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Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

Minhui Zhu, Nigel Goldenfeld

TL;DR

The paper develops a unified framework combining self-consistent expansion (SCE) with renormalization-group (RG) ideas to tackle singular perturbation problems in deterministic PDEs, focusing on Barenblatt's nonlinear diffusion equation. By formulating Barenblatt's problem both in a PDE framework and a field-theoretic Martin–Siggia–Rose (MSR) setting, the authors derive an anomalous scaling exponent (anomalous dimension) that governs long-time behavior and demonstrate that a first-order SCE can outperform standard perturbative RG, especially at strong coupling. The work shows that the SCE can be implemented with a variational parameter constrained by a Callan–Symanzik-type equation, yielding consistent results across formulations and enabling systematic higher-order improvements. The results suggest broader applicability of SCE to singular perturbation problems, including potential extensions to boundary-layer theory, multi-scale analyses, and numerical methods, with implications for physics-informed computation and beyond.

Abstract

The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics and high order perturbation theory for the anharmonic oscillator. Here we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first order self-consistent expansion improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems, and illustrate its application using the example of Barenblatt's equation. The scope of our results on the combination of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.

Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

TL;DR

The paper develops a unified framework combining self-consistent expansion (SCE) with renormalization-group (RG) ideas to tackle singular perturbation problems in deterministic PDEs, focusing on Barenblatt's nonlinear diffusion equation. By formulating Barenblatt's problem both in a PDE framework and a field-theoretic Martin–Siggia–Rose (MSR) setting, the authors derive an anomalous scaling exponent (anomalous dimension) that governs long-time behavior and demonstrate that a first-order SCE can outperform standard perturbative RG, especially at strong coupling. The work shows that the SCE can be implemented with a variational parameter constrained by a Callan–Symanzik-type equation, yielding consistent results across formulations and enabling systematic higher-order improvements. The results suggest broader applicability of SCE to singular perturbation problems, including potential extensions to boundary-layer theory, multi-scale analyses, and numerical methods, with implications for physics-informed computation and beyond.

Abstract

The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics and high order perturbation theory for the anharmonic oscillator. Here we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first order self-consistent expansion improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems, and illustrate its application using the example of Barenblatt's equation. The scope of our results on the combination of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.

Paper Structure

This paper contains 17 sections, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the SCE calculation for Polymer problem
  3. Barenblatt's equation: a singular perturbation problem
  4. Divergence of the usual perturbation theory
  5. The origin of divergence
  6. Self-similar solution at the long-time asymptotic regime
  7. Perturbative RG
  8. Application of self-consistent expansion to Barenblatt's equation
  9. SCE in PDE framework
  10. Comparison of methods
  11. RG and SCE in field-theoretic framework
  12. From dynamical systems to field theory -- the MSR formalism
  13. Perturbation theory using generating functionals and Feynman diagrams
  14. The Callan-Symanzik equation
  15. Perturbative RG
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Estimates of the anomalous dimension $\alpha$ as a function of the perturbation strength $\epsilon$ as calculated by renormalization group (first and second order in $\epsilon$), the self-consistent expansion at lowest order, and the exact value by solving Eq. (\ref{['exact_anom1']}) and (\ref{['exact_anom2']}) using a shooting method.