Universality-class crossover by a nonorder field introduced to the pair contact process with diffusion

Universality-class crossover by a nonorder field introduced to the pair contact process with diffusion

Su-Chan Park (박수찬) Department of Physics, The Catholic University of Korea, Bucheon 14662, Republic of Korea
July 17, 2019
Abstract

The one-dimensional pair contact process with diffusion (PCPD), an interacting particle system with diffusion, pair annihilation, and creation by pairs, has defied a consensus about the universality class that it belongs to. An argument by Hinrichsen [H. Hinrichsen, Physica A 361, 457 (2006)] claims that freely diffusing particles in the PCPD should play the same role as frozen particles, when it comes to the critical behavior. Therefore, the PCPD is claimed to have the same critical phenomena as a model with infinitely many absorbing states that belongs to the directed percolation (DP) universality class. To investigate if diffusing particles are really indistinguishable from frozen particles in the sense of the renormalization group, we numerically study a variation of the PCPD by introducing a nonorder field associated with infinitely many absorbing states. We find that a crossover from the PCPD to the DP occurs due to the nonorder field. Since, by studying a similar model, we exclude the possibility that mere introduction of a nonorder field to one model can entail a nontrivial crossover to another model in the same universality class, we attribute the observed crossover to the difference of the universality class of the PCPD from the DP class.

pacs:
05.70.Ln,05.70.Jk,64.60.Ht
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I Introduction

The pair contact process with diffusion (PCPD) is an interacting particle system with diffusion, pair annihilation (), and creation by pairs (), which exhibits an absorbing phase transition. The PCPD was first studied by Grassberger in 1982 Grassberger (1982), but it had not attracted much interest until Howard and Täuber Howard and Täuber (1997) introduced a so-called ‘bosonic’ PCPD. Since dynamics in the bosonic PCPD drives particle density in the active phase to infinity in finite time, a more controllable model was desired to study the phase transition. Carlon et al. Carlon et al. (2001) introduced a present form of the PCPD on a lattice with hard core exclusion to keep the density from diverging. The conclusion in Ref. Carlon et al. (2001) was already controversial, which has triggered a lot of numerical and analytical studies of the PCPD Hinrichsen (2001a); Ódor (2000); Park et al. (2001); Hinrichsen (2001b); Park and Kim (2002); Dickman and de Menezes (2002); Kockelkoren and Chaté (2003); Ódor (2003); Barkema and Carlon (2003); Janssens et al. (2004); Park and Park (2005a, b); Hinrichsen (2006); Park and Park (2006); Kwon and Kim (2007); Smallenburg and Barkema (2008); Park and Park (2009); Schram and Barkema (2012); Gredat et al. (2014); Park (2014).

It turned out that the PCPD, especially in one dimension, is influenced by strong corrections to scaling. This, in turn, makes it difficult for numerical studies to lead to a consensus about the universality class of the one dimensional PCPD. Accordingly, many scenarios had been suggested in the early stage of research (for a review of early discussions, see Henkel and Hinrichsen (2004); Park and Park (2008a)). The controversy remains unabated, but by now only two competing theories have survived.

In one theory, the PCPD is claimed to form a different universality class from the directed percolation (DP) universality class Kockelkoren and Chaté (2003); Park and Park (2005a, b, 2006, 2009); Park (2014). This theory is supported by the following facts: First, the upper critical dimension of the PCPD is 2 Ódor et al. (2002), while that of the DP is 4. Second, diffusion bias changes the universality class of the one-dimensional PCPD, whereas such a bias can be asymptotically removed by a Galilean transformation in the DP class Park and Park (2005a). Third, there are nontrivial crossover behaviors between the PCPD and the DP models Park and Park (2006, 2009). In the other theory, the PCPD is claimed to belong to the DP class Barkema and Carlon (2003); Hinrichsen (2006); Smallenburg and Barkema (2008); Schram and Barkema (2012). An argument to support this theory was put forward by Hinrichsen Hinrichsen (2006), which will be called the DP argument for later reference.

