We introduce and solve a model of hardcore particles on a one-dimensional periodic lattice which undergoes an active-absorbing-state phase transition at finite density. In this model, an occupied site is defined to be active if its left neighbor is occupied and the right neighbor is vacant. Particles from such active sites hop stochastically to their right. We show that both the density of active sites and the survival probability vanish as the particle density is decreased below half. The critical exponents and spatial correlations of the model are calculated exactly using the matrix product ansatz. Exact analytical study of several variations of the model reveals that these nonequilibrium phase transitions belong to a new universality class different from the generic active-absorbing-state phase transition, namely, directed percolation.
We introduce and solve a model of hardcore particles on a one-dimensional periodic lattice which undergoes an active-absorbing-state phase transition at finite density. In this model, an occupied site is defined to be active if its left neighbor is occupied and the right neighbor is vacant. Particles from such active sites hop stochastically to their right. We show that both the density of active sites and the survival probability vanish as the particle density is decreased below half. The critical exponents and spatial correlations of the model are calculated exactly using the matrix product ansatz. Exact analytical study of several variations of the model reveals that these nonequilibrium phase transitions belong to a new universality class different from the generic active-absorbing-state phase transition, namely, directed percolation.
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 1D 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.
We study diffusion, split and reconstitution of hard k-mers on a one-dimensional periodic lattice, subjected to a constraint that the lengths of the k-mers do not increase beyond a fixed value n, however initially, at t=0 the k-mers can have lengths larger than n. For higher values of packing fractions (i.e. higher ρ, where , N is number of sites covered by the k-mers, L is system size) the system gets stuck into absorbing state. On the other hand, for lower , the syst We study diffusion, split and reconstitution of hard k-mers on a one-dimensional periodic lattice, subjected to a constraint that the lengths of the k-mers do not increase beyond a fixed value n, however initially, at t=0 the k-mers can have lengths larger than n. For higher values of packing fractions (i.e. higher ρ, where , N is number of sites covered by the k-mers, L is system size) the system gets stuck into absorbing state. On the other hand, for lower , the system remains in the active state. In this article it is shown that in this class of models, the system continuously undergoes an active to absorbing state phase transition when the density crosses a critical threshold . The active steady state of these models can be exactly solved using matrix product ansatz, and it is shown that the density of active rods (where, Na is number of active rods) vanishes at the transition point as . em remains in the active state. In this article it is shown that in this class of models, the system continuously undergoes an active to absorbing state phase transition when the density crosses a critical threshold . The active steady state of these models can be exactly solved using matrix product ansatz, and it is shown that the density of active rods (where, Na is number of active rods) vanishes at the transition point as .
We report on some extensive analyses of a recently proposed model [A. Lipowski, Phys. Rev. E 60, 6255 (1999)] with infinitely many absorbing states. By performing extensive Monte Carlo simulations, we have determined critical exponents and shown strong evidence that this model is not in the directed percolation universality class. The conjecture that this two-dimensional model exhibits a dimensional reduction (behaving as one-dimensional directed percolation) is firmly disproven. The reason for the model not exhibiting standard directed percolation scaling behavior is traced back to the existence of what we call superabsorbing sites, i.e., absorbing sites that cannot be directly activated by the presence of neighboring activity in one or more than one direction. Supporting this claim we present two strong evidences: (i) in one dimension, where superabsorbing sites do not appear at the critical point, the system behaves as directed percolation, and (ii) in a modified two-dimensional variation of the model, defined on a honeycomb lattice, for which superabsorbing sites are very rarely observed, directed percolation behavior is recovered. Finally, a parallel updating version of the model exhibiting a nonequilibrium first-order transition is also reported.
At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.
We investigate the scaling properties of phase transitions between survival and extinction (active-to-absorbing-state phase transition, AAPT) in a model that by itself belongs to the directed percolation (DP) universality class, interacting with a spatiotemporally fluctuating environment having its own nontrivial dynamics. We model the environment by (i) a randomly stirred fluid, governed by the Navier-Stokes (NS) equation, and (ii) a fluctuating surface, described either by the Kardar-Parisi-Zhang (KPZ) or the Edward-Wilkinson (EW) equations. We show, by using a one-loop perturbative field theoretic setup that, depending upon the spatial scaling of the variance of the external forces that drive the environment (i.e., the NS, KPZ, or EW equations), the system may show weak or strong dynamic scaling at the critical point of active-to-absorbing-state phase transitions. In the former case AAPT displays scaling belonging to the DP universality class, whereas in the latter case the universal behavior is different.
We study in further detail particle models displaying a boundary-induced absorbing state phase transition [Deloubrière and van Wijland Phys. Rev. E 65, 046104 (2002) and Barato and Hinrichsen, Phys. Rev. Lett. 100, 165701 (2008)]. These are one-dimensional systems consisting of a single site (the boundary) where creation and annihilation of particles occur, and a bulk where particles move diffusively. We study different versions of these models and confirm that, except for one exactly solvable bosonic variant exhibiting a discontinuous transition and trivial exponents, all the others display nontrivial behavior, with critical exponents differing from their mean-field values, representing a universality class. Finally, the relation of these systems with a (0+1)-dimensional non-Markovian process is discussed.