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Statistical Mechanics: Rigorous Results
- Publisher
- Springer International Publishing
- Copyright
- © The Author(s) 2017
- ISBN
- 978-3-319-55022-0
- Pages
- 25 –48
- DOI
- 10.1007/978-3-319-55023-7_3
- Publisher site
- See Chapter on Publisher Site
Abstract
[In this Chapter, we review the fundamental theoretical tools, starting with the space of disordered configurations and its associated dynamical systems, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal A}_d$$\end{document} of the physical observables, together with its Fourier and differential calculus. The latter is provided by a set of commuting derivations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial $$\end{document} and a trace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal T}$$\end{document}. The triple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal A}_d,\partial ,{\mathcal T})$$\end{document} defines a non-commutative manifold known as the non-commutative Brillouin torus. We reformulate the topological invariants and other response functions in this new framework. We also introduce the magnetic derivations and investigate the behavior of the correlation functions w.r.t. the magnetic fields. This Chapter also fixes the notation and defines the precise settings for the rest of our calculations.]
Published: Mar 18, 2017
Keywords: Fourier Coefficient; Gibbs Measure; Functional Calculus; Differential Calculus; Bernoulli Shift