Crossovers from parity conserving to directed percolation universality
The crossover behavior of various models exhibiting phase transition to absorbing phase with parity conserving class has been investigated by numerical simulations and cluster mean-field method. In case of models exhibiting symmetric absorbing phases (the NEKIMCA and Grassberger’s A stochastic cellular automaton) the introduction of an external symmetry breaking field causes a crossover to kink parity conserving models characterized by dynamical scaling of the directed percolation (DP) and the crossover exponent: . In case an even offspringed branching and annihilating random walk model (dual to NEKIMCA) the introduction of spontaneous particle decay destroys the parity conservation and results in a crossover to the DP class characterized by the crossover exponent: . The two different kinds of crossover operators can’t be mapped onto each other and the resulting models show a diversity within the DP universality class in one dimension. These ’sub-classes’ differ in cluster scaling exponents.
pacs:PACS numbers: 05.70.Ln, 82.20.Wt
The study of nonequilibrium phase transitions is an important task of statistical physics. Genuinely nonequilibrium transitions can be observed most easily in models exhibiting transition from an active to an “absorbing” state, where the fluctuations are negligible, hence no return is possible. The exploration of critical phenomena and universality classes of simple, one component models has been started Marro and Dickman (1999); Hinrichsen (2000); Ódor (2004a); Lübeck (2004) and important steps towards a full classification have been done Elgart and Kamenev (2006); Ódor (2008).
For a long time it was a common belief that all continuous, nonequilibrium phase transitions belong to class of the directed percolation (DP) and a hypothesis advanced by Janssen and Grassberger Janssen (1981); Grassberger (1982); Grinstein et al. (1989). This states that in one component systems exhibiting continuous phase transitions to a single absorbing state (without extra symmetry and inhomogeneity or disorder) short ranged interactions can generate DP class transition only. Despite the robustness of this class experimental observation is rare, owing to the high sensitivity to disorder and long-range interactions. A very recent experimental study Takeuchi et al. (2007) has reported clear and comprehensive experimental evidence of DP criticality. Later it was discovered that in systems with infinitely many frozen absorbing states (IM type) Jensen and Dickman (1993a); Jensen (1993); Jensen and Dickman (1993b); Mendes et al. (1994) like in the pair contact process (PCP) the static exponents coincide with those of the DP. The dynamical cluster spreading behavior is different owing to the long time memory generated by the frozen monomers Ódor et al. (1998). In the literature the transition type of PCP is often called DP type. Very recently Park and Park (2007) have investigated in dimension the crossover from PCP to DP type of models by introducing different absorbing state reduction mechanisms and nontrivial exponents have been found. This finding confirms that strictly speaking the universality class of PCP and DP is different.
The first example for clearly non-DP critical behavior was found among stochastic cellular automata (SCA) by Grassberger Grassberger et al. (1984). These models exhibit symmetric absorbing states and an effective kink dynamics, which follows an even offspringed branching and annihilating random walk (BARWe). In reaction-diffusion (RD) particle models, with BARWe dynamics (, ) such phase transition class was discovered by Takayasu and Tretyakov (1992); Jensen (1994). Since this is different from odd offspringed branching and annihilating random walks, in which DP class transition occurs, the name “parity conserving” (PC) was introduced to denote this class 111However other different names for this class like “Directed Ising”(DI) or “Generalized voter (GV)” or “BARW” can also be found in the literature.
An important example of the PC class behavior was discovered in one-dimensional kinetic Ising models with combined zero temperature spin-flip and finite temperature spin-exchange dynamics (NEKIM) Menyhárd (1994). Here the domain walls between up and down spins follow BARWe dynamics and an exact duality transformation in one dimension between the NEKIM and the BARWe particle model was established by Mussawisade et al. (1998) 222Note that such mapping is not possible in higher dimensions, but since the upper critical dimension is it is not important.. Naturally the NEKIM exhibits two, symmetric absorbing states. This universality class has not been observed in nature yet, however we pointed out Menyhárd and Ódor (2006) that this model is very insensitive to the quenched disorder, therefore it is a good candidate for experimental verification. In fact it’s inactive phase, where annihilating random walk dominates has been observed Kroon et al. (1993) unless very strong disorder fractures the medium.
