Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem

We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 \rho}) in Fourier space, as a function of \rho and the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + \rho) are genuinely different which could lead to a re-interpretation of results in the literature.Comment: Latex, 7 pages, eps files for two figures as well as Europhys. Lett. style files included; slightly expanded reincarnatio

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

Generating Function for Particle-Number Probability Distribution in Directed Percolation

We derive a generic expression for the generating function (GF) of the particle-number probability distribution (PNPD) for a simple reaction diffusion model that belongs to the directed percolation universality class. Starting with a single particle on a lattice, we show that the GF of the PNPD can be written as an infinite series of cumulants taken at zero momentum. This series can be summed up into a complete form at the level of a mean-field approximation. Using the renormalization group techniques, we determine logarithmic corrections for the GF at the upper critical dimension. We also find the critical scaling form for the PNPD and check its universality numerically in one dimension. The critical scaling function is found to be universal up to two non-universal metric factors.Comment: (v1,2) 8 pages, 5 figures; one-loop calculation corrected in response to criticism received from Hans-Karl Janssen, (v3) content as publishe

Theory of Branching and Annihilating Random Walks

A systematic theory for the diffusion--limited reaction processes $A + A \to 0$ and $A \to (m+1) A$ is developed. Fluctuations are taken into account via the field--theoretic dynamical renormalization group. For $m$ even the mean field rate equation, which predicts only an active phase, remains qualitatively correct near $d_c = 2$ dimensions; but below $d_c' \approx 4/3$ a nontrivial transition to an inactive phase governed by power law behavior appears. For $m$ odd there is a dynamic phase transition for any $d \leq 2$ which is described by the directed percolation universality class.Comment: 4 pages, revtex, no figures; final version with slight changes, now accepted for publication in Phys. Rev. Let

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka-Volterra Models

We study the general properties of stochastic two-species models for predator-prey competition and coexistence with Lotka-Volterra type interactions defined on a $d$-dimensional lattice. Introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical, deterministic mean-field picture. Already within mean-field theory, however, spatial constraints, modeling locally limited resources, lead to the emergence of a continuous active-to-absorbing state phase transition. Field-theoretic arguments, supported by Monte Carlo simulation results, indicate that this transition, which represents an extinction threshold for the predator population, is governed by the directed percolation universality class. In the active state, where predators and prey coexist, the classical center singularities with associated population cycles are replaced by either nodes or foci. In the vicinity of the stable nodes, the system is characterized by essentially stationary localized clusters of predators in a sea of prey. Near the stable foci, however, the stochastic lattice Lotka-Volterra system displays complex, correlated spatio-temporal patterns of competing activity fronts. Correspondingly, the population densities in our numerical simulations turn out to oscillate irregularly in time, with amplitudes that tend to zero in the thermodynamic limit. Yet in finite systems these oscillatory fluctuations are quite persistent, and their features are determined by the intrinsic interaction rates rather than the initial conditions. We emphasize the robustness of this scenario with respect to various model perturbations.Comment: 19 pages, 11 figures, 2-column revtex4 format. Minor modifications. Accepted in the Journal of Statistical Physics. Movies corresponding to Figures 2 and 3 are available at http://www.phys.vt.edu/~tauber/PredatorPrey/movies

Ageing in the contact process: Scaling behavior and universal features

We investigate some aspects of the ageing behavior observed in the contact process after a quench from its active phase to the critical point. In particular we discuss the scaling properties of the two-time response function and we calculate it and its universal ratio to the two-time correlation function up to first order in the field-theoretical epsilon-expansion. The scaling form of the response function does not fit the prediction of the theory of local scale invariance. Our findings are in good qualitative agreement with recent numerical results.Comment: 20 pages, 3 figure

Crossover from Isotropic to Directed Percolation

Percolation clusters are probably the simplest example for scale--invariant structures which either are governed by isotropic scaling--laws (``self--similarity'') or --- as in the case of directed percolation --- may display anisotropic scaling behavior (``self--affinity''). Taking advantage of the fact that both isotropic and directed bond percolation (with one preferred direction) may be mapped onto corresponding variants of (Reggeon) field theory, we discuss the crossover between self--similar and self--affine scaling. This has been a long--standing and yet unsolved problem because it is accompanied by different upper critical dimensions: $d_c^{\rm I} = 6$ for isotropic, and $d_c^{\rm D} = 5$ for directed percolation, respectively. Using a generalized subtraction scheme we show that this crossover may nevertheless be treated consistently within the framework of renormalization group theory. We identify the corresponding crossover exponent, and calculate effective exponents for different length scales and the pair correlation function to one--loop order. Thus we are able to predict at which characteristic anisotropy scale the crossover should occur. The results are subject to direct tests by both computer simulations and experiment. We emphasize the broad range of applicability of the proposed method.Comment: 19 pages, written in RevTeX, 12 figures available upon request (from [email protected] or [email protected]), EF/UCT--94/2, to be published in Phys. Rev. E (May 1994

Viability of competing field theories for the driven lattice gas

It has recently been suggested that the driven lattice gas should be described by a novel field theory in the limit of infinite drive. We review the original and the new field theory, invoking several well-documented key features of the microscopics. Since the new field theory fails to reproduce these characteristics, we argue that it cannot serve as a viable description of the driven lattice gas. Recent results, for the critical exponents associated with this theory, are re-analyzed and shown to be incorrect.Comment: 4 pages, revtex, no figure

Novel non-equilibrium critical behavior in unidirectionally coupled stochastic processes

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d_c = 4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A -> A + A, A + A -> A, and A -> \emptyset. We study a hierarchy of such DP processes for particle species A, B,..., unidirectionally coupled via the reactions A -> B, ... (with rates \mu_{AB}, ...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents \beta_i which are markedly reduced at each hierarchy level i >= 2. This scenario can be understood on the basis of the mean-field rate equations, which yield \beta_i = 1/2^{i-1} at the multicritical point. We then include fluctuations by using field-theoretic renormalization group techniques in d = 4-\epsilon dimensions. In the active phase, we calculate the fluctuation correction to the density exponent for the second hierarchy level, \beta_2 = 1/2 - \epsilon/8 + O(\epsilon^2). Monte Carlo simulations are then employed to determine the values for the new scaling exponents in dimensions d<= 3, including the critical initial slip exponent. Our theory is connected to certain classes of growth processes and to certain cellular automata, as well as to unidirectionally coupled pair annihilation processes. We also discuss some technical and conceptual problems of the loop expansion and their possible interpretation.Comment: 29 pages, 19 figures, revtex, 2 columns, revised Jan 1995: minor changes and additions; accepted for publication in Phys. Rev.

Non-equilibrium stationary state of a two-temperature spin chain

A kinetic one-dimensional Ising model is coupled to two heat baths, such that spins at even (odd) lattice sites experience a temperature $T_{e}$ ($$). Spin flips occur with Glauber-type rates generalised to the case of two temperatures. Driven by the temperature differential, the spin chain settles into a non-equilibrium steady state which corresponds to the stationary solution of a master equation. We construct a perturbation expansion of this master equation in terms of the temperature difference and compute explicitly the first two corrections to the equilibrium Boltzmann distribution. The key result is the emergence of additional spin operators in the steady state, increasing in spatial range and order of spin products. We comment on the violation of detailed balance and entropy production in the steady state.Comment: 11 pages, 1 figure, Revte