Field theoretic renormalization group for a nonlinear diffusion equation

• Published in: arXiv.org
• Main Authors: Antonov, N V, Honkonen, Juha
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 02.07.2002
• Subjects: Approximation; Field theory; Green's functions; Mathematical analysis; Mathematical models; Nonlinearity; Partial differential equations; Self-similarity
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0207006

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 \(\phi^{\alpha}\), 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.