The introduction of a symmetry breaking external magnetic field, in NEKIM, which favors one of the absorbing states (but preserves the BARWe dynamics) was shown to change the type of transition from PC to DP type Menyhárd and Ódor (1996). Such crossover mechanism has been observed in other symmetric models as well Park and Park (1995); Bassler and Browne (1996); Hinrichsen (1997); Hwang et al. (1998); Kwon et al. (1999). On the other hand simulations Kwon and Park (1995) and field theory Cardy and Täuber (1996); Canet et al. (2005) proved that a PC breaking reaction in the BARWe model should also change the type of phase transition from PC to DP type.
The more detailed study of crossover behavior among nonequilibrium universality classes has been intensified in the recent years. The universal crossover exponent, defined by the shape of phase boundary () as one introduces a relevant scaling field () (see for example Lawrie and Sarbach (1984)),
has been determined in case of DP to the compact DP class Mendes et al. (1996); Janssen (2005); Lübeck (2006); Dantas et al. (2008). Crossover between DP and isotropic percolation was investigated by field theory Frey et al. (1993); Fröjdh and den Nijs (1997); Janssen and Stenull (2000) and simulations Inui et al. (1999). Very recently the numerical exploration of the crossover behavior from the diffusive pair contact process (PCPD) to DP Park and Park (2006) has strengthened the existence of the independent PCPD class theory. In IM type models ’nontrivial crossovers to DP class’ have been found Park and Park (2007). The crossover behavior from DP to mean-field, generated by long-range diffusion, following the early studies Ódor et al. (1993); Ódor and Szabó (1994), has currently been re-examined via numerical techniques Dantas et al. (2007) and field theory Messer and Hinrichsen (2008) and estimates for the exponent have been provided. Similar, diffusion driven crossover in case of PC class was determined within the framework of NEKIM Menyhárd and Ódor (1995) long ago.
In the present paper we determine the crossover exponent from PC to DP class in a SCA version of the NEKIM model Menyhárd and Ódor (1996) (NEKIMCA) and compare it with other realizations of the PC class. In particular we confirm the universality of in case of symmetry breaking fields by simulating Grassberger’s A model. We compare this crossover behavior with the outcome of parity conservation breaking in a BARWe model.
Ii Crossover of NEKIMCA in an external field
The NEKIM exhibiting PC class transition was suggested by Menyhárd (1994) as a generalization of the Glauber Ising model Glauber (1963). It is defined by the alternating application of a spin-flip sweep and a Kawasaki spin-exchange update of a one dimensional lattice. While the spin-flip dynamics generates annihilating random walk of kinks, the spin-exchange introduces a parity conserving branching () of the domain walls. Tuning the relative strengths of the reactions one can get a phase transition from an active to a kink-free, adsorbing state (the order parameter is the density of kinks).
It was realized in Menyhárd and Ódor (1996) that branching reactions appear automatically in the SCA version of the NEKIM, if the spin-flip update is done synchronously due to overlaps. With this dynamics we can obtain a very simple model, with PC type of critically, which can be implemented on a computer by efficient bit-coding. The NEKIMCA spin updates, represented by bit field operations of a computer word, generating BARWe reactions of the kinks are the followings:
Random walk of domain walls ():
generated by a spin-flip between oppositely oriented spins, with probability ,
Annihilation of a pair of kinks:
generated by a spin-flip between a pair of same oriented spins, with probability ,
Branching of a kink:
can be generated by two overlapping spin-flips (with the probability ) such that it causes a spin-exchange.
There are two independent parameters related to the original parametrization of Glauber:
In the present study we investigated the and cases. The corresponding PC critical points, without external field, are located at and respectively.
