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I need a good reference on the subject Statistical Mechanics with a rigorous point of view.

Almost all physics book on this subject does not care about definitions. They differentiate discrete function without any worry. I guess this subject has widely studied in mathematics literature such as the dynamical system and information theory. So I have the question:

Is there any good reference studies the subject rigorously with more precise backgrounds?

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The mathematical framework for the theory of statistical mechanics is a very large and very important branch of mathematical physics and has been major developments in the field. Some of the most important rigorous references are below.

Statistical Mechanics for Mathematicians

  1. Statistical Mechanics A Short Treatise. Giovanni Gallavotti. Springer Verlag, (1999).
  2. Statistical Mechanics of Disorder Systems - A Mathematical Perspective. Anton Bovier. Cambridge Series in Statistical and Probabilistic Mathematics, (2006).
  3. Statistical Mechanics: Rigorous Results. David Ruelle. World Scientific, (1999).
  4. Entropy, Large Deviations, and Statistical Mechanics. Richard S. Ellis. Classics in Mathematics. Springer-Verlag, (2006).
  5. Entropy and equilibrium states in Classical Statistical Mechanics. O.E. Lanford. Lecture notes in Physics, vol. 20, Springer-Verlag, (1973).

Gibbs Measures

  1. Gibbs measures and phase transitions. Hans-Otto Georgii. (2 edition). (2011)
  2. A Course on Large Deviations with an Introduction to Gibbs Measures. Firas Rassoul-Agha and Timo Seppalainen, (2010).

Ising Model

  1. Le modèle d'Ising. Yvan Velenik, (2009).

Ergodic Theory

  1. An Introduction to Ergodic Theory. P. Walters. (GTM-Springer). (1982)
  2. Ergodic Theory. K. Petersen. (Cambridge Studies in Advances Mathematics 2). (1983)
  3. Topics in Ergodic Theory. W. Parry. (Cambridge Tracts in Mathematics) (1983)
  4. Invitation to Ergodic Theory. C. E. Silva. (Student Mathematical Library vol 42) (AMS). (2007)
  5. An outline of Ergodic Theory. S. Kalikow and R. McCutcheon. (Cambridge Studies in Advances Mathematics). (2010)
  6. Introduction à la théorie ergodique. Thierry de la Rue.
  7. Ergodic Theory: with a view towards Number Theory. Manfred Einsiedler and Thomas Ward. Springer (GTM Vol. 259) (2011).
  8. Dynamical Systems and Ergodic Theory. Lectures Notes of Corinna Ulcigrai (University of Bristol). (2011)

These are books used for undergraduates and graduates courses level and for graduate research level but they also require a high level of mathematical knowledge. Such as measure theory, dynamical theory and other main important fields that display key rules in concepts.

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    $\begingroup$ @Kiarash : Let me also mention our book. It extends considerably my (old) lecture notes that are referred to in this answer. $\endgroup$ – Yvan Velenik Mar 22 '17 at 8:55
  • $\begingroup$ Can I add this book to my list? I'm riding it and it is quite useful and very close to a mathematical discussion (and also a physical meaning). This old notes are alredy great. $\endgroup$ – Rafael Wagner Mar 25 '17 at 17:45
  • $\begingroup$ Of course, that would be great :) . And thanks for your feedback! $\endgroup$ – Yvan Velenik Mar 26 '17 at 8:06
  • $\begingroup$ @YvanVelenik Thank you! that's great $\endgroup$ – Kiarash Apr 16 '17 at 17:22

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