Universality classes in anisotropic non-equilibrium growth models

We study the effect of generic spatial anisotropies on the scaling behavior in the Kardar-Parisi-Zhang equation. In contrast to its "conserved" variants, anisotropic perturbations are found to be relevant in d > 2 dimensions, leading to rich phenomena that include novel universality classes and the possibility of first-order phase transitions and multicritical behavior. These results question the presumed scaling universality in the strong-coupling rough phase, and shed further light on the connection with generalized driven diffusive systems.Comment: 4 pages, revtex, 2 figures (eps files enclosed

The role of diffusion in branching and annihilation random walk models

Different branching and annihilating random walk models are investigated by cluster mean-field method and simulations in one and two dimensions. In case of the A -> 2A, 2A -> 0 model the cluster mean-field approximations show diffusion dependence in the phase diagram as was found recently by non-perturbative renormalization group method (L. Canet et al., cond-mat/0403423). The same type of survey for the A -> 2A, 4A -> 0 model results in a reentrant phase diagram, similar to that of 2A -> 3A, 4A -> 0 model (G. \'Odor, PRE {\bf 69}, 036112 (2004)). Simulations of the A -> 2A, 4A -> 0 model in one and two dimensions confirm the presence of both the directed percolation transitions at finite branching rates and the mean-field transition at zero branching rate. In two dimensions the directed percolation transition disappears for strong diffusion rates. These results disagree with the predictions of the perturbative renormalization group method.Comment: 4 pages, 4 figures, 1 table include

Influence of local carrying capacity restrictions on stochastic predator-prey models

We study a stochastic lattice predator-prey system by means of Monte Carlo simulations that do not impose any restrictions on the number of particles per site, and discuss the similarities and differences of our results with those obtained for site-restricted model variants. In accord with the classic Lotka-Volterra mean-field description, both species always coexist in two dimensions. Yet competing activity fronts generate complex, correlated spatio-temporal structures. As a consequence, finite systems display transient erratic population oscillations with characteristic frequencies that are renormalized by fluctuations. For large reaction rates, when the processes are rendered more local, these oscillations are suppressed. In contrast with site-restricted predator-prey model, we observe species coexistence also in one dimension. In addition, we report results on the steady-state prey age distribution.Comment: Latex, IOP style, 17 pages, 9 figures included, related movies available at http://www.phys.vt.edu/~tauber/PredatorPrey/movies

Kinetics of phase-separation in the critical spherical model and local scale-invariance

The scaling forms of the space- and time-dependent two-time correlation and response functions are calculated for the kinetic spherical model with a conserved order-parameter and quenched to its critical point from a completely disordered initial state. The stochastic Langevin equation can be split into a noise part and into a deterministic part which has local scale-transformations with a dynamical exponent z=4 as a dynamical symmetry. An exact reduction formula allows to express any physical average in terms of averages calculable from the deterministic part alone. The exact spherical model results are shown to agree with these predictions of local scale-invariance. The results also include kinetic growth with mass conservation as described by the Mullins-Herring equation.Comment: Latex2e with IOP macros, 28 pp, 2 figures, final for

Critical behavior of the two dimensional 2A->3A, 4A->0 binary system

The phase transitions of the recently introduced 2A -> 3A, 4A -> 0 reaction-diffusion model (G.Odor, PRE 69 036112 (2004)) are explored in two dimensions. This model exhibits site occupation restriction and explicit diffusion of isolated particles. A reentrant phase diagram in the diffusion - creation rate space is confirmed in agreement with cluster mean-field and one-dimensional results. For strong diffusion a mean-field transition can be observed at zero branching rate characterized by $\alpha=1/3$ density decay exponent. In contrast with this for weak diffusion the effective 2A ->3A->4A->0 reaction becomes relevant and the mean-field transition of the 2A -> 3A, 2A -> 0 model characterized by $\alpha=1/2$ also appears for non-zero branching rates.Comment: 5 pages, 5 figures included, small correction