A single Monte Carlo step (MCS) consists of updating all sites at once as described in Menyhárd and Ódor (2000) (throughout the paper the time is measured by MCS). The simulations were reformed on sized lattices, with periodic boundary conditions up to MCS. The change of critical point has been determined for several values of external field. The transition probabilities are modified in the presence of an external magnetic field as:
Fig. 1 shows the phase diagram in the plane, which is similar to the one we obtained in Menyhárd and Ódor (1996) by simulations and cluster mean-field technique. We have applied only random initial state simulations to find the points of the line of phase transition exhibiting kink density decay exponent of the DP Ódor (2004a).
A clear power-law scaling of the critical point shift as the function of can be fitted with the form (1) in the region with crossover exponent. The error-bars on Fig. 1 come from our numerical estimates for the critical point shifts and the least squares error estimate fitting procedure. This value is close to the early numerical estimates: by Bassler and Browne (1996) and by Kwon et al. (1999).
Iii Crossover of the Grassberger-A SCA model
The PC conserving SCA by Grassberger is realized by the following range-1 update rule (we show the configurations at and the probability of getting ’’ at time ):
t-1: 100 001 101 110 011 111 000 010 t: 1 1 0 1-p 1-p 0 0 1
The time evolution pattern in dimension, for small evolves towards a stripe-like ordered steady state (with double degeneration), while for the kinks (the ’’ and ’’ pairs) survive. According to the classification of Wolfram Wolfram (1983) for we have the Rule-94, class 1 CA, while the limit is the chaotic Rule-22 deterministic CA. This model has been investigated from damage spreading point of view by us Ódor and Menyhárd (1998).
Now we extend this model in such a way that a symmetry breaking occurs in the absorbing state. This can be achieved by favoring one of the absorbing phases ( or ) shifted by a single site. Therefore we modified the transition rates
t-1: 100 001 101 110 011 111 000 010 t: 1 1 0 1-p’ 1-p’ 0 0 1
such that at odd and at even sites. The simulations were run on sized rings up to MCS. The location of the critical point is shown on Fig. 2.
To get more precise estimates for the crossover exponent we determined the local slopes of the data points:
This is plotted in the insert of Fig.2 and in the limit one can read-off the linear extrapolation value (with least squares error estimates) in agreement with the value for NEKIMCA+h. As the figure shows the correction to scaling can not be neglected.
Iv Crossover in a BARWe model
To study the PC to DP crossover in an other way we take a simple version of a BARWe particle model introduced in Zhong et al. (2003) and generalized in Ódor and Szolnoki (1996) (ZAMb model) to allow phase transition at finite branching rate. This model is defined on the one dimensional lattice as follows. An occupied site is chosen randomly and is tried for diffusion, with probability , or branching with probability ); while the time is increased by , where is the number of occupied (active) sites. In a diffusion step the particle jumps to its randomly chosen nearest neighbor sites. If the site is occupied both particles are annihilated with probability . On the other hand the jump is rejected with probability . The branching process involves the creation of two new particles around the neighborhood. If either, or both neighboring sites is previously occupied the target site(s) become empty with probability . Otherwise the lattice remains unaltered with probability .
To induce a crossover to DP we introduced a spontaneous particle removal or a coagulation with a small probability to the above reactions. The simulations were done on lattices of size with periodic boundary conditions for diffusion rate (the convergence of GMF approximations was found to be rather good for this diffusion rate in Ódor and Szolnoki (1996)). For we used the critical point value determined in Ódor and Szolnoki (1996): . As we increase the critical point shifts as shown on Figure 3.
A simple power-law fitting for the data in the range of the form 1 resulted in . A more precise estimate, which takes the corrections to scaling into account can be obtained by calculating the local slopes (7). By plotting the effective exponent as the function of a linear extrapolation resulted in both for the and the parity breaking reactions 333Note that the sign of correction to scaling is different for the two crossover versions.
