Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems

TL;DR

This paper reviews how field-theoretic renormalization group methods are applied to analyze fluctuations and universal behavior in reaction-diffusion systems, including complex multi-species reactions and phase transitions.

Contribution

It provides a comprehensive pedagogical overview of applying field-theoretic RG techniques to reaction-diffusion problems, highlighting new insights into universality and nonequilibrium phase transitions.

Findings

Calculation of universal density decay exponents and amplitudes

Application of RG methods to multi-species and disordered systems

Analysis of directed percolation and branching-annihilating random walks

Abstract

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 k A -> l A (l < 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 eps = d_c - d expansions with respect to the upper critical dimension d_c. 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 analyze field-theoretic approaches to nonequilibrium 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.