Nonperturbative Renormalization-Group Study of Reaction-Diffusion Processes

Abstract

We generalize nonperturbative renormalization group methods to nonequilibrium critical phenomena. Within this formalism, reaction-diffusion processes are described by a scale-dependent effective action, the flow of which is derived. We investigate branching and annihilating random walks with an odd number of offspring. Along with recovering their universal physics (described by the directed percolation universality class), we determine their phase diagrams and predict that a transition occurs even in three dimensions, contrarily to what perturbation theory suggests.

What can be learnt from the nonperturbative renormalization group?

“…However, all these differences do not prevent us from constructing an analog of the effective average action method for out of equilibrium systems [41,42]. The nontrivial point is to choose a R k function.…”

“…We have also chosen to take it time-independent so that the time integral in the RG equation can be trivially performed [41]. It is not known whether this corresponds to a "good" choice from the accuracy point of view.…”

Reaction–diffusion processes and non-perturbative renormalization group

“…To construct Γ k , one needs to freeze the slow modes. This is achieved by adding a scale-dependent mass-like term to the original action [5,1,15]:…”

“…(8) ensures that Γ k has the proper limit at k = Λ: Γ k=Λ ∼ S [5,15]. An exact functional differential equation governs the RG flow of Γ k under an infinitesimal change of the scale s = log(k/Λ) [5,1]:…”

“…Indeed, this method has lead to great successes for systems at equilibrium during the last decade, appearing as a well-adapted tool to tackle strong coupling problems [4,5]. Moreover, the NPRG formalism has been recently extended to non-equilibrium systems [1] and has given rise to important progress in the field of reaction-diffusion processes, unveiling genuinely non-perturbative effects [1,2,3], which we review in this paper. This paper is organised as follows.…”