Reentrant phase diagram of branching annihilating random walks with one and two offspring

Emerging extreme value and Fermi-Dirac distributions in the Lévy branching and annihilating process

We study the dynamics of the branching and annihilating process with long-range interactions. Static particles generate an offspring and annihilate upon contact. The branching distance is supposed to follow a Lévy-like power-law distribution with P(r)∝1/r^{α}. We analyze the long term behavior of the mean particles number and its fluctuations as a function of the parameter α that controls the range of the branching process. We show that the dynamic exponent associated with the particle number fluctuations varies continuously for α<4 while the particle number exponent only changes for α<3. A crossover from extreme value Frechet (at α=3) and Gumbell (for 2<α<3) distributions is developed, similar to the one reported in recent experiments with cw-pumped random fiber lasers presenting underlying gain and Lévy processes. We report the dependence of the relevant dynamical power-law exponents on α showing that explosive growth takes place for α≤2. Further, the average occupation number distribution is shown to evolve from the standard Fermi-Dirac form to the generalized one within the context of nonextensive statistics.

Branching annihilating random walks with long-range attraction in one dimension

We introduce and numerically study the branching annihilating random walks with long-range attraction (BAWL). The long-range attraction makes hopping biased in such a manner that particle's hopping along the direction to the nearest particle has larger transition rate than hopping against the direction. Still, unlike the Lévy flight, a particle only hops to one of its nearest-neighbor sites. The strength of bias takes the form x^{-σ} with non-negative σ, where x is the distance to the nearest particle from a particle to hop. By extensive Monte Carlo simulations, we show that the critical decay exponent δ varies continuously with σ up to σ=1 and δ is the same as the critical decay exponent of the directed Ising (DI) universality class for σ≥1. Investigating the behavior of the density in the absorbing phase, we argue that σ=1 is indeed the threshold that separates the DI and non-DI critical behavior. We also show by Monte Carlo simulations that branching bias with symmetric hopping exhibits the same critical behavior as the BAWL.

Why patchy diffusion-limited aggregation belongs to the directed-percolation universality class

We present a possible link between nonequilibrium phase transition observed in patchy diffusion-limited aggregation (DLA) [M. J. Kartha and A. Sayeed, Phys. Lett. A 380, 2791 (2016)10.1016/j.physleta.2016.06.036] and directed bond percolation (DP). A system of directed percolation with patchy particles (patchy DP) in which the bond connectivity is established depending on patch size p is analyzed. It is observed that patchy DP starting from a single seed shows a nonequilibrium phase transition. Below a critical value of the patch size p_{c}, the system reaches an absorbing state above which is a fluctuating active state as observed in the DP system. The value of this nonuniversal parameter p_{c} is observed to be slightly higher than the value observed in patchy DLA. Close to the critical value, the order parameter P(∞)∼(p-p_{c})^{β} where β=0.272±0.010, which is consistent with the directed-percolation universality class. Therefore the intrinsic nature of patchy DP is responsible for the phase transition in patchy DLA. This study reveals that the estimated critical value of patch size p_{c}=0.80625±0.00020 in patchy DP is different from the critical bond probability p_{c}=0.6447 in the DP system. This elucidates that the bond probability in DP is not equivalent to the patch probability of a particular site. Our work also gives an insight into the problem related with formation of an extended network of pentagon subunits in connection with the virus capsid.

Absorbing phase transition in a conserved lattice gas model with next-nearest-neighbor hopping in one dimension

The absorbing phase transition of the modified conserved lattice gas (m-CLG) model was investigated in one dimension. The m-CLG model was modified from the conserved lattice gas (CLG) model in such a way that each active particle hops to one of the nearest-neighbor and next-nearest-neighbor empty sites. The order parameter exponent, the dynamic exponent, and the correlation length exponent were estimated from the power-law behavior and finite-size scaling of the active particle densities. The exponents were found to differ considerably from those of the ordinary CLG model and were also distinct from those of the Manna model, suggesting that next-nearest-neighbor hopping is a relevant factor that alters the critical behavior in the one-dimensional CLG model.

Universality class of the conserved Manna model in one dimension

The nonequilibrium absorbing phase transition of the discrete conserved Manna model was studied via Monte Carlo simulations on a one-dimensional chain, using the natural initial states with a sequential update. The critical density of the particles was found to be smaller than the recently reported value, and the order-parameter exponent was considerably different from the directed percolation (DP) value. The influence of quenched disorder was also studied on a diluted strip of L_{x}×L_{y} lattice sites with L_{x}≫L_{y}, and the results were compared with those of the contact process (CP). It was found that the Manna model and the CP exhibited distinctly different behaviors; the CP exhibited nonuniversal power-law decreases of active-site densities in the Griffith phase, whereas the Manna model showed a standard critical behavior. These results consistently suggest that the Manna model belongs to a universality class that is different from the DP class.