Since the DP argument has motivated the present work, we repeat it here for completeness. The DP argument is based on a numerical observation that the dynamic exponent of the PCPD in one dimension is smaller than 2. Since critical clusters spread as and isolated particles spread diffusively as , a comparison of these two scales suggests that the critical spreading will eventually dominate the critical behavior over the diffusive spreading and, accordingly, diffusion is irrelevant in the sense of the renormalization group (RG). Although it is in principle possible for a pair to be formed purely by diffusion of isolated particles, the irrelevance of diffusion implies that such events will hardly occur in the long time limit. Thus, it does not matter whether a particle can diffuse or not, as far as critical phenomena are concerned. If particles are not allowed to diffuse, the model is the pair contact process (PCP) Jensen (1993), which has infinitely many absorbing states (IMAS) and belongs to the DP class. In this sense, diffusion can at best affect corrections to scaling and the one-dimensional PCPD should exhibit the same universal behavior as the PCP or, in the context of universality, as the DP.

The aim of this paper is to figure out if diffusing particles in the PCPD are really indistinguishable in the RG sense from the frozen isolated particles of the PCP, as claimed in the DP argument. If isolated particles can be regarded as frozen particles of the PCP, introducing dynamics which make diffusing particles frozen should not yield any other singular behavior than the typical DP critical behavior. Motivated by this idea, we introduce and study numerically a variation of the PCPD in which diffusing particles can mutate irreversibly to immobile species with rate . The case with will correspond to the PCPD and the case with nonzero , which includes the PCP (), has IMAS. The exact definition of the model will be given in Sec. II. We will present simulation results of the model in Sec. III, focusing on the existence of a crossover around . Section IV summarizes and concludes this work.

Ii Model

This section introduces a one-dimensional two-species lattice model with periodic boundary conditions. The system size will be denoted by , and each species will be denoted by and , respectively. Each site can be one of the three possible states; -occupied, -occupied, and vacant states. No multiple occupancy is allowed. The system evolves stochastically under the following rules:

(1)

where stands for a vacant site; () represents a site occupied by species (); and can be any of and ; and the variables above the arrows mean the transition rates of the corresponding events. In all simulations in this paper, we set and with two parameters and ( and ). We always use the configuration with all sites occupied by species as an initial condition and all discussions are based on this initial condition. The extension to higher dimensions is straightforward, but most discussions in this paper are limited to the one dimensional model.

For the case with , species cannot be generated by the rules, so this case is identical to the PCPD studied in Ref. Park and Park (2005a). When is strictly positive, particles can appear by the dynamics and any configuration only with isolated particles are absorbing. Hence, the model with nonzero has infinitely many absorbing states, which is characterized by the nonzero density of species in all phases. Following the convention, we will call the density of species, denoted by , a nonorder field. In particular, if , isolated particles can at best mutate to particles, so the two species play the exactly same role as particles in the PCP Jensen (1993). Since our main concern is the regime where , we will call this model the PCPD with a nonorder field (PNF).

To simulate the model, we made a list of active pairs. By an active pair we mean two consecutive sites that take one of 6 forms, , , , , , and . The list contains the information as to where active pairs are located. The size of the list at time is denoted by . At when all sites are occupied by , the size of the list is the same as the system size . At time , we choose one of the active pairs at random with equal probability. If only one site is occupied (by by definition) in the chosen active pair, this particle mutates to with probability or hops to the empty site inside the chosen active pair with probability . With probability , however, nothing happens. If both sites of the chosen active pair are occupied, with probability these two sites become vacant () or with probability there will be an attempt to branch an particle to one of its nearest neighbors, to be called a target site, which is chosen at random with equal probability. If the target site is empty, particle will be placed there. Otherwise, nothing happens. After the above step, time increases by . The above procedure will continue until either the system falls into one of the absorbing states or time exceeds the preassigned observation time. In all simulations whose results will be presented in the next section, however, the system size is so large that no simulation ends up with an absorbing state at the end of the preassigned observation time.

We are mainly interested in the behavior of pair density defined as

(2)

where means the state (one of , , and ) at site at time , is the Kronecker delta symbol, and stands for the average over all realizations. The order parameter is then defined as

(3)

which is zero (nonzero) in the absorbing (active) phase. Note that defined in Eq. (3) is the order parameter used in both the PCP and the PCPD.