Three-fold way to extinction in populations of cyclically competing species

Species extinction occurs regularly and unavoidably in ecological systems. The time scales for extinction can broadly vary and inform on the ecosystem's stability. We study the spatio-temporal extinction dynamics of a paradigmatic population model where three species exhibit cyclic competition. The cyclic dynamics reflects the non-equilibrium nature of the species interactions. While previous work focusses on the coarsening process as a mechanism that drives the system to extinction, we found that unexpectedly the dynamics to extinction is much richer. We observed three different types of dynamics. In addition to coarsening, in the evolutionary relevant limit of large times, oscillating traveling waves and heteroclinic orbits play a dominant role. The weight of the different processes depends on the degree of mixing and the system size. By analytical arguments and extensive numerical simulations we provide the full characteristics of scenarios leading to extinction in one of the most surprising models of ecology

Facilitated spin models in one dimension: a real-space renormalization group study

We use a real-space renormalization group (RSRG) to study the low temperature dynamics of kinetically constrained Ising chains (KCICs). We consider the cases of the Fredrickson-Andersen (FA) model, the East model, and the partially asymmetric KCIC. We show that the RSRG allows one to obtain in a unified manner the dynamical properties of these models near their zero-temperature critical points. These properties include the dynamic exponent, the growth of dynamical lengthscales, and the behaviour of the excitation density near criticality. For the partially asymmetric chain the RG predicts a crossover, on sufficiently large length and time scales, from East-like to FA-like behaviour. Our results agree with the known results for KCICs obtained by other methods.Comment: 13 pages. Extended East model RG to arbitrary block sizes. To appear in Phys. Rev.

Nonequilibrium critical dynamics of the relaxational models C and D

We investigate the critical dynamics of the $n$-component relaxational models C and D which incorporate the coupling of a nonconserved and conserved order parameter S, respectively, to the conserved energy density rho, under nonequilibrium conditions by means of the dynamical renormalization group. Detailed balance violations can be implemented isotropically by allowing for different effective temperatures for the heat baths coupling to the slow modes. In the case of model D with conserved order parameter, the energy density fluctuations can be integrated out. For model C with scalar order parameter, in equilibrium governed by strong dynamic scaling (z_S = z_rho), we find no genuine nonequilibrium fixed point. The nonequilibrium critical dynamics of model C with n = 1 thus follows the behavior of other systems with nonconserved order parameter wherein detailed balance becomes effectively restored at the phase transition. For n >= 4, the energy density decouples from the order parameter. However, for n = 2 and n = 3, in the weak dynamic scaling regime (z_S <= z_rho) entire lines of genuine nonequilibrium model C fixed points emerge to one-loop order, which are characterized by continuously varying critical exponents. Similarly, the nonequilibrium model C with spatially anisotropic noise and n < 4 allows for continuously varying exponents, yet with strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium perturbations leads to genuinely different critical behavior with softening only in subsectors of momentum space and correspondingly anisotropic scaling exponents. Similar to the two-temperature model B the effective theory at criticality can be cast into an equilibrium model D dynamics, albeit incorporating long-range interactions of the uniaxial dipolar type.Comment: Revtex, 23 pages, 5 eps figures included (minor additions), to appear in Phys. Rev.

Critical scaling and aging near the flux-line-depinning transition

We utilize Langevin molecular dynamics simulations to study dynamical critical behavior of magnetic flux lines near the depinning transition in type-II superconductors subject to randomly distributed attractive point defects. We employ a coarse-grained elastic line Hamiltonian for the mutually repulsive vortices and purely relaxational kinetics. In order to infer the stationary-state critical exponents for the continuous non-equilibrium depinning transition at zero temperature T = 0 and at the critical driving current density j_c, we explore two-parameter scaling laws for the flux lines' gyration radius and mean velocity as functions of the two relevant scaling fields T and j - j_c. We also investigate critical aging scaling for the two-time height auto-correlation function in the early-time non-equilibrium relaxation regime to independently measure critical exponents. We provide numerical exponent values for the distinct universality classes of non-interacting and repulsive vortices

Cluster mean-field study of the parity conserving phase transition

The phase transition of the even offspringed branching and annihilating random walk is studied by N-cluster mean-field approximations on one-dimensional lattices. By allowing to reach zero branching rate a phase transition can be seen for any N <= 12.The coherent anomaly extrapolations applied for the series of approximations results in $\nu_{\perp}=1.85(3)$ and $\beta=0.96(2)$.Comment: 6 pages, 5 figures, 1 table included, Minor changes, scheduled for pubication in PR