Critical behavior of absorbing phase transitions for models in the Manna class with natural initial states

The critical behavior of absorbing phase transitions for two typical models in the Manna universality class, the conserved Manna model and the conserved lattice gas model, both on a square lattice, was investigated using the natural initial states. Various critical exponents were estimated using the static and dynamic simulations. The exponents characterizing dynamics of active particles differ considerably from the known exponents obtained using the random initial states, whereas those associated with the steady-state quantities remain the same. The critical exponents for both models were consistent with errors of less than 1% and satisfied the known scaling relations; thus, the known violation of scaling relations for models with a conserved field was resolved using the natural initial states. The results differed by 7%∼12% from the directed percolation values.

Influence of quenched disorder on absorbing phase transitions in the conserved lattice gas model

Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.

Discontinuous phase transitions of conserved threshold transfer process with deterministic hopping

The deterministic conserved threshold transfer process, which is a variant of the conserved threshold transfer process modified in a way as that hopping of a particle is to be deterministic, is proposed. The critical behavior of the model is investigated in one, two, and four dimensions. It is found that the order parameter yields a discontinuous transition; i.e., the transition appears to be first ordered in all dimensions considered. The origin of such a discontinuous transition is investigated, considering clustering of active sites and accumulation of critical sites just before the steady state is reached.

Crossover from the parity-conserving pair contact process with diffusion to other universality classes

The pair contact process with diffusion (PCPD) with modulo 2 conservation (PCPD2) [ 2A-->4A , 2A-->0 ] is studied in one dimension, focused on the crossover to other well established universality classes: the directed Ising (DI) and the directed percolation (DP). First, we show that the PCPD2 shares the critical behaviors with the PCPD, both with and without directional bias. Second, the crossover from the PCPD2 to the DI is studied by including a parity-conserving single-particle process (A-->3A) . We find the crossover exponent 1/varphi_{1}=0.57(3) , which is argued to be identical to that of the PCPD-to-DP crossover by adding A-->2A . This suggests that the PCPD universality class has a well-defined fixed point distinct from the DP. Third, we study the crossover from a hybrid-type reaction-diffusion process belonging to the DP [ 3A-->5A , 2A-->0 ] to the DI by adding A-->3A . We find 1/varphi_{2}=0.73(4) for the DP-to-DI crossover. The inequality of varphi_{1} and varphi_{2} further supports the non-DP nature of the PCPD scaling. Finally, we introduce a symmetry-breaking field in the dual spin language to study the crossover from the PCPD2 to the DP. We find 1/varphi_{3}=1.23(10) , which is associated with a new independent route from the PCPD to the DP.

Validity of scaling relations in absorbing phase transitions with a conserved field

The two scaling relations in absorbing phase transitions, nu_||=beta/theta and z=nu_||/nu_(perpendicular), are studied for a conserved lattice gas model. The critical indices calculated elaborately from the all-sample average density of active particles appear to satisfy both relations. However, the exponent nu_(perpendicular) calculated from the surviving samples does not appear to be consistent with the value in the thermodynamic limit. This is in contrast with earlier observations [M. Rossi, Phys. Rev. Lett. 85, 1803 (2000); S. Lübeck and P. C. Heger, Phys. Rev. E. 68, 056102 (2003)], in that the former scaling relation was claimed to be violated.

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 Z_2 symmetric absorbing phases (the cellular automaton version of the nonequilibrium kinetic Ising model (NEKIMCA) and a stochastic cellular automaton invented by Grassberger, Krause, and von der Twer [J. Phys. A 17, L105 (1984)]) 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: 1/phi approximately equal to 0.53(2) . In the case of a branching and annihilating random walk model with an even number of offspring (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: 1/phi approximately equal to 0.205(5) . The two different kinds of crossover operators cannot be mapped onto each other and the resulting models show a diversity within the DP universality class in one dimension. These subclasses differ in cluster scaling exponents.

Three different routes from the directed Ising to the directed percolation class

The scaling nature of absorbing critical phenomena is well understood for the directed percolation (DP) and the directed Ising (DI) systems. However, a full analysis of the crossover behavior is still lacking, which is of our interest in this study. In one dimension, we find three different routes from the DI to the DP classes by introducing a symmetry-breaking field (SB), breaking a modulo 2 conservation (CB), or making channels connecting two equivalent absorbing states (CC). Each route can be characterized by a crossover exponent, which is found numerically as phi=2.1+/-0.1 (SB), 4.6+/-0.2 (CB), and 2.9+/-0.1 (CC), respectively. The difference between the SB and CB crossover can be understood easily in the domain wall language, while the CC crossover involves an additional critical singularity in the auxiliary field density with the memory effect to identify itself independent.