Before presenting simulation data, we would like to argue that the PNF with nonzero should belong to the DP class, mimicking the DP argument with a slight modification. As long as is nonzero, an isolated particle can mutate to within finite time of for any . Since the correlation time diverges at the critical point, diffusion of particles which should be terminated in finite time is irrelevant in the RG sense. Thus, the critical behavior of the PNF with nonzero cannot be distinguishable from the PCP which belongs to the DP class.

Note that, in the original DP argument, the dynamic exponent has to be assumed to be smaller than 2, which makes the extension to higher dimensional systems problematic. In our argument, however, we do not have to assume the value of the dynamic exponent, so it can also be applicable to higher dimensions.

Iii Results

This section presents simulation results of the PNF defined by the rules in Eq. (1). We use as a tuning parameter with fixed and the critical point will be denoted by . The system size is and no realization has ended up with an absorbing state within the observation time, which minimally guarantees that the finite size effect is not important.

Figure 1: (Color online) Semilogarithmic plots of vs. around the critical point for , where is the critical decay exponent of the DP. For (middle curve), the curve becomes flat from for about three log decades and the other curves veer up () or down (). Thus, we conclude that the critical point is and the model belongs to the DP class.
0 111From Ref. Park and Park (2005a). 0.1 0.103 635(5)
0.0001 0.133 050(5) 0.2 0.093 545(5)
0.0003 0.132 595(5) 0.3 0.087 872(2)
0.0008 0.131 83(1) 0.4 0.084 265(5)
0.001 0.131 580(5) 0.5 0.081 815(5)
0.002 0.130 54(1) 0.6 0.080 090(5)
0.004 0.128 92(1) 0.7 0.078 860(5)
0.008 0.126 45(1) 0.8 0.077 990(5)
0.01 0.125 410(5) 0.9 0.077 410(5)
0.05 0.112 56(1) 1 222From Ref. Park and Park (2007).
Table 1: Critical points of the PNF for various ’s. The numbers in parentheses indicate the errors of the last digits.

We first show the behavior of around the critical point for . In Fig. 1, we plot against around the critical point, where is the critical decay exponent of the DP class. In what follows, the symbol is reserved for the DP critical decay exponent. The numbers of independent runs are 16, 24, and 24 for , , and , respectively. At , the curve is almost flat for more than three log decades, while two other curves veer up or down in the long time limit. Hence, we conclude that this case indeed belongs to the DP class with , where the number in parentheses is the uncertainty of the last digit. We also studied the critical behavior for various values of in a similar manner. The resulting critical points are summarized in Table 1.

Since both the PCP and the PNF with belong to the DP universality class, there should not be a significant change due to the variation of when . To make this point strong, we present the behaviors of at the corresponding critical points to various values of in Fig. 2. Indeed, the critical decays of for are barely discernible.

Figure 2: (Color online) Double-logarithmic plots of vs. at the critical points for various ’s from to (bottom to top). Three curves corresponding to and 1 are barely discernible, whereas difference among curves corresponding to are conspicuous.

A symptom of crossover is already observed in Fig. 2 for . When , ’s at transiently follow the PCPD behavior and then eventually deviate from the PCPD curve. In particular, the time when the critical PNF starts to deviate from the PCPD curve increases as gets smaller and there is no symptom of saturation up to . This behavior seems consistent with the crossover scaling ansatz Park and Park (2006)

(4)

where with to be the critical point of the model with (that is, the critical point of the PCPD), is the critical decay exponent of the PCPD, is the correlation time exponent of the PCPD, is the crossover exponent, and is a scaling function.

Since the scaling ansatz suggests a power-law behavior of the phase boundary for small in such a manner that

(5)

we can estimate by analyzing the behavior of the phase boundary for small without resorting to the values of and . Although was estimated as in Ref. Park and Park (2005a), which is larger than the more accurate estimate in Ref. Park (2014), one should note that the accuracy of in Ref. Park and Park (2005a) was attained without resorting much to the accuracy of ; the critical point was actually estimated by finding two ’s belonging to the absorbing and active phases, respectively. Thus, we can safely use the value from Ref. Park and Park (2005a) to find the crossover exponent .