V GMF+CAM calculations
The simulation results were complemented by analytical cluster mean-field approximations and coherent anomaly extrapolations. The generalized (cluster) mean-field method (GMF) suggested for SCA by Gutowitz et al. (1987) and for dynamical RD models by Dickman (1988) has been shown to be successful approach for exploring the phase diagram of nonequilibrium models (see for example Ódor and Szolnoki (2005); Ódor (2004b) and the references in Ódor (2004a)). It’s extension with the coherent anomaly (CAM) extrapolation Suzuki (1986) enables one to extract the true scaling behavior.
In GMF we set up equations for the steady state of the system based on -point block probabilities. Correlations with a range greater than are neglected. By increasing from (traditional mean-field) step by step we take into account more and more correlations and get better approximations. One can set up a master equations for the block probabilities as
where the site variables take the values: . During the solution of these equations one estimates larger than sized block probabilities by the maximum overlap approximation:
Taking into account the spatial symmetries and the conservation of probability for the maximal, approximation of this work we had to find the solution of a set of nonlinear equations of 272 independent variables.
The steady state solutions of the -cluster approximations for the NEKIMCA () and the ZAMb model () have been determined and the corresponding densities are calculated numerically. The phase transition points are obtained for several values of the crossover parameter in both cases. In case of the ZAMb model (at ) they are plotted on Fig.4
As one can see for (site mean-field) the critical point exhibits linear relation without corrections to scaling. This also true for the ZAMB+sink case. This means the mean-field value for both kinds of crossover.
For the curves pick up type of corrections to scaling (the leading order singularity is expected to remain mean-field like). Assuming a scaling with the correction form
which is natural if we swap the axes and , one can determine the amplitudes () of the leading order term. Following the envelope scaling hypothesis of Suzuki (1986) the Coherent Anomaly Method (CAM) extrapolation can be performed on the amplitude data in the limit.
According to CAM the amplitudes of the cluster mean-field singularities in the leading order scale as
allowing to estimate the exponent of the true singular behavior (Eq.(1)). The amplitudes were determined by a next leading order fitting form (10) on the the data in the neighborhood of for the ZAMb model. In case of the NEKIMCA the same procedure has been applied for the crossover critical point GMF results.
The highest amplitudes with a CAM fitting form (which takes into account possible scaling corrections Ódor and Szolnoki (2005))
where for NEKIMCA and for the ZAMb are plotted on Fig. 5. In this form the nonuniversal fitting parameters are: , and the anomaly exponent is : . By the simulations (apart from the constants and ) we have an expected behavior for (12) both for ZAMb and NEKIMCA. This is plotted on Fig. 5 by the dashed lines. As one can see our GMF data can be fitted with those lines, (with the values: , for ZAMb and , for NEKIMCA) but the amplitudes for just start to converge towards the asymptotic scaling curves. This suggests that larger -cluster approximations should be determined. However the numerical instability of root finding of the GMF method (within the space of more than variable, nonlinear system) prevented us to go further. Still the different behavior for NEKIMCA and ZAMb can be justified by this figure.
Vi Conclusions and discussion
We have performed simulations and GMF+CAM approximations for various versions of the PC class model crossovers towards the ”DP class”. By determining the crossover exponents we have found and outstanding difference between the effect of a symmetry breaking field () and the parity conservation breaking (). Although the NEKIMCA and the BARWe are dual models, the two types of crossover operators can’t be mapped onto each other. Furthermore they can’t be mapped onto a local one in the dual system. The process in the spin language would mean flipping of all spins from a given site (see Fig.6). Such transformation can’t even be done in finite systems with periodic boundary conditions.
A symmetry breaking in the BARWe model favors the ’’ (up) or ’’ (down) oriented spins depending its sign. In case of a ’’ preference (see Fig.6) this means that ’’domains broaden on the expense of ’’ domains, hence odd-even kinks (the corresponding particle pairs in BARWe) attract and even-odd pairs repel each other (shown by the horizontal arrows). Therefore that particle model becomes an effectively two-component, parity conserving one, and the sufficient conditions of the DP hypothesis do not hold. Still one can see a DP type of decay in the global order parameter because calling a ’’ domain a macroscopic particle ’’, an effective DP process: , , describes it’s dynamics.
