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Module MA3443: Statistical Physics I

Credit weighting (ECTS)

5 credits

Semester/term taught

Michaelmas term 2016-17

Lecturer

Prof Dmytro Volin.

Contact Hours

• 3 hours per week - lectures and problem solving under lecturer's supervision.
• 1 hour per week - problem solving under supervision of a PhD student devoted mostly to solution of home works.
• Problem solving sessions will be additionally helped by 4th year students.
• 5 extra informal meetings with lecturer to discuss selected topics in thermodynamics, calculus, combinatories, and symbolic programming (these informal meetings are optional for the students and won't be assessed)

Course Outline

 In this course we will study macroscopic systems, i.e. systems that consist of very large number of smaller ones. Because of the large number of components, they acquire new significant properties which do not exist on the level of single particles. The main such property is existence of the function called entropy and of the second law of thermodynamics which states that entropy always increases. Our main goal would be to master these concepts and understand how they emerge from the microscopic description, i.e. from classical or quantum mechanics, when it is applied to the system with very large number of components. On the way we will learn a lot of interesting math, encounter several exciting physical phenomena, and do a bit of computer simulations for ideal gas and random processes.

MA3443 for Mathematics students

To understand statistical physics, we are going to learn a considerable amount of mathematical tools which have applications in diverse areas of science. Hence, you may perceive a large part of this course as an advanced calculus and introduction to combinatorics/probability/information theory.  There are many computational assignments during the term. One motivation to take MA3443 is to improve your practical skills in the mentioned mathematical topics.

Module Content
The detailed syllabus and tentative schedule are given here , only the key elements are listed below;

• Thermodynamics: 0th,1st,2nd,3rd laws. Concept of temperature and entropy (geometric approach a-la Carathéodory). Thermodynamic potentials. Ideal gases. Heat engines. Reversible vs irreversible processes.
• Calculus: Gauss integrals. Gamma function (generalisation of factorial). Saddle-point approximation. Euler-Maclaurin resummation.
• Combinatorics: Multinomial coefficients, distribution of balls among boxes (Fermi/Bose), generating function.
• Probability theory: Discrete and continuous random systems. Probability density function. Central limit theorem. Random walk.
• Information entropy and the optimal encoding.
• Foundations of statistical physics. Probabilistic description of statistical systems. Microcanonical, canonical, and grand canonical ensembles. Partition function. Derivation of thermodynamics from statistical physics. Boltzmann and Gibbs approaches to entropy. 
• Boltzmann's H-theorem and the 2nd law. Time arrow.
• Review lecture about information and entropy in physical systems: Maxwell's demon, Szilard engine, Gibbs mixing paradox; thermodynamics of Black holes, Bekenstein bound, idea of holography.

Literature/ways of study:

There is no particular book which we will follow closely. The course is largely based on problem solving. Learning the corresponding practical skills through constant exercise is the key element for successful accomplishment of the course.

• HW's will be followed by comprehensive solutions. Study them

• Lecture notes by D.Volin available on the blackboard contain the key theoretical concepts covered. 

• Any standard book, to use for background reading, there is no shortage of choices. E.g: Kerson Huang, 'Introduction to Statistical Physics'

• L.D. Landau, E.M. Lifshitz, 'Statistical Physics' (good to read starting from December and later on as MA3444 follow-up)

Module Prerequisite

Compulsory: 
MA2341/2 - Advanced Classical Mechanics
strong students may seek lecturer's permission to attend the course without MA2341/2 studied.
Desired: 
Thermodynamics part of PY2T10 - classical physics for theoretical physics.
ST2351/2 - probability and theoretical statistics (or other module on similar topic). 
thermodynamics/statistics pieces of MA3443 will be self-consistently covered but prior knowledge will facilitate learning

Assessment Detail

This module will be examined in a 2-hour examination in Trinity term. The continuous assessment contributes 25% to the final annual mark, which consists of 20% for homework submissions and 5% for the multiple-choice test at the end of the term. 
Supplemental exam, if required, will consist of 100% exam.