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Mathematical physics and ergodic theory of lattice systems - Ising model, quasicrystals

General data

Course ID:	1000-1M22MIK
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Mathematical physics and ergodic theory of lattice systems - Ising model, quasicrystals
Name in Polish:	Fizyka matematyczna i teoria ergodyczna układów sieciowych - model Isinga, kwazikryształy
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	6.00 Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective monographs
Learning outcomes:	Knowledge and skills: 1. Knows the ferromagnetic Ising model, can calculate magnetization in simple lattice models. 2. Can formulate variation rules. 3. Can present simple lattice-gas models without periodic ground states. Social competence: Can talk to physicists.
Assessment methods and assessment criteria:	Passing Criteria: Homework 50% Short project 50%

Classes in period "Summer semester 2023/24" (in progress)

Time span:	2024-02-19 - 2024-06-16	Selected timetable range: weekly course term Navigate to timetable MO TU WYK CW W TH FR
Type of class:	Classes, 30 hours more information Lecture, 30 hours more information
Coordinators:	Jacek Miękisz
Group instructors:	Jacek Miękisz
Students list:	(inaccessible to you)
Examination:	Examination
Short description:	The lecture will be devoted to the study of mathematical models of systems of interacting particles placed on the nodes of a regular lattice. We will discuss the Ising model of interacting spins and the percolation theory. These are sections of modern probability theory, in particular, it is a topic developed by Stanislav Smirnov and Hugo Dominil - Copin, winners of the Fields Medal, respectively in 2010 and 2022. We will discuss their results. We will discuss Hilbert's 18-th problem and its relation to quasicrystals and the ergodic theory of symbolic dynamical systems. We do not assume knowledge of physics or mathematics beyond courses in the first two years of study.
Full description:	The lecture will be devoted to the study of mathematical models of systems of interacting particles located on the nodes of regular lattices. As an example illustrating the existence of magnets, the Ising model of interacting spins will be presented. We will prove spontaneous symmetry breaking - the existence of a phase transition. We will discuss Hilbert's 18-th problem and its relation to quasicrystals - microscopic models of interacting particles for which the energy functional minimum is reached only at non-periodic configurations. Non-periodic tiling planes and their connections with the ergodic theory of symbolic dynamical systems will be presented. We will also deal with one-dimensional systems - Thue-Morse and Fibonacci sequences and Sturm systems in general. Fundamental open problems will be presented: the existence of non-periodic Gibbs measures and the existence of one-dimensional non-ergodic cellular automata. We do not assume knowledge of physics or mathematics beyond courses in the first two years of study. Lecture schedule 1. Why do magnets exist? Ising model of interacting spins 2. Spontaneous symmetry breaking in the ferromagnetic Ising model 3. Minimization of the free energy functional 4. The exact solution of the one-dimensional Ising model 5. Generalizations of the Ising model - classical lattice-gas models 6. Percolation 7. Non-periodic tilings - Hilbert's 18-th problem 8. Microscopic models of quasicrystals - systems without non-periodic ground states 9. Non-periodic Gibbs measures 10. Symbolic dynamic systems - Thue-Morse and Fibonacci sequences 11. Ergodic theory of non-periodic systems 12. Topology of non-periodic systems 13. One-dimensional systems of interacting particles without periodic ground states 14. Cellular automata
Bibliography:	1. Sacha Friedli and Yvan Velenik, Statistcal Mechanics of Lattice Systems - A Concrete Mathematical Introduction, Cambridge University Press, 2018 Available on-line https://www.unige.ch/math/folks/velenik/smbook/ 2. Jean Bricmont, Making Sense of Statistical Mechanics, Springer, 2022 https://link.springer.com/book/10.1007/978-3-030-91794-4#toc 3. Michael Baake and Uwe Grimm, Aperiodic Order, vol 1, A Mathematical Invitation, Cambridge University Press, 2013

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