Universality split in absorbing phase transition with conserved field on fractal lattices

The universality split in absorbing phase transition between the conserved lattice gas (CLG) model and the conserved threshold transfer process (CTTP) is investigated on a checkerboard fractal and on a Sierpinski gasket. The critical exponents theta, beta, nu||, and z, which are associated with, respectively, the density of active particles in time, the order parameter, the temporal correlation length, and the dynamics of active particles, are elaborately measured for two models on selected fractal lattices. The exponents for the CLG model are found to be distinctly different from those of the CTTP model on a checkerboard fractal, whereas the two models exhibit the same critical behavior on a Sierpinski gasket, indicating that the universality split between the two models occurs only on a checkerboard fractal. Such a universality split is attributed from the dominant hopping mechanisms caused by the intrinsic properties of the underlying fractal lattice.

Absorbing phase transition in a conserved lattice gas model in one dimension

We investigated the nonequilibrium phase transition of the conserved lattice gas model in one dimension using two update methods: i.e., the sequential update and the parallel update. We measured the critical indices of theta, beta, nu(parallel), and nu(perpendicular), and found that, for a parallel update, the exponents were delicately influenced by the hopping rule of active particles. When the hopping rule was designed to be symmetric, the results were found to be consistent with those of the sequential update. The exponents we obtained were precisely the same as the corresponding results of a recently presented lattice model of two species of particles with a conserved field in one dimension, in contrast with the authors' claim. We also found that one of the scaling relations known for absorbing phase transition is violated.

Absorbing phase transition in conserved lattice gas model on fractal lattices

We study the continuous phase transition of the conserved lattice gas (CLG) model from an active phase into an absorbing phase on two fractal lattices, i.e., on a checkerboard fractal and on a Sierpinski gasket. In the CLG model, a particle is assumed to be active if any of the neighboring sites are occupied by a particle and inactive if all neighboring sites are empty. We estimate critical exponents theta, beta, nu||, and nu perpendicular, characterizing, respectively, the density of active particles in time, the order parameter, the temporal and spatial correlation lengths near the critical point, and the results are confirmed by off-critical scaling and finite size scaling. The order parameter exponent beta on a checkerboard fractal appears to lie between the one-dimensional (1D) value and two-dimensional (2D) value of the CLG model, while that on a Sierpinski gasket lies between the 1D and 2D values of the conserved threshold transfer process. Such a difference is manifested based on the intrinsic properties of the underlying fractal lattices.

Double domain structure of the pair contact process with diffusion

We investigate the domain structure of the pair contact process with diffusion (PCPD). PCPD is a stochastic reaction-diffusion model which evolves by the competition of two binary reactions, 2A-->3A and 2A-->0 . In addition, each particle diffuses isotropically, which leads to the bidirectional coupling between solitary particles and pairs. The spreading domain formed from localized activities in vacuum consists of two regions: The coupled region of size R{p} where pairs and solitary particles coexist and the uncoupled region of size R{U} where only solitary particles exist, respectively. The size of the whole domain R is given as R=R{p}+R{U} . At criticality, R{p} and R{U} scale as R{p} approximately t(1/Z{p}) and R{U} approximately t(1/Z{U}) with Z{U}>Z{p}. We estimate Z{p}=1.61(1) and Z{U}=1.768(8) . Hence the correction to the scaling of R,Q=R{U}/R{p} slowly decays extremely, which makes it practically impossible to identify the asymptotic scaling behavior of R. The double domain structure is another reason for the extremely slow approach to the asymptotic scaling regime of PCPD.

Scaling of spreading in models unidirectionally coupled to source particles

We investigate the spreading behavior of evolving clusters using unidirectionally coupled two-level hierarchies in one spatial dimension. In the hierarchy, while only two source particles (A) hop away from each other without branching its offspring on the bottom level, different species of particles (B) evolve according to given dynamics belonging to one of known universality classes on the top level. Two levels are unidirectionally coupled from the bottom to the top level by the branching A-->A+2B. We derive the spreading exponent zU of the uncoupled region of size RU(t) approximately tzU up to the first order correction in terms of the spreading exponent of source particles (zA) and that of given dynamics of the top level (zo) as zU=(1--zA)zo/(1-zo). From the relation, zA and zU always satisfy the inequality zU< or =zA for zA> or =zo. The inequality confirms that the scaling of the spreading in the slave level should follow the scaling of the source in unidirectionally coupled systems. We numerically confirm the relation for three different B-particle dynamics; annihilating random walks, branching annihilating random walks with one and two offspring which belong to the directed percolation, and the parity conserving universality class respectively.