However there are certain operators which do not exhibit DP type behavior. For example in case of cluster spreading it is easy to see that there are two sectors in this model. An odd parity one, with final survival probability defined as
and an even parity one, with normal, DP exponent. As a consequence the hyper-scaling relation connecting the cluster spreading exponents and the rapidity reversal symmetry of DP is not satisfied for this critical behavior. The universality class of the DP is split into sub-classes. The subclass of models with BARWe dynamics with broken symmetry (let’s call it DP-2) is different from that of the ordinary 1+1 dimensional DP. Since the difference is manifested in the cluster spreading behavior, which is a consequence of a special initial condition, similarly to the terminology used in equilibrium models with different surface classes we do not claim the splitting of the DP class itself.
This sub-class behavior is similar to the one, which is observed in the pair contact process, where the frozen monomers cause different cluster behavior from that of the simple contact process. Also that kind of DP-2 behavior should arise, when one introduces spin-anisotropy in the NEKIM Menyhárd and Ódor (2002, 2003) and the effective two-component BARWe model exhibits DP type of phase transition for finite branching rate in the global order parameter, but different scaling behavior occurs in terms the cluster spreading. Furthermore this model exhibits a re-entrant phase diagram and at zero branching rate one finds another phase transition belonging to the two-component BARWe model Cardy and Täuber (1996), but the cluster spreading behavior, which is sensitive to the spin anisotropy is different again. Therefore such sub-classes are not at all rare among low-dimensional reaction-diffusion models.
We would like to point out that by using an alternative definition for universality classes a different interpretation can also be given for the same numerical results presented here. Some authors define a university class by the field-theoretic action of the model (without specifying the fixed point) instead of the models exhibiting the same set of known critical exponents as we do. Since the action of BARWe particles Cardy and Täuber (1996) and the action of the NEKIM spin model Canet et al. (2005) (called GV model) do not agree, they should belong to different universality classes, which ”intersect” in one dimension only Hinrichsen (2000). According to this picture the applied symmetry breaking to the PC and GV class models ( breaking in GV and particle removal in BARWe) should end up in DP class somehow.
However we think that our definition of universality class, and therefore our interpretation, is more precise. It can describe the critical behavior of (particle) system for which no proper field theoretical action has been found. Even within a field theory multiple fixed point solutions (corresponding to multiple critical points) can exist and the relevancy of terms affecting the stability of a fixed point is not clear in many cases.
Furthermore a coarse grained field theory may not capture all scaling details of a particle model. In a discrete particle model diffusive annihilating particles can die out within finite time (even in an infinite sized system) due to the recurrence relation of random walks in one dimension 444Note that this is not a finite size effect. The particle survival probability decays asymptotically as and the final survival probability scales as (13). As a consequence a thermodynamic limit, in which the particle density are kept finite can’t be established.
On the other hand in field theories of continuous variables the survival probability is always unity and it had been unclear if the exponent and could be defined sensibly. To overcome this discrepancy a reinterpretation of survival probability was proposed in Muñoz et al. (1997). This alternative definition of in field theory resulted in correct scaling exponents and hyper-scaling relations corresponding to symmetries.
Since in our case the “thermodynamic limit” can’t be restricted to infinite particle number the cluster spreading behavior starting from finite number of particles is relevant from the universality point of view as in many papers investigating absorbing phase transition via this approach (see Grassberger and de la Torre (1979); Jensen and Dickman (1993a); Jensen (1993); Marro and Dickman (1999); Hinrichsen (2000); Ódor (2004a); Lübeck (2004). The cluster spreading behavior (the finite survival probability, pair-connectedness functions, avalanche distributions … etc. ) are sensitive to the initial parity of particles, hence strictly speaking the DP-2 subclass is not identical to the DP class in 1+1 dimension.
The authors thank Su-Chan Park and Hyunggyu Park for the useful communications. Support from Hungarian research funds OTKA (Grant No. T046129) is acknowledged. The authors thank the access to the HUNGRID and LCG-GRID.
