Field theoretic renormalization group for a nonlinear diffusion equation

Abstract

The paper is an attempt to relate two vast areas of the applicability of the renormalization group (RG): field theoretic models and partial differential equations. It is shown that the Green function of a nonlinear diffusion equation can be viewed as a correlation function in a field-theoretic model with an ultralocal term, concentrated at a spacetime point. This field theory is shown to be multiplicatively renormalizable, so that the RG equations can be derived in a standard fashion, and the RG functions (the $\beta$ function and anomalous dimensions) can be calculated within a controlled approximation. A direct calculation carried out in the two-loop approximation for the nonlinearity of the form ${\varphi }^{\alpha }$, where $\alpha >1$ is not necessarily integer, confirms the validity and self-consistency of the approach. The explicit self-similar solution is obtained for the infrared asymptotic region, with exactly known exponents; its range of validity and relationship to previous treatments are briefly discussed.

8 pages, 2 figures, RevTeX

References in corpus (2)

• Field Theory of Branching and Annihilating Random Walks
• Group analysis and renormgroup symmetries

Cited by in corpus (3)

• A Renormalization Group Approach to Spontaneous Stochasticity
• Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
• Critical dynamics in systems controlled by fractional kinetic equations