Nonequilibrium Statistical Mechanics
General Information
Course start: January 18, 2019, at 13:00 in the AlbaNova main building.
Course format: 12 lectures (12x2 hours).
Course material: Lecture notes.
Credits: 7.5p.
Examination: Homework examination consisting of 3 sets of problems, each with 24 problems. Oral examination at end of course. For PhD students there is in addition a written project.
Prerequisites: Introductory thermodynamics and statistical physics, and some quantum mechanics.
Lecturer: Jens H