Elliptic Extensions in Statistical and Stochastic Systems

Hermite's theorem reveals three levels of mathematical frames for a valid simple addition formula: rational, q-analogue, and elliptic-analogue. This theorem has led to productive studies in enumerative combinatorics, theory of special functions, representation theory, and integrable systems. This monograph discusses the recent progress in elliptic extensions of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory. It uses special functions such as Jacobi's theta functions, Weierstrass elliptic functions, and Jacobi's elliptic functions, and discusses non-colliding Brownian bridges on a one-dimensional torus and an interval, as well as determinantal point processes on a two-dimensional torus. These limit transitions are considered mathematical realizations of thermodynamic or hydrodynamic limits in statistical mechanics.

Format: Paperback / softback
Length: 125 pages
Publication date: 07 April 2023
Publisher: Springer Verlag, Singapore

Hermite's theorem reveals that a simple addition formula holds validity across three distinct mathematical frameworks: rational, q-analogue, and elliptic-analogue. These frameworks provide the foundation for extensive research in various fields, including enumerative combinatorics, theory of special functions, representation theory, and the study of integrable systems. The origins of these formulas can be traced back to a paper published in 1987 by Date, Jimbo, Kuniba, Miwa, and Okado, which explored the application of theta function identities in precisely solvable statistical mechanics models. Subsequently, these formulas have been extended to the elliptic level in numerous mathematical and theoretical physics research areas.

In this monograph, we present a comprehensive overview of the recent developments in the elliptic extensions of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory. At the elliptic level, a wide range of special functions are employed, including Jacobi's theta functions, Weierstrass elliptic functions, Jacobi's elliptic functions, and others. While this monograph does not serve as a comprehensive handbook of mathematical formulas for these elliptic functions, it focuses on utilizing a simplified version of the four kinds of Jacobi's theta functions. This allows for the definition of seven systems of orthogonal theta functions, which are written using polynomials of the argument multiplied by a single theta function or pairs of such functions. These systems were introduced by Rosengren and Schlosser in 2006, in conjunction with the seven irreducible reduced affine root systems.

By employing Rosengren and Schlosser's theta functions, researchers have been able to explore non-trivial connections and relationships between different mathematical structures. These developments have opened up new avenues for understanding complex systems and have significant implications in fields such as physics, mathematics, and computer science. As our understanding of these formulas continues to evolve, we can expect to see further advancements in our understanding of statistical mechanics, probability theory, and other related disciplines.

Weight: 226g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811995262
Edition number: 1st ed. 2023