Uwe C. Tauber

Theory of Branching and Annihilating Random Walks

Nonequilibrium models with an extensive number of degrees of freedom whose dynamics violates detailed balance occur in studies of many biological, chemical and physical systems. Like equilibrium systems, their stationary states may exhibit phase transitions which in many cases appear to fall into distinct classes characterized by universal quantities such as critical exponents. One of the most common such classes is that exemplified by directed percolation (DP) [1]. This represents a transition from a nontrivial ‘active’ steady state to an absorbing ‘inactive’ state with no fluctuations. Many nonequilibrium phase transitions appear to belong to this universality class, e.g., the contact process [2], the dimer poisoning problem in the ZGB model [3], and auto–catalytic reaction models [4]. The universal properties of the DP transition are theoretically well understood in the context of a renormalization group (RG) analysis based on an expansion around mean field theory below the upper critical dimension dc = 4 [5]. More recently a class of models has been studied which, in certain cases, appear as exceptions to the general rule that such transitions should fall into the DP universality class. These include a probabilistic cellular automaton model [6], certain kinetic Ising models [7,8], and an interacting monomer–dimer model [9]. In one dimension the dynamics of these is equivalent to a class of models called branching and annihilating random walks (BARWs) [10–12], which also have a natural generalization to higher dimensions. In the language of reaction– diffusion systems, BARWs describe the stochastic dynamics of a single species of particles A undergoing three basic processes: diffusion, often modeled by a random walk on a lattice and characterized by a diffusion coefficient D; an annihilation reaction A + A → ⊘ when particles are close (or on the same site), at rate λ; and a branching process A → (m + 1)A (where m is a positive integer), at rate σm. The above–mentioned one– dimensional models all correspond to the case m = 2. For the kinetic Ising model, the particles A are to be identified with the domain walls, and the transition to the inactive state corresponds to the ordering of the Ising spins [7,8]. In general, this new universality class has been observed in d = 1 for even values of m, when the number of particles is locally conserved modulo 2. When m is odd, the DP values of the exponents appear to be realized. (It should be remarked that several of the models which have been studied do not contain three independent parameters corresponding to D, λ, and σm so that it may occur that the actual transition is inaccessible. This appears to be so for the simplest lattice BARW model with m = 2, which is always in the inactive phase [10].) Besides the appearance of a new universality class, another issue which clearly requires theoretical explanation is the occurrence of a transition at a finite value of σm. For the mean field rate equation for the average density u n(t) = −2λ n(t) 2 + mσm n(t) (1)

Applications of field-theoretic renormalization group methods to reaction-diffusion problems

We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction–diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction–diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA → �A (� < k ). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization and Callan–Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative � = dc − d expansions with respect to the upper critical dimension dc. With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, L´ evy flights, persistence problems and the influence of spatial boundaries. We also analyse field-theoretic approaches to non-equilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

Field Theory of Branching and Annihilating Random Walks

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A → ∅ and A → (m + 1) A, where m ≥ 1. Starting from the master equation, a field-theoretic representation of the problem is derived, and fluctuation effects are taken into account via diagrammatic and renormalization group methods. For d > 2, the mean-field rate equation, which predicts an active phase as soon as the branching process is switched on, applies qualitatively for both even and odd m, but the behavior in lower dimensions is shown to be quite different for these two cases. For even m, and d near 2, the active phase still appears immediately, but with nontrivial crossover exponents which we compute in an expansion in ∈ = 2 − d, and with logarithmic corrections in d = 2. However, there exists a second critical dimension d′c ≈ 4/3 below which a nontrivial inactive phase emerges, with asymptotic behavior characteristic of the pure annihilation process. This is confirmed by an exact calculation in d = 1. The subsequent transition to the active phase, which represents a new nontrivial dynamic universality class, is then investigated within a truncated loop expansion, which appears to give a correct qualitative picture. The model with m = 2 is also generalized to N species of particles, which provides yet another universality class and which is exactly solvable in the limit N → ∞. For odd m, we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d ≤ 2. In this case, the transition to the active phase is in the directed percolation universality class. Finally, we study the modification when the annihilation reaction is 3A → ∅. When m = 0 (mod 3) the system is always in its active phase, but with logarithmic crossover corrections for d = 1, while the other cases should exhibit a directed percolation transition out of a fluctuation-driven inactive phase.

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.

Two-loop renormalization-group analysis of the Burgers–Kardar-Parisi-Zhang equation

A systematic analysis of the Burgers--Kardar-Parisi-Zhang equation in d+1 dimensions by dynamic renormalization-group theory is described. The fixed points and exponents are calculated to two-loop order. We use the dimensional regularization scheme, carefully keeping the full d dependence originating from the angular parts of the loop integrals. For dimensions less than

Fluctuations and correlations in lattice models for predator-prey interaction.

{\mathit{d}}_{\mathit{c}}

Field-Theory Approaches to Nonequilibrium Dynamics

=2 we find a strong-coupling fixed point, which diverges at d=2, indicating that there is nonperturbative strong-coupling behavior for all d\ensuremath{\ge}2. At d=1 our method yields the identical fixed point as in the one-loop approximation, and the two-loop contributions to the scaling functions are nonsingular. For dg2 dimensions, there is no finite strong-coupling fixed point. In the framework of a 2+\ensuremath{\epsilon} expansion, we find the dynamic exponent corresponding to the unstable fixed point, which described the nonequilibrium roughening transition, to be z=2+O(

Spatial rock-paper-scissors models with inhomogeneous reaction rates

{\mathrm{\ensuremath{\epsilon}}}^{3}

Mode-coupling and renormalization group results for the noisy Burgers equation.

), in agreement with a recent scaling argument by Doty and Kosterlitz [Phys. Rev. Lett. 69, 1979 (1992)]. Similarly, our result for the correlation length exponent at the transition is 1/\ensuremath{\nu}=\ensuremath{\epsilon}+O(

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

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