Scale invariance and dynamic phase transitions in diffusion-limited reactions

TL;DR

This paper explores the critical behavior and universality classes of diffusion-limited reactions, analyzing phase transitions and scaling properties using renormalization group techniques and path-integral methods.

Contribution

It provides a comprehensive overview of universality classes in diffusion-limited reactions, including new insights into parity conservation effects and the role of occupation restrictions.

Findings

Directed percolation universality class applies to basic reactions.

Parity conservation leads to a distinct universality class in BARW.

Occupation number restrictions significantly influence reaction dynamics.

Abstract

Many systems that can be described in terms of diffusion-limited `chemical' reactions display non-equilibrium continuous transitions separating active from inactive, absorbing states, where stochastic fluctuations cease entirely. Their critical properties can be analyzed via a path-integral representation of the corresponding classical master equation, and the dynamical renormalization group. An overview over the ensuing universality classes in single-species processes is given, and generalizations to reactions with multiple particle species are discussed as well. The generic case is represented by the processes A <-> A + A, and A -> 0, which map onto Reggeon field theory with the critical exponents of directed percolation (DP). For branching and annihilating random walks (BARW) A -> (m+1) A and A + A -> 0, the mean-field rate equation predicts an active state only. Yet BARW with odd m display a DP transition for d <= 2. For even offspring number m, the particle number parity is conserved locally. Below d_c' = 4/3, this leads to the emergence of an inactive phase that is characterized by the power laws of the pair annihilation process. The critical exponents at the transition are those of the `parity-conserving' (PC) universality class. For local processes without memory, competing pair or triplet annihilation and fission reactions k A -> (k - l) A, k A -> (k+m)A with k=2,3 appear to yield the only other universality classes not described by mean-field theory. In these reactions, site occupation number restrictions play a crucial role.