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PHASE TRANSITIONS: EXACT (OR ALMOST EXACT) RESULTS FOR THE VARIOUS MODELS-CHAPTER 12

Authors
Publisher
Elsevier Ltd
Identifiers
DOI: 10.1016/b978-075062469-5/50014-1
Disciplines
  • Chemistry
  • Mathematics

Abstract

Publisher Summary The onset of a phase transition in a given physico-chemical system is characterized by features whose qualitative nature is determined by the universality class to which the system belongs. This chapter proposes considering a variety of model systems belonging to different universality classes and analyzing them theoretically to find out how these features arise and how they vary from class to class. The chapter starts off by analyzing the Ising model in one dimension, followed by a study of the general n-vector models again in one dimension. This is important for several reasons. Firstly, phenomena that can be looked upon as one-dimensional nearest-neighbor problems exist. Secondly, it helps in the evolution of mathematical techniques for treating lattices in higher dimensions, which is essential for understanding the critical behavior of a variety of physical systems met with in nature. Thirdly, it helps in estimating the status of the Bethe approximation as a possible theory of the Ising model, because it mathematically demonstrates that at least this approximation leads to exact results in one dimension. The chapter also considers a generalization of the Ising chain in which the spin variable is an n-dimensional vector of magnitude unity whose components can vary continuously over the range –1 to +1. The chapter demonstrates that the vector models with n > 2, while differing quantitatively from one another, differ rather qualitatively from the scalar models (for which n = 1).

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