Novel phase transition in a model with absorbing states

We study a two-dimensional model which undergoes a transition between active and absorbing phase. The transition point in this model is of novel type: it combines some features of a continuous ((2+1)DP) and a discontinuous transition. Some arguments supported by Monte Carlo simulations prompted us to predict the exact location of the transition point. Typeset using REVTEX 1 Recently, nonequilibrium phase transitions are intensively studied in variety of models [1]. In addition to some potential applications, the motivation to study these transitions comes from the belief that they can be categorized into universality classes similarly to equilibrium phase transitions. Of particular interest in this context are models which exhibit transitions between active and absorbing phases [2]. There exists already a substantial numerical evidence that phase transitions for such models indeed can be classified into some universality classes. For example it is believed that models with unique absorbing states should belong to the so-called directed-percolation (DP) universality class [3]. Moreover, models with double (symmetric) absorbing states or with some conservation law in their dynamics, belong to another universality class [4]. Similarly to equilibrium systems, continuous phase transitions are not the only possibility some models are known to undergo discontinuous transitions [5–7]. Although such transitions are not classified into universality classes, they might be more relevant since, discontinuous transitions are, at present, the only type of transitions which can be observed experimentally. On the contrary, experimental realization of continuous phase transitions still remains an open problem [8]. One reason for a relatively good understanding of equilibrium phase transitions is a wealth of exactly solvable models in this field [9]. With this respect the situation is much worse for nonequilibrium phase transitions. None of the models with absorbing states and with continuous or discontinuous transition was solved exactly and all results in this field concerning the critical exponents or location of a transition point are only numerical. In the present paper we study a two-dimensional model with infinitely many absorbing states. At a certain value of a control parameter r = rc, the model undergoes a transition between active and absorbing phases. But the interesting point is a novel type of this transition: it seems to combine some features of both discontinuous and continuous transitions. Namely, at r = rc an order parameter jumps discontinuously to zero but in addition to that the order parameter has a power-law singularity upon approaching the transition point from the active phase. The exponent β of this singularity is close to 0.583 which suggests that 2 this power-law singularity is related to (2+1) directed percolation (DP). Moreover, some elementary arguments, supported by Monte Carlo simulations prompted us to predict the exact location of the transition point, namely, rc = 0. If our predictions are correct, this might be the first model of DP universality class with exactly known critical point. Our model is a certain modification of a model introduced in a context of modelling biological evolution [10,11]. It is defined on a two-dimensional square lattice where with each bond between nearest-neighbouring sites i and j we introduce a bond variables wi,j ∈ (−0.5, 0.5). Introducing the parameter r, we call the site i active when ∏ j wi,j < r, where j runs over all nearest neighbours of i. Otherwise, the site is called nonactive. The model is driven by random sequential dynamics and when the active site i is selected we assign anew, with uniform probability, four bond variables wi,j, where j is one of nearest neighbours of i. Nonactive sites are not updated, but updating a certain (active) site might change the status of its neighbours. An important quantity characterizing this model is the steady-state density of active sites ρ. How does ρ change with the control parameter r? Of course, for r ≥ (0.5) = 0.0625 all sites are active (ρ = 1) for any distribution of bond variables wi,j. It is natural to expect that for r < 0.0625, and not too small there will be a certain fraction of active sites (ρ > 0) and this fraction will decrease when r decreases. Since the dynamical rules imply that the model has absorbing states with all sites nonactive (ρ = 0), one can expect that at a certain r the model undergoes a transition between active and absorbing phase. On general grounds one expects that this transition might be either continuous and presumably of (2+1)DP universality class [12] or discontinuous. Existence of a transition is confirmed in Fig.1 which shows the density ρ as a function of r obtained using Monte Carlo simulations. Simulations were made for the linear system size L = 300 and we have checked that the presented results are, within small statistical error, size-independent. After relaxing the random initial configuration for trel = 10 4 we made measurements during runs of t = 10 (the unit of time is defined as a single on average update/lattice site). From this figure one can also see that the transition point rc is located 3 very close to r = 0 and in the following we are going to show that it is very likely that in this model rc = 0 (exactly). First, we show that for r < 0 the model is in the absorbing phase. The argument for that is elementary and based on the following observation: for r < 0 there exist a finite probability that after updating, a given site will become nonactive forever. Indeed, when one of the selected anew bond satisfies |wi,j| < −r/(0.5) , (1) then sites i and j become permanently nonactive (i.e., no matter what are the other bonds attached to these sites, they will always remain nonactive). For r < 0 there is a finite probability to satisfy Eq. 1 and the above mechanism leads to the rapid decrease of active sites and hence the system reaches an absorbing state. The above mechanism is not effective for r ≥ 0 since there is no value which would ensure permanent nonactivity of a certain site. To confirm that for r < 0 the system is in the absorbing phase we present in Fig. 2 the time evolution of ρ for r = −10 and −10. Although these values are very close to r = 0, one can clearly see that the system evolves toward the absorbing state. (For r smaller than these values the approach to the absorbing state would be even faster.) As we already mentioned, for r ≥ the mechanism, which generates permanently nonactive sites is not effective. Most likely, this has important consequences: as shown in Fig. 2, even for r = 0 the system does not evolve toward an absorbing state but remains in the active phase. These results indicate that at r = 0 the model undergoes a first order transition between active and absorbing phases characterized by a discontinuity of the order parameter ρ. However, the most interesting feature of the model is the fact that upon approaching the first-order transition point r = 0 the order parameter exhibits a power-law singularity, which usually signals a continuous transition. This singularity, which is already visible in the inset of Fig. 1 is also presented in the logarithmic plot in Fig. 3. The parameter ρ0 = 0.3590 (i.e., the density of active sites for r = 0) in Fig. 3 was obtained from the least square analysis of small-r (r ≤ 10) data shown in Fig. 1 using the formula 4 ρ(r) = ρ0 + Ar , (2) where we assumed that the critical point is located at r = 0 [14]. From the slope of the data in Fig 3 we estimate β = 0.581(5) which indicates that the exponent β for that model might be the same as in the (2+1)DP [13,14]. As we already mentioned, transitions in two-dimensional models with absorbing states are likely to belong to (2+1)DP universality class and our model seems to conform to this rule. However, a characteristic feature of models of DP universality class is that at the transition point the model falls into an absorbing state. With this respect our model is very unusual since at the transition point (r = 0) the model is not in the absorbing phase (see Fig. 2). However, the model enters the absorbing phase as soon as r becomes negative. The above model has a transition point which, clearly, has both discontinuous and continuous features. This is a novel type of behaviour and certainly worth further studies. For example, a characteristic feature of continuous transitions is a divergence of time and length scales. These quantities usually remain finite in first-order transitions. Studying a size dependence of ρ at r = 0 we observed that results quickly saturates with the system size which suggests that the characteristic length scale of the system is finite. This would be in agreement with the first-order-transition scenario. However, it gives rise to an important question: how can the system with finite length scale build nontrivial power-law singularity. Are there any indications that such transitions might take place in real systems? In our opinion, one of the possible applications might be related with phase transitions in nuclear physics. Indeed, there are some indications that multifragmentation of heavy nuclei resembles a phase transition which has both firstand second-order features [15]. Such systems were already modeled using Ising-like models. However, such an approach implicitly assume a thermalization of the system which is not obvious in these multifragmentation processes. Models with absorbing states might provide an alternative description of such processes. As a final remark, let us note that the model is characterized by very large gauge-like