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Regular version of the site
Master 2020/2021

## Statistical Mechanics: Algorithms and Computations

Type: Elective course (Materials. Devices. Nanotechnology)
Area of studies: Electronics and Nanoelectronics
When: 2 year, 3 module
Mode of studies: distance learning
Instructors: Renat Ikhsanov
Master’s programme: Материалы. Приборы. Нанотехнологии
Language: English
ECTS credits: 3

### Course Syllabus

#### Abstract

In this course a student will learn a whole lot of modern physics (classical and quantum) from basic computer programs that you will download, generalize, or write from scratch, discuss, and then hand in. A student will find out about algorithms, and about the deep insights into science that one can obtain by the algorithmic approach. #### Learning Objectives

• Objectives of mastering the discipline "Statistical Mechanics: Algorithms and Computations": • give students an idea of modern physics (classical and quantum) from basic computer programs that you will download, generalize, or write from scratch, discuss, and then hand in;
• • give students an understanding of the essential concepts of Monte Carlo techniques (detailed balance, irreducibility, and a-periodicity), and Metropolis algorithm. #### Expected Learning Outcomes

• Knowledge: the essential concepts of Monte Carlo techniques.
• Skills: to program Metropolis algorithm.
• Possess: python programming.
• Knowledge: - hard-disk model; - difference between direct sampling and Markov-chain sampling.
• Skills: to make the connection of Monte Carlo and Molecular Dynamics algorithms.
• Possess: Newtonian mechanics and statistical mechanics.
• Knowledge: entropic interactions concept.
• Skills: to make perfect algorithm to sample configurations.
• Knowledge: sampling, and its connection with integration.
• Skills: to use the Maxwell and Boltzmann distributions of velocities and energies.
• Knowledge: quantum mechanics.
• Skills: to use the Trotter approximation.
• Possess: density matrices and path integrals techniques.
• Knowledge: the properties of bosons.
• Skills: to do standard sampling techniques.
• Possess: the Lévy construction.
• Knowledge: the Bose-Einstein condensation phenomenon.
• Skills: to program the sampling algorithm.
• Possess: the path-integral technique.
• Knowledge: - the local algorithm; - the heat-bath algorithm.
• Skills: to program the Ising model.
• Knowledge: a dynamic Monte Carlo algorithm.
• Knowledge: the central limit theorem. #### Course Contents

• Topic 1. Monte Carlo algorithms (Direct sampling, Markov-chain sampling).
This lecture covers Monte Carlo algorithms. In the tutorial we will use the 3x3 pebble game to understand the essential concepts of Monte Carlo techniques (detailed balance, irreducibility, and a-periodicity), and meet the celebrated Metropolis algorithm.
• Topic 2. Hard disks: From Classical Mechanics to Statistical Mechanics
This lecture covers the hard-disk model, which was first simulated by Molecular Dynamics in the 1950's. The tutorial includes the difference between direct sampling and Markov-chain sampling, and the connection of Monte Carlo and Molecular Dynamics algorithms, that is, the interface between Newtonian mechanics and statistical mechanics. The tutorial also includes classical concepts from statistical physics (partition function, virial expansion et. al.).
• Topic 3. Entropic interactions and phase transitions
This lecture covers the entropic interactions, coming only from statistical-mechanics considerations.
• Topic 4. Sampling and integration
This lecture covers the sampling, and its connection with integration, the Maxwell and Boltzmann distributions of velocities and energies.
• Topic 5. Density matrices and Path integrals (Quantum Statistical mechanics 1/3)
This lecture covers quantum statistical mechanics. The lecture starts from learning about density matrices and path integrals, fascinating tools to study quantum systems. In many cases, the Trotter approximation will be useful to consider non-trivial systems, and also to follow the time evolution of a system.
• Topic 6. Lévy Quantum Paths (Quantum Statistical mechanics 2/3).
This lecture covers the properties of bosons, indistinguishable particles with peculiar statistics. At the same time, it will also go further by learning a powerful sampling algorithm, the Lévy construction.
• Topic 7. Bose-Einstein condensation (Quantum Statistical mechanics 3/3)
This lecture covers the Bose-Einstein condensation phenomenon, theoretically predicted in the 1920's and observed in the 1990's in experiments with ultracold atoms.
• Topic 8. Ising model - Enumerations and Monte Carlo algorithms.
This lecture covers the Ising model, which captures the essential physics of a set of magnetic spins. This is also a fundamental model for the development of sampling algorithms.
• Topic 9. Dynamic Monte Carlo, simulated annealing.
This lecture covers dynamic Monte Carlo algorithm which runs faster than the clock. This is easily devised for a single-spin system, and can also be generalized to the full Ising model. In the tutorial we move towards the simulated-annealing technique, a physics-inspired optimization method with a very broad applicability.
• Topic 10. The Alpha and the Omega of Monte Carlo, Review, Party.
This lecture includes the experiment of Buffon's needle, already performed in the 18th century and its connection with the central limit theorem. #### Assessment Elements

• Экзамен (тест)
If a student misses the exam because of some valid reason, s/he receives «absence» grade. The grade for the course is calculated on the course page on the basis of the student’s number of points that are awarded to the student for answering questions of the proposed tests. Контрольные работы и экзамен по курсу проводятся в письменной форме на платформе Coursera (https://www.coursera.org/learn/statistical-mechanics). Во время написания контрольных и экзаменационных работ студентам запрещено: общаться с кем-либо, пользоваться конспектами и подсказками. Кратковременным нарушением связи во время контрольной работы или экзамена считается нарушение связи менее часа. Долговременным нарушением связи считается нарушение связи в течение часа и более. При долговременном нарушении связи студент не может продолжить участие в контрольной или экзамене. Процедура пересдачи аналогична процедуре сдачи.
• Самостоятельная работа #### Interim Assessment

• Interim assessment (3 module)
0.4 * Самостоятельная работа + 0.6 * Экзамен (тест) #### Recommended Core Bibliography

• Baxter, R. J. (2007). Exactly Solved Models in Statistical Mechanics (Vol. Dover ed). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152951
• Ландау Л.Д., Лифшиц Е.М. - Курс теоретической физики. Статистическая физика - Издательство "Физматлит" - 2001 - 616с. - ISBN: 978-5-9221-0054-0 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/2230
• Теоретическая физика. Т.5, Ч. 1: Статистическая физика, Ландау, Л. Д., Лифшиц, Е. М., 2005