Methods and applications of ultrametric and p-adic analysis: From wavelet theory to biophysics

Abstract

This paper is devoted to new methods of p-adic and ultrametric analysis and their applications in physics and molecular biology. Although the basic definitions are given in the text, it is recommended that the reader be familiar with the material of the first two chapters in Vladimirov, Volovich and Zelenov’s book p-Adic Analysis andMathematical Physics [1]. Our purpose in this paper is to present some new methods and applications of p-adic and ultrametric analysis developed from 19941 till 2007. The overlap of material between this paper and [1] is minimized. In particular, we do not discuss p-adic quantum mechanics and string theory. This paper covers several topics of p-adic and ultrametric analysis and applications. Most of the results given here are new and have been presented earlier only in journal articles. The first topic is purely mathematical; it is related to the development of the theory of ultrametric wavelets and the study of it relationship to the spectral analysis of ultrametric pseudodifferential operators. The role of ultrametric pseudodifferential operators (such as the Vladimirov operator of p-adic fractional differentiation) in ultrametric analysis is similar to that played by the differentiation operator in real analysis. The use of wavelet bases makes it possible to calculate spectra of ultrametric pseudodifferential operators. In general, in the ultrametric case, wavelet theory is substantially simpler and more powerful and convenient than in the real case. We consider here both the p-adic case and the case of general locally compact ultrametric spaces. An important place in the paper is occupied by a discussion of applications of ultrametric analysis. We discuss applications to the theory of spin glasses, dynamics of macromolecules, and genetics. A common feature of the applications of ultrametric analysis considered here is that they describe collective effects in complex systems. The main destination of physics is describing phenomena based on first principles (physical laws). In many cases, such a description involves difficulties. Thus, it is important to develop various approximations relying on patterns of the effective behavior of systems under consideration in various regimes. A well-known example is the passage from a microscopic description of a many-particle system to a kinetic and, then, a hydrodynamic description [2]. Another example is the stochastic limit of quantum theory, in which the dynamics of a quantum system interacting with the environment is approximated by some quantum random process [3]. In this paper, we consider three examples of application to complex systems, where an important role in the description of a system is played by the ultrametric specifics. In general, complexity, which is often discussed in applications to biological systems (as well as in different contexts), does not lead to farreaching conclusions by itself. The examples considered in what follows show that, as supposed in [1], ultrametric analysis makes it possible to give a simple description of some complex systems, for which methods of real analysis are essentially insufficient. Certainly, we do not claim that ultrametric methods