- Marro and Dickman (1999) J. Marro and R. Dickman, Nonequilibrium phase transitions in lattice models (Cambridge University Press, Cambridge, 1999).
- Hinrichsen (2000) H. Hinrichsen, Adv. Phys. 49, , 815 (2000).
- Ódor (2004a) G. Ódor, Rev. Mod. Phys 76, 663 (2004a).
- Lübeck (2004) S. Lübeck, Int. J. of Mod. Phys. B 18, 3977 (2004).
- Elgart and Kamenev (2006) V. Elgart and A. Kamenev, Phys. Rev. E 74, 041101 (2006).
- Ódor (2008) G. Ódor, Universality In Nonequilibrium Lattice Systems (World Scientific, 2008).
- Janssen (1981) H. K. Janssen, Z. Phys. B 42, 151 (1981).
- Grassberger (1982) P. Grassberger, Z. Phys. B 47, 365 (1982).
- Grinstein et al. (1989) G. Grinstein, Z.-W. Lai, and D. A. Browne, Phys. Rev A 40, 4820 (1989).
- Takeuchi et al. (2007) K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano, Phys. Rev. Lett. 99, 234503 (2007).
- Jensen and Dickman (1993a) I. Jensen and R. Dickman, Phys. Rev. E 48, 1710 (1993a).
- Jensen (1993) I. Jensen, Phys. Rev. Lett. 70, 1465 (1993).
- Jensen and Dickman (1993b) I. Jensen and R. Dickman, Phys. Rev. E 48, 1710 (1993b).
- Mendes et al. (1994) J. F. F. Mendes, R. Dickman, M. Henkel, and M. C. Marques, J. Phys. A 27, 3019 (1994).
- Ódor et al. (1998) G. Ódor, J. F. Mendes, M. A. Santos, and M. C. Marques, Phys. Rev. E 58, 7020 (1998).
- Park and Park (2007) S. Park and H. Park, Phys. Rev. E 76, 051123 (2007).
- Grassberger et al. (1984) P. Grassberger, F. Krause, and T. von der Twer, J. Phys. A 17, L105 (1984).
- Takayasu and Tretyakov (1992) H. Takayasu and A. Y. Tretyakov, Phys. Rev. Lett. 68, 3060 (1992).
- Jensen (1994) I. Jensen, Phys. Rev. E 50, 3623 (1994).
- Menyhárd (1994) N. Menyhárd, J. Phys. A 27, 6139 (1994).
- Mussawisade et al. (1998) K. Mussawisade, J. E. Santos, and G. M. Schütz, J. Phys. A 31, 4381 (1998).
- Menyhárd and Ódor (2006) N. Menyhárd and G. Ódor, Phys. Rev. E 73, 036130 (2006).
- Kroon et al. (1993) R. Kroon, H. Fleurent, and R. Sprik, Phys. Rev E 47, 2462 (1993).
- Menyhárd and Ódor (1996) N. Menyhárd and G. Ódor, J. Phys. A. 29, 7739 (1996).
- Park and Park (1995) H. Park and H. Park, Physica A 221, 97 (1995).
- Bassler and Browne (1996) K. E. Bassler and D. A. Browne, Phys. Rev. Lett. 77, 4094 (1996).
- Hinrichsen (1997) H. Hinrichsen, Phys. Rev. E 55, 219 (1997).
- Hwang et al. (1998) W. Hwang, S. Kwon, H. Park, and H. Park, Phys. Rev. E 57, 6438 (1998).
- Kwon et al. (1999) S. Kwon, W. Hwang, and H. Park, Phys. Rev. E 59, 4949 (1999).
- Kwon and Park (1995) S. Kwon and H. Park, Phys. Rev. E 52, 5955 (1995).
- Cardy and Täuber (1996) J. L. Cardy and U. C. Täuber, Phys. Rev. Lett. 77, 4780 (1996).