Figure 3: Phase boundary of the PNF in the -plane. Symbols represent the critical points obtained by simulations (see Table 1) and lines which connect two nearest symbols are for guides to the eyes. The phase boundary approaches the ordinate continuously with infinite slope, while it approaches the PCP critical point (corresponding to ) with finite slope. Inset: Plot of vs. on a double logarithmic scale. For comparison, Eq. (5) with is also drawn.

In Fig. 3, we depict the phase boundary of the PNF in the -plane. The phase boundary approaches the ordinate with infinite slope, which suggests that in Eq. (5) is smaller than 1. The fitting of the phase boundary gives ; see the inset of Fig. 3. Hence, we conclude that there is indeed a crossover behavior for and the time for the critical PNF to deviate from the critical decay of the PCPD diverges as gets smaller.

Although the number of absorbing states in itself does not depend on for any , the so-called natural density Jensen and Dickman (1993) of the nonorder field is expected to vary with . Since the discontinuity of the natural density of a nonorder field can give a crossover even within the same universality class Park and Park (2007), we need to check if the natural density changes abruptly at .

At the critical point, the density of species is expected to have asymptotic behavior as Ódor et al. (1998)

(6)

where is the natural density at the critical point for given . If we plot as a function of , a straight line is expected and the extrapolation of the straight line to can give . In Fig 4, we show the result of this extrapolation for . Applying the same method, we obtained for other values of . In the inset of Fig. 4, we show how behaves with . As can be easily guessed, is an increasing function of . Furthermore, it turns out that exhibits a power-law behavior as with an estimate , which suggests that is continuous at .

Figure 4: Plot of vs. at the critical point for . Close to the ordinate, the curve becomes almost straight. By a linear extrapolation using Eq. (6), we found the natural density . The fitting result is drawn as a straight line. Inset: Plot of vs. on a double logarithmic scale. for small is fitted by a power function with a result .

We now argue that is exactly 0.5. First, we take the existence of nonzero for granted. Now, consider a situation where an isolated particle appears (by the given dynamics) in the sea of particles. This isolated particle will either mutate to in time of or meet another particle in time of ( with to be the order of mean distance between and the nearest particle in this situation). If , the chance of meeting a particle before mutation occurs is very low. Thus, we can regard isolated particles as frozen particles. In this case, is purely determined by the dynamics of pairs and should not depend on , which contradicts to the assumption . If , diffusing particle will find a frozen particle before it mutates and the newly formed pair begins so-called defect dynamics. Since the system is at the critical point, the defect dynamics are eventually terminated and the number of particles is likely to decrease by 1 at the end of the defect dynamics when it is compared with the number of particles at the time a pair is formed by diffusion. As the number of particles can only decrease as long as is finite, should decrease to zero, which again is contradictory to the assumption . Thus, a consistent conclusion is attainable only for .

Finally, we would like to check if mere presence of a nonorder field is enough to result in a crossover even within the same universality class. To this end, we slightly modify the PCP (that is, the PNF with ) in such a way that whenever isolated particles appear by a pair annihilation, each isolated particle is removed with probability and it remains there with probability . Note that up to two isolated particles can appear by a single pair-annihilation event (if two particles in the middle of a local configuration are removed by pair annihilation, two ’s at the end of this cluster become isolated). Once an isolated particle which appears right after a pair-annihilation event is decided not to disappear, it remains frozen until another particle come to one of its nearest neighbors. We use the same transition rates as in Eq. (1) for creation and annihilation events. For convenience, we will call this model the modified PCP (MPCP).

The MPCP with is just the PCP. When , there is no isolated particles and this becomes a model with a single absorbing state (we take a configuration with every site occupied by a particle as an initial state). Just like the PNF, the number of absorbing states does not depend on , though the natural density of a nonorder field should depend on . For any value of , MPCP is expected to belong to the DP universality class.

Using the DP exponent , we estimated critical points by studying the behavior of as in Fig. 1. The critical points are summarized in Table 2. To see whether there is a nontrivial crossover for small , we plotted critical decay behavior for various ’s in Fig. 5. Unlike the PNF around , there is no symptom of a diverging time scale for . Furthermore, of the MPCP has finite slope at unlike the PNF at . Thus, we conclude that mere introduction of a nonorder field is not sufficient to trigger a crossover.

