in the active phase if q is close to 1,the stability of the absorbing state changes again at, say, q ( q > q ), which explains the existence of reentrantphase transitions.In mathematics literature, a model is called attrac-tive if, roughly speaking, more particles induce longer lifetime (and/or larger chance of survival) of the system. Atypical example of an attractive model is the contact pro-cess [26, 27]. The BAW is not attractive [1] and neither isthe BAWL. Thus, larger branching rate does not neces-sarily imply more chance of survival. In the BAWLR, wehave shown that larger branching rate sometimes makesthe system fall into an absorbing state. To our knowl-edge, the BAWLR is the first model that exhibits a reen-trant phase transition due to the lack of attractiveness.It is worth commenting that the reentrant phase tran-sition observed in Ref. [28] should be attributed to theabsence of the active phase in the branching annihilat-ing process [2], because the reentrant phase transitiondisappears when dynamic branching is employed [28]. V. TWO OFFSPRING
In this section, ℓ is always set to be 2 and we do notexplicitly mention the value of ℓ when we refer to themodel. If σ < ε > σ = 1 and 2 ε >
1, that is,if P s >
0, then the absorbing state is dynamically unsta-ble and the critical phenomena are completely describedby the scaling functions (17) and (18). Hence, this sec-tion is interested in phase transitions in the cases with P s = 0.For σ <
1, the condition P s = 0 is applicable onlyto the BAWLA. Recall that the BAWLA with nonzero ε does not belong to the DI class [18, 19]. Thus, thereshould be a crossover behavior for small | ε | , which is thetopic of Sec. V A.For σ = 1, P s can be zero even if ε > ε . These reentrant phase transitionswill be studied in Sec. V B. . . . . . . (a)(b) N t ρ t δ t FIG. 6. (a) Semilogarithmic plots of N vs. t for ε = 0 and ℓ = 2 around the critical point. The curve for q = 0 .
489 532(top) veers up for large t , while the curve for q = 0 .
489 53(bottom) veers down, which yields q = 0 .
489 531(1). (b)Semilogarithmic plot of ρt δ vs. t for the BAWLA with ℓ = 2around the critical point for σ = 0 and ε = − .
16 with δ =0 . q = 0 .
612 29 (top) veers up for large t , while the curve for q = 0 .
612 29 (bottom) veers down. Weconclude q c = 0 .
612 27(2).
A. Crossover in the BAWLA with σ = 0 The crossover behavior is described by a scaling func-tion ρ ( t ; q, ε ) = t − δ DI R h t | ε | ν k /φ , ( q − q ) t /ν k i , (24)where q is the critical point for ε = 0; δ DI and ν k are thecritical exponent of the DI class; and φ is the crossoverexponent. In a recent study [29], ν k is found to be 3.55and we will use this value for a data-collapse plot.To find the crossover exponent, we analyze how thephase boundary behaves when | ε | is small. If we denotethe critical point for negative ε by q c , Eq. (24) gives | q c − q | ∼ | ε | /φ , (25)which will be used to find the crossover exponent φ .We first find q for ε = 0. In the literature, the value of q is available (for example see Refs. [28, 30]), but we willprovide more accurate value in this paper. To compareour estimate to the literature, one should notice that p is usually used to denote the hopping probability, whichis 1 − q in this paper.To find q , we performed simulations with the two-particle initial condition. At the critical point, the num-ber N ( t ) of particles averaged over all ensemble behavesfor large t as N ( t ) ∼ t η . (26) TABLE II. Critical points q c of the BAWLA with ℓ = 2 for σ = 0. Numbers in parentheses indicate uncertainty of thelast digits. ε q c .
489 531(1) − . . − .
005 0 .
497 15(15) − .
01 0 . − .
02 0 . − .
04 0 . − .
08 0 . − .
16 0 .
612 27(2)
Since η ≈ N ( t ) drawn against t should veer up (down) if the system is in the active(absorbing) phase. Thus, we will find q by plotting N as a function of t on a semilogarithmic scale around thecritical point.In Fig. 6(a), we present the simulation results. Thenumber of independent runs for each parameter is 5 × .Since the graph for q = 0 .
489 532 (0 .
489 53) veers up(down) in the long-time limit, we conclude that q =0 .
489 531(1). Note that this estimate is more accuratethan previous estimates in the literature (for example, inRef. [30], the critical point was estimated as 1 − p c =0 .
