# PC5202, Advanced Statistical Mechanics, Jan 2020

Lecturer: Prof. Wang Jian-Sheng

Schedule: Tue/Thu 12:00-2:00 (noon), at S1A-02-17.

Final exam: Tue 30 April 2020, 1:00pm; midterm, Thu 5 Mar, 12:00 (close book).

Reference Books: "Thermodynamics and an introduction to thermostatistics" 2nd ed, by H. B. Callen; "Statistical Mechanics" 2nd ed, by K. Huang; "Statistical Mechanics of Phase Transitions", by J. M. Yeomans; "Nonequilibrium Statistical Mechanics", by R. Zwanzig; "Nonequilibrium Statistical Physics", by N. Pottier.

Syllabus: This module presents an introduction to phase transitions and fluctuations. For phase transitions, the course starts with the treatment of Landau and mean-field. Exact Ising model results are then discussed. Critical exponents are introduced and their relations obtained using the scaling hypothesis of Kadanoff's scheme. Real space renormalization is then used to show how the critical exponents can be calculated. For fluctuations, Langevin, Fokker-Planck equations will be used. Time dependence and fluctuation dissipation theorem then follow. Brownian motion will be used as an example. This module is targeted at physics graduate students with at least one year of statistical mechanics.

Course Outline:

Week 1: 14,16, Jan, introduction, thermodynamics (Callen, Ch.1 to 3), assignment 1

Week 2: 21, 23 Jan, dynamics, foundations of statistical mechanics

Week 3: 28, 30 Jan, microcanonical, canonical and grand-canonical ensembles (Huang, Ch.6-8), assignment 2

Week 4: 4, 6 Feb, example: coupled oscillators, quantum systems

Week 5: 11, 13 Feb, phase diagrams, van der Waals equation (Callen, Ch. 9, 10). assignment 3

Week 6: 18, 20 Feb, mean-field theory (J. M. Yeomans),

Week recess, no classes.

Week 7: 3, 5 Mar (past midterm test on Thursday noon), Ising model, exact solutions (Huang Ch.14, 15) assignment 4

Week 8: 10, 12 Mar, duality, critical exponents, scaling theory (Huang Ch.16)

Week 9: 17, 19 Mar, Brownian motion, Langevin equations (Zwanzig, Ch.1) assignment 5.

Week 10: 24, 26 Mar, Fokker-Planck equations (Pottier, Ch.10-11),

Week 11: 31 Mar, 2 Apr, linear response, Green-Kubo formula, Jarzynski equality (Pottier, Ch.13), assignment 6.

For the rest time until the end of semester, everything will be online:

Week 12: 7, 9 Apr, Boltzmann equations (Huang, Ch.3,4)

Week 13: no class, reading week 1.

Past final exams with answers: 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020.

Lectures notes (typed textbook is posted at LumiNUS): stat-mech-1, 2, 3, 4, 5.

Callen's postulates sheet.

Sites of similar courses: Univ Delaware, Univ of London, a set of lecture notes by Vilfan and notes by D. Cohen. Read the derivation of Sackur-Tetrode formula, a mathematically more precise treatment of Brownian motion, white noise, etc.; see also Lebowitz's article on irreversibility and an article on ergodicity and fundation of equilibrium statisticsl mechanics.