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)