489 65(5), but this estimate was based on somewhatwrong value of the critical decay exponent).When ε < σ = 0, the BAWLA does not belongto the DI class [18, 19]. For example, the critical decayexponent for this case is δ = 0 . δ DI = 0 . σ = 0 indeed forms a universality class. InFig. 6(b), we present simulation results for ε = − .
16 at q = 0 .
612 25, 0.612 27, and 0.612 29. The system size inthe simulation is L = 2 and the numbers of indepen-dent runs are 160, 320, 160 for q = 0 .
612 25, 0.612 27,0.612 29, respectively. We depict ρt δ as a function of t .At q = 0 .
612 27, the curve is almost flat for more thantwo logarithmic decades, which gives q c = 0 .
612 27(2)and confirms that the critical behavior of the BAWLAwith σ = 0 is indeed universal.We found critical points by extensive Monte Carlo sim-ulations for various ε ’s, which are tabulated in Table II.In Fig. 7(a), we plot q c − q as a function of | ε | on adouble logarithmic scale. We fit the data for small | ε | using a power-law function to get 1 /φ = 0 . φ = 1 . q is fixed to be q , the scaling function (24) suggeststhat plots of ρt δ DI against t | ε | ν k /φ should collapse ontoa single curve for small | ε | . Using ν k = 3 .
55 [29], weget ν k /φ ≈ .
15. A scaling-collapse plot can be used tocheck the consistency of the estimate of φ . Figure 7(b)shows such a scaling-collapse plot for ε = − × − , − − , − × − , and − × − . The system size inthe simulation is L = 2 and the number of independentruns for the smallest | ε | is 400. Indeed, the data arewell collapsed onto a single curve, which also indirectly − − − − − − − − − − − (a)(b) q c − q | ε | ρ t δ D I t | ε | ν k /φ FIG. 7. (a) Plot of q c − q vs. | ε | for the BAWLA with σ = 0and ℓ = 2 on a double logarithmic scale. The size of errorbars are smaller than the symbol size. The straight line withslope 0.89 shows the result of power-law fitting. (b) Scalingcollapse plot of ρt δ DI vs. t | ε | ν k /φ at q = q = 0 .
489 531 for ε = − × − , − − , − × − , and − × − on a doublelogarithmic scale. All data collapse nicely onto a single curvefor large t . supports the recent estimate of ν k in Ref. [29].It is worthwhile to comment on another form of attrac-tion that can be found in the literature. The bias dueto long-range attraction with σ = 0 superficially lookssimilar to a bias due to a symmetry breaking field intro-duced to a DI model [31]. To be concrete, let us considera mapping from the BAW with ℓ = 2 to a nonequilibriumkinetic Ising model (NEKIM) such that a particle in theBAW is interpreted as a domain wall in the kinetic Isingmodel. Details of the mapping are summarized in Ta-ble III. We only consider the initial condition with order-ing . . . A + A − A + A − . . . , which is preserved by dynamics.Note that the BAW in this paper corresponds to the rate D = D = p/ λ = (1 − p ) / D > D , then hoppingof A + ( A − ) particle are more likely to hop to the right(left), which effectively enlarges a domain of upspins inthe NEKIM. This bias can be interpreted as an effectof external magnetic field coupled to Ising spins, whichbreaks the symmetry between the two absorbing states.In the presence of the symmetry breaking field, the modelnow belongs to the DP class and its crossover exponentis 1 /φ ≈ .
47 or φ ≈ . TABLE III. A mapping of the BAW ( ℓ = 2) to a nonequilib-rium kinetic Ising model. ↑ and ↓ stand for Ising spins withspin 1 and −
1, respectively. A + and A − are interpreted asdomain walls ↑↓ and ↓↑ , respectively. In branching events, aparent is indicated by a bold face symbol. If D = D , A + and A − are dynamically indistinguishable.BAW NEKIM transition rate A + ∅ ⇒ ∅ A + ↑↓↓⇒↑↑↓ D ∅ A − ⇒ A − ∅ ↓↓↑⇒↓↑↑ D A − ∅ ⇒ ∅ A − ↓↑↑⇒↓↓↑ D ∅ A + ⇒ A + ∅ ↑↑↓⇒↑↓↓ D A + A − ⇒ ∅∅ ↑↓↑⇒↑↑↑ D A − A + ⇒ ∅∅ ↓↑↓⇒↓↓↓ D A + ∅∅ ⇔ A + A − A + ↑↓↓↓⇔↑↓↑↓ λ A − ∅∅ ⇔ A − A + A − ↓↑↑↑⇔↓↑↓↑ λ A + A − ∅ ⇔ A + ∅ A − ↑↓↑↑⇔↑↓↓↑ λ A − A + ∅ ⇔ A − ∅ A + ↓↑↓↓⇔↓↑↑↓ λ ∅∅ A + ⇔ A + A − A + ↑↑↑↓⇔↑↓↑↓ λ ∅∅ A − ⇔ A − A + A − ↓↓↑↑⇔↓↑↓↑ λ ∅ A − A + ⇔ A − ∅ A + ↓↓↑↓⇔↓↑↑↓ λ ∅ A + A − ⇔ A + ∅ A − ↑↑↓↑⇔↑↓↓↑ λ This can be clearly explained by the different values ofthe corresponding crossover exponents, 1 /φ = 0 .
