Renormalization and Scaling in Applied Mathematics

By Gunduz Caginalp
Renormalization and Scaling in Applied Mathematics
This tutorial is based upon lectures that were given in Bonn-Rottgen, Germany during August 2004. Approximately thirty participants attended this Summer School that was made possible by a grant of the German Research Foundation (DFG) entitled: Priority Program 1095 "Analysis, Modeling and Simulation of Multiscale Problems," and organized by Dr. Christof Eck and Prof. Heike Emmerich.

The lectures are intended to be self-contained, and no special prerequisite material should be necessary in order to understand the material. There are several exercises with solutions that can be used to test one’s understanding of the material.

The tutorial is divided into four parts. The first lecture introduces the idea of renormalization group in mathematical structures (e.g., Cantor sets, Koch trianges) that exhibit exact self-similarity. This approach -- rather than the historical order in which the research evolved -- facilitates an understanding of the concepts particularly for a mathematical audience. The second lecture focuses on the problem of percolation which can be stated (though not solved) easily in mathematical terms. The problem is that each lattice (e.g., square lattice in two-dimensional space) is occupied with a probability, p, and unoccupied with probability 1-p. The issue is to determine the critical probability at which clusters of occupied lattice points become infinite in size, and to determine the exponents that govern the divergence of related quantities, such as size. These two lectures largely follow the approach of the text by Creswick, Farach and Poole, "Introduction to Renormalization Group Methods in Physics."

Methods under the umbrella of "Renormalization and Scaling (RG)" constitute a common philosophy rather than a specific methodology. In a few mathematical cases, as discussed in Lecture 1, there is complete self-similarity. In other situations, the self similarity is either approximate or in some asymptotic form. An example of the latter is in the decay, blow-up or finite time extinction of solutions to nonlinear differential equations. In the third lecture, the simplest of these is discussed: a heat equation with a nonlinear source term. The methodology involves a union of classical asymptotic analysis and a systematic approach to RG. The basic work is in (G Caginalp: Phys. Rev. E. 53, 66-73,1996), while more recent research has been published in (G Caginalp and H. Merdan:Discrete and Continuous Systems: Series B, 3, 565 – 588, 2003).

Finally, in the fourth lecture, an application is made to interface problems. The issue of initial instability has been studied extensively for such problems. Yet there are few results or methods, except in specific exact solutions, for the situation for large time. This lecture demonstrates how one can obtain the key exponent associated with the basic length scale of the interface as a function of time. This lecture is based mainly on the following papers: G Caginalp: SIAM Applied Mathematics 62, 424-432 (2001); G Caginalp and H. Merdan: Physica D, 198, 136-147 (2004)Nonlinear Analysis (to appear).

Exercises and solutions appear at the end of the lecture material.

Author Institutional Affiliation
Gunduz Caginalp
Department of Mathematics
University of Pittsburgh
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Tutorial LevelAdvanced Tutorial
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