- Canet et al. (2005) L. Canet, H. C. B. Delamotte, I. Dornic, and M. A. Muñoz, Phys. Rev. Lett. 95, 100601 (2005).
- Lawrie and Sarbach (1984) I. Lawrie and S. Sarbach, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic Press, London, 1984), vol. 9.
- Mendes et al. (1996) J. F. F. Mendes, R. Dickman, and H. Hermann, Phys. Rev. E 54, R3071 (1996).
- Janssen (2005) H. K. Janssen, J. Phys.: Cond. Mat. 17, S1973 (2005).
- Lübeck (2006) S. Lübeck, J. Stat. Mech. p. P09009 (2006).
- Dantas et al. (2008) W. G. Dantas, M. J. de Oliveira, and J. F. Stilck, Braz. J. Phys 38, 94 (2008).
- Frey et al. (1993) E. Frey, U. C. Täuber, and F. Schwabl, Phys. Rev. E 49, 5058 (1993).
- Fröjdh and den Nijs (1997) P. Fröjdh and M. den Nijs, Phys. Rev. Lett. 78, 1850 (1997).
- Janssen and Stenull (2000) H. K. Janssen and O. Stenull, Phys. Rev. E 62, 3173 (2000).
- Inui et al. (1999) N. Inui, H. Kakuno, A. Y. Tretyakov, G. Komatsu, and K. Kameoka, Phys. Rev. E 59, 6513 (1999).
- Park and Park (2006) S. Park and H. Park, Phys. Rev. E 73, 025105(R) (2006).
- Ódor et al. (1993) G. Ódor, N. Boccara, and G. Szabó, Phys. Rev. E 48, 3168 (1993).
- Ódor and Szabó (1994) G. Ódor and G. Szabó, Phys. Rev. E 49, 2764 (1994).
- Dantas et al. (2007) W. G. Dantas, M. J. de Oliveira, and J. F. Stilck, J. Stat. Mech. p. P08009 (2007).
- Messer and Hinrichsen (2008) A. Messer and H. Hinrichsen (2008), arXiv:0802.2789v2.
- Menyhárd and Ódor (1995) N. Menyhárd and G. Ódor, J. Phys. A. 28, 4505 (1995).
- Glauber (1963) R. J. Glauber, J. Math. Phys. 4, , 191 (1963).
- Menyhárd and Ódor (2000) N. Menyhárd and G. Ódor, Braz. J. of Phys. 30, 113 (2000).
- Wolfram (1983) S. Wolfram, Rev. Mod. Phys. 55, 601 (1983).
- Ódor and Menyhárd (1998) G. Ódor and N. Menyhárd, Phys. Rev. E 57, 5168 (1998).
- Zhong et al. (2003) D. Zhong, D. Avraham, and M. A. Muñoz, Eur. Phys. J B 35, 505 (2003).
- Ódor and Szolnoki (1996) G. Ódor and A. Szolnoki, Phys. Rev. E 53, 2231 (1996).
- Gutowitz et al. (1987) H. Gutowitz, J. Victor, and B. Knight, Physica D 28, 18 (1987).
- Dickman (1988) R. Dickman, Phys. Rev. A 38, 2588 (1988).
- Ódor and Szolnoki (2005) G. Ódor and A. Szolnoki, Phys. Rev. E 71, 066128 (2005).
- Ódor (2004b) G. Ódor, Phys. Rev. E 70, 066122 (2004b).
- Suzuki (1986) M. Suzuki, J. Phys. Soc. Jpn. 55, 4205 (1986).
- Menyhárd and Ódor (2002) N. Menyhárd and G. Ódor, Phys. Rev. E 66, 016127 (2002).
- Menyhárd and Ódor (2003) N. Menyhárd and G. Ódor, Phys. Rev. E 68, 056106 (2003).
- Muñoz et al. (1997) M. A. Muñoz, G. Grinstein, and Y. Tu, Phys. Rev. E 56, 5101 (1997).
- Grassberger and de la Torre (1979) P. Grassberger and A. de la Torre, Ann. Phys. 122, 373 (1979).