89 forthe long-range attraction and 1 /φ = 0 .
47 for the sym-metry breaking field. Besides, changing the directionof the external magnetic field ( h
7→ − h ) does not in-duce new phenomena, because the dynamics are invari-ant under the simultaneous transformations h
7→ − h and A + ↔ A − . This should be contrasted with sign change of ε in the BAWL, which has a drastic effect as we observed. B. Reentrant phase transition for σ = 1 Now we move on to the BAWLR with σ = 1 and ε < .
5. As shown in Fig. 4, this case belongs to theDI class. An intriguing phenomenon was observed as q varies for ε = 0 .
35 and µ = 0 .
2. In Fig. 8, we depict thebehavior of the density for q = 10 − , 10 − , 0.1, and 0.2.Since we have found q c ≈ .
165 in Fig. 4, it is natural toexpect that the density decays as t − . for q = 0 . q = 0 .
2, which is indeed the case. A rathersurprising result is the behavior for q = 10 − , which hasnonzero steady-state density. Since the absorbing stateis stable for very small q as the curve for q = 10 − inFig. 8 shows, there should be different transitions pointsin three regions, 0 < q < .
01, 0 . < q < .
1, and0 . < q < .
2. We did not dare to find the phase bound-ary with accuracy. Our preliminary simulations showedthat the reentrant phase transitions occur for some nar-row region of ε , which is roughly 0 . < ε < . q , the discussion in Sec. IIIshows that the absorbing state is stable. As q increases,the steady-state density starts to be nonzero from a cer- − − − ρ t . . − − FIG. 8. Double-logarithmic plots of ρ vs. t for the BAWLRwith σ = 1, ℓ = 2, µ = 0 . ε = 0 .
35 for q = 10 − (dot-dashedline), 10 − (dotted line), 0 . tain q . And then competition between the nonattrac-tiveness and the branching results in a reentrant phasetransitions in the region q > q . But we do not have agood quantitative argument about the region of ε , wherethe reentrant phase transitions occur. We would like toleave this intriguing question to later research. VI. SUMMARY
In this paper, we have studied absorbing phase tran-sitions in the branching annihilating random walk withlong-range interaction with main focus on the effect ofrepulsion. The results in Ref. [21], especially P s , areimportantly relied on to understand the behavior of the BAWL.When the number ℓ of offspring per branching is 1,the condition that the absorbing state is stable againstsmall perturbation of branching is found to be P s ≤ .Even though P s > , there can be reentrant phase tran-sitions from the active phase, to the absorbing phase,and back to the active phase as the branching rate q increases from zero. This reentrant phase transitionsare explained, based on the nonattractiveness [1] of theBAWLR. The critical phenomena at nontrivial transitionpoint are described by the DP scaling. We also arguedthat a similar conclusion should be arrived at for any oddnumber of ℓ .When ℓ = 2, unlike the case with ℓ = 1, there is onlythe active phase for q > P s >
0. Only when P s = 0,the transition point q c can be nonzero. For the case with P s = 0, we studied two cases; σ = 0 with ε < σ = 1with ε < . For the first case, we studied the crossoverfrom the DI class to the class with σ = 0. We foundthe crossover exponent to be φ = 1 . q c = 0, we suggested the scaling functionsaround q = 0. The scaling function for σ < σ = 1 in that a logarithmic scaling appearsfor σ <
1; see Eqs. (18) and (17) for σ < σ = 1,respectively. These scaling functions are confirmed bynumerical simulations. ACKNOWLEDGMENTS
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