0 0.064 162(1) 0.8 0.074 046(1)
0.05 0.064 6925(5) 0.9 0.075 5325(15)
0.1 0.065 234(1) 0.95 0.076 303(1)
0.2 0.066 3465(5) 1 0.077 0905(5)111From Ref. Park and Park (2007).
0.5 0.069 957(1)
Table 2: Critical points of the MPCP for various ’s. The numbers in parentheses indicate the errors of the last digits.

Iv Summary and discussion

We studied a variation of the pair contact process with diffusion (PCPD) by introducing a nonorder field assocated with infinitely many absorbing states; see Eq. (1). We called this model the PCPD with a nonorder field (PNF). We analyzed absorbing phase transitions for various values of . When is nonzero, the PNF has infinitely many absorbing states (IMAS) and was found to belong to the directed percolation (DP) universality class. When is close to 1, critical decay of the pair density does not show any significant change from the pair contact process (PCP). This is also manifest by the finite slope of the phase boundary at ; see Fig. 3 around . On the other hand, a singular behavior representing a crossover appears when nonorder field is introduced to the PCPD (the PNF with ). This crossover is described by the crossover exponent , whose numerical value was found as from the analysis of the phase boundary at (see Fig. 3 around and its inset). We argued that the natural density at the critical point changes continuously as when , which is consistent with simulation results. To support that this crossover is originated from the difference of the universality classes, we also presented simulation results of another model which belongs to the DP class irrespective of whether the number of absorbing states is infinite or finite. Unlike the PNF around , no crossover was observed in this case.

Figure 5: (Color online) Double-logarithmic plots of vs. at the critical point of the MPCP for various ’s from to (top to bottom). As a guide to the eyes, a line segment with slope is also drawn. Unlike the PNF, the critical behavior for small is hardly discernible from that for ; see also Fig. 2. Inset: Plot of as a function of for the MPCP. Unlike the PNF, the slope at is finite.

In Refs. Park and Park (2006, 2009), the existence of the nontrivial crossover from the PCPD to the DP was invoked to be the evidence of the existence of the fixed point of the PCPD distinct from the DP in one dimension. The same conclusion is also arrived at in this paper. Furthermore, our work clarifies the difference between diffusing particles in the PCPD and frozen particles in the PCP, unlike the anticipation from the DP argument.

It is worth while to discuss the role of a nonorder field which sometimes triggers a crossover within the same universality class Park and Park (2007) and sometimes not. This difference can be understood as follow: The nonorder field in this paper cannot play a role of an order parameter for all cases, whereas the nonorder field in Ref. Park and Park (2007) becomes an order parameter as soon as crossover dynamics are introduced. That is, the crossover dynamics in Ref. Park and Park (2007) make the irrelevant nonorder field a relevant order parameter, which is the origin of the crossover within the same universality class. observed in Ref. Park and Park (2007) reflects that a nonorder field becomes relevant in the RG sense by crossover dynamics while absence of role change of nonorder field in this paper does not give a nontrivial crossover behavior within the same universality class.

As a final remark, we would like to compare the three different crossovers in this work and in Ref. Park and Park (2006, 2009) to the also three different crossovers from the directed Ising (DI) class to the DP class Bassler and Browne (1996); Kwon et al. (1999); Odor and Menyhard (2008); Park and Park (2008b). Note that three different mechanisms from the DI to the DP are qualitatively identical to those from the PCPD to the DP. In this regard, the crossover in this paper seems different from that in Ref. Park and Park (2006), though the values of crossover exponents are quite close; 0.62 in this paper and 0.57 in Ref. Park and Park (2006).

Acknowledgements.
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (Grant No. 2014R1A1A2058694) and by the Catholic University of Korea, research fund 2017. The author furthermore thanks the Regional Computing Center of the University of Cologne (RRZK) for providing computing time on the DFG-funded High Performance Computing (HPC) system CHEOPS as well as support. The author would also like to thank Korea Institute for Advanced Study (KIAS) for its support and hospitality during his stay there on sabbatical leave (2016-2017).

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