Critical behavior of the pair contact process

Abstract

I study a nonequilibrium lattice model, the pair contact process, in which pairs of particles annihilate with probability p or else create a particle at a vacant nearest neighbor. The model exhibits a continuous phase transition from an active state, with an ongoing production of particles, to an absorbing state without pairs. The model has infinitely many absorbing states. Computer simulations in ID yielded critical exponents consistent with directed percolation, for the first time placing a model with infinitely many absorbing states firmly in the directed-percolation universality class.PACS numbers: 05.50.+q, 05.70.Ln The study of nonequilibrium many-particle systems is an important problem in many branches of physics, chemistry, biology, and even sociology [1,2]. Much attention has been given to nonequilibrium models exhibiting a continuous phase transition from an active steady state to a unique absorbing state (a state in which the system is trapped). One of the major achievements in the study of nonequilibrium phase transitions is the discovery that a wide variety of models exhibiting this kind of transition belong to the same universality class. Among these models the best known are probably the contact process [3-5], Schlogl's first and second models [6-8], directed percolation (DP) [9-12], Reggeon field theory (RFT) [13][14][15][16], and the ZGB (Ziflf-Gulari-Barshad) model [17][18][19]. The study of these and many other models [20][21][22][23][24][25][26] demonstrates the robustness of DP critical behavior in spite of quite dramatic differences in the evolution rules of the various models. Presently there is thus substantial evidence in favor of the hypothesis that models with a scalar order parameter exhibiting a continuous transition to a unique absorbing state generically belong to the universality class of directed percolation. This DP conjecture was first put forward by Grassberger [8] and Janssen [7] and later extended by Grinstein, Lai, and Browne [18] to multicomponent models such as the ZGB model.Whereas the universality of DP critical behavior for models with a single absorbing state seems well established, the study of models with more than one absorbing state is still in its very beginning. That models with more than one absorbing state can exhibit new critical behavior was first demonstrated by Grassberger, Krause, and von der Twer [27] in a study of a model involving the processes A'-• 3X and 2X-^ 0. This model is very similar to a class of models known as branching annihilating walks (BAW's) [24]. In the BAW a particle jumps, with probability p, to a nearest neighbor and if this site is occupied both particles annihilate. With probability 1 -p the particle produces n offspring which are placed on the neighboring sites. If an offspring is created on a site which is already occupied, it annihilates with the occupying particle leaving an empty site. For n even these models have non-DP behavior [25] whereas the behavior for n odd is compatible with DP [26]. Note that in both the mode...

Stochastic dynamics and logistic population growth

“…On the other hand, if a and n 0 are both even or if a is odd, the absorbing state is accessible and the population becomes extinct. It is worth noting that these kinetic rules can be implemented as dynamical lattice models or interacting particle systems, for example as a contact process or a branchingannihilating random walk (BARW) [49][50][51][52][53][54][55] and that parity conservation, or the lack thereof, also plays a crucial role in the dynamics of these spatially extended systems. They can display a nonequilibrium transition from a nontrivial fluctuating steady state to an absorbing state with no fluctuations [53].…”

Static critical behavior in the inactive phase of the pair contact process

“…In the case of PCP, the field responsible for the dynamics (the density of pairs of particles, ρ 2 ) is coupled to another field (the density of isolated particles, ρ 1 ). This background of isolated particles is responsible for the non-universality of some dynamic properties of the system at criticality [3,4]. Recently the one-dimensional PCP with particle diffusion (known as PCPD or annihilation/fission model) has received a lot of attention and was at the center of some controversy [5,6,8,7,9]; nevertheless its critical behaviour is not yet fully clarified.…”

Universality class of absorbing transitions with continuously varying critical exponents

“…Any state with only isolated particles is an absorbing state. But there is no infinite dynamic barrier between them and the transition belongs to the DP class [13].…”

“…For instance, the PCP [13] and the modified interacting monomerdimer (IMD-IMA) model [24] have infinitely many absorbing states. In the PCP, a frustration between any of the absorbing states can disappear locally, so the absorbing transition falls into the DP class.…”

“…the Domany-Kinzel cellular automaton [10], the Ziff-Gulari-Barshad model for a surface catalytic reaction [11], the branching-annihilating random walks with an odd number of offspring [12], and the pair contact process (PCP) [13]. Unlike others, the PCP has infinitely many absorbing states, which leads to controversial transient behaviors, i.e., nonuniversal scaling [13,14,15,16] versus absence of scaling [17,18]. But its stationary critical behavior still belongs to the DP class at low dimensions [19].…”