Course Outline |
Fall 2021
Jul 10, 2025 |
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| McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see Academic Integrity for more information).
L'université McGill attache une haute importance à l’honnêteté académique. Il incombe par conséquent à tous les étudiants de comprendre ce que l'on entend par tricherie, plagiat et autres infractions académiques, ainsi que les conséquences que peuvent avoir de telles actions, selon le Code de conduite de l'étudiant et des procédures disciplinaires (pour de plus amples renseignements, veuillez consulter le site Academic Integrity. |
| Topics in Mathematics and Statistics
. - 7411 - MATH 594 - 001 |
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Associated Term: Fall 2021
Downtown Campus Topics Course Schedule Type Topics: Prerequisites. Honours Analysis 3-Math 454, Honours Probability-Math 356, and willingness to pick up on the pre-requisite topics (which are of independent interest) as we proceed. The references, and in some cases pre-recorded videos with pre-requisites, will be provided. In exceptional cases (and this in particular applies to the Joint Honours Math. Phys students), the course can be taken with Math 454 and Math 356 or Phys 362 as co-prerequisites. If you are interested to do so, please contact the instructor. This topic course concerns the interplay between the Large Deviation Principles (LDP) of probability theory and the mathematical foundations of statistical mechanics (SM). Although this fundamental link goes back to the pioneering work of Boltzmann and has played a central role in the development of both subjects, it is rarely discussed at the introductory level. The goal of the course is to describe the basic theory of LDP and SM with an emphasis on the foundational link between them. LDP. Cramér’s theorem in the i.i.d. setting. General structure of LDP. Gärtner-Ellis theorem. Boltzmann-Sanov theorems. Method of Ruelle-Lanford functions. Varadhan’s Lemma. Applications SM of Lattice Gasses. Interactions and pressure. Entropy. Boltzmann and Gibbs equilibrium states. Equivalence of ensembles. Theory of Gibbs states. Hausdorff dimension and Boltzmann entropy. Information theory perspective. Beyond Gibbsianity. Additional topics will include: LDP and SM in the general dynamical systems setting. Thermodynamic formalism of dynamical systems. Rotators, dynamics, and the 0 Law of Thermodynamics. Required Readings & Materials: References. A perhaps closest reference to the spirit of the course is [Pf02]. The classical references for LDP are [DZ10, Ell85, Va84]. Our exposition of Statistical Mechanics of Lattice Gasses is influenced by monographs [Is79, Ru69, Ru78, Si93] and Section 2 in [ERS93]; see also [FV17] for a pedagogical introduction to the topic. It is important, however, to keep in mind that the course presentation will be self-contained and will follow its own path. The above references can (and should) be consulted for the perspective and additional information. References [DZ10] Dembo, A., and Zeitouni, O.: Large Deviations Techniques and Applications. Springer, New York, 1998. [Ell85] Ellis, R. S.: Entropy, Large Deviations and Statistical Mechanics. Grundlehren der mathemaischen Wissenschaften, 271, Springer, Berlin, 1985. [ERS93] Enter, A.C.D., Fernandez, R., and Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72, 879-1167 (1993). [FV17] Friedli, S., and Velenik, Y.: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge University Press, 2017. [Is79] Israel, R.B.: Convexity in the Theory of Lattice Gases. Princeton University Press, 1979. [Pf02] Pfister, C.E.: Thermodynamical aspects of classical lattice systems. In: Sidoravicius V. (eds) In and Out of Equilibrium. Progress in Probability 51, Birkhauser, Boston MA, 2002. [Ru69] Ruelle, D.: Statistical Mechanics. Rigorous Results. W.A. Benjamin Inc, 1969. [Ru78] Ruelle, D.: Thermodynamic Formalism. Cambridge University Press, 1978. [Si93] Simon, B.: The Statistical Mechanics of Lattice Gases, Vol I. Princeton University Press, 1993. [Va84] Varadhan, S.R.S.: Large Deviations and Applications. SIAM, Philadelphia, 1984. Method of Evaluation: Evaluation. Take home final exam. Course URL: Office Hours:
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