Absorbing phase transitions of unidirectionally coupled nonequilibrium systems

We investigate the change of critical behavior of two-level hierarchy systems in which the second level (B) is unidirectionally coupled to the first (A) by the coupling dynamics A --> A + nB with n = 1 or 2. The first level belongs to the directed percolation or the parity-conserving (PC) universality class, the second to PC. If both levels are critical, the active region of the second level becomes heterogeneous. In the so-called coupled region the first level feeds particles to the second, while in the uncoupled region the second level evolves autonomously. Measuring dynamic critical exponents in both regions, we show to what extent the critical behavior of the second level depends on the universality class of the first. These results suggest a simple criterion for the emergence of unusual critical behavior of unidirectionally coupled nonequilibrium systems.

Branching and annihilating Lévy flights

We consider a system of particles undergoing the branching and annihilating reactions A-->(m+1)A and A+A-->Ø, with m even. The particles move via long-range Lévy flights, where the probability of moving a distance r decays as r(-d-sigma). We analyze this system of branching and annihilating Lévy flights using field theoretic renormalization group techniques close to the upper critical dimension d(c)=sigma with sigma<2. These results are then compared with Monte Carlo simulations in d=1. For sigma close to unity in d=1, the critical point for the transition from an absorbing to an active phase occurs at zero branching. However, for sigma bigger than about 3/2 in d=1, the critical branching rate moves away from zero with increasing sigma, and the transition lies in a different universality class, inaccessible to controlled perturbative expansions. We measure the exponents in both universality classes and examine their behavior as a function of sigma.

Critical branching-annihilating random walk of two species

The effect of blocking between different species occurring in one dimension is investigated here numerically in the case of particles following branching and annihilating random walk. It is shown that two-dimensional simulations confirm the field theoretical results with logarithmic corrections. In one dimension, however, if particles exhibit hard core interaction I confirm the very recent predictions of Kwon et al. [Phys. Rev. Lett. 85, 1682 (2000)] that there are two different universality classes depending on the spatial symmetry of the offspring production characterized by beta(S)=0.5 and beta(A)=2. Elaborate analysis of simulation data shows that the order parameter exponent beta does not depend on initial conditions or on diffusion rates of species but strong correction to scaling is observed. By systematic numerical simulations the critical point properties have been explored and initial condition dependence of the dynamical exponents Z and alpha is shown. In the case of a random initial state the particle-density decay at the critical point follows the t(-1/4) law with logarithmic corrections with two offsprings.

Does hard core interaction change absorbing-type critical phenomena?

It has been generally believed that hard core interaction is irrelevant to absorbing-type critical phenomena because the particle density is so low near an absorbing phase transition. We study the effect of hard core interaction on the N-species branching annihilating random walks with two offspring and report that hard core interaction drastically changes the absorbing-type critical phenomena in a nontrivial way. Through a Langevin equation-type approach, we predict analytically the values of the scaling exponents, nu( perpendicular) = 2, z = 2, alpha = 1/2, and beta = 2 in one dimension for all N>1. Direct numerical simulations confirm our prediction. When the diffusion coefficients for different species are not identical, nu( perpendicular) and beta vary continuously with the ratios between the coefficients.

Surface critical behavior in systems with nonequilibrium phase transitions

We study the surface critical behavior of branching-annihilating random walks with an even number of offspring (BARW) and directed percolation (DP) using a variety of theoretical techniques. Above the upper critical dimensions d(c), with d(c)=4 (DP) and d(c)=2 (BARW), we use mean field-theory to analyze the surface phase diagrams using the standard classification into ordinary, special, surface, and extraordinary transitions. For the case of BARW, at or below the upper critical dimension d</=d(c), we use field theoretic methods to study the effects of fluctuations. As in the bulk, the field-theory suffers from technical difficulties associated with the presence of a second critical dimension. However, we are still able to analyze the phase diagrams for BARW in d=1 and 2, which turn out to be very different from their mean field analog. Furthermore, for the case of BARW only (and not for DP), we find two independent surface beta(1) exponents in d=1, arising from two distinct definitions of the order parameter. Using an exact duality transformation on a lattice BARW model in d=1, we uncover a relationship between these two surface beta(1) exponents at the ordinary and special transitions. Many of our predictions are supported using Monte Carlo simulations of two different models belonging to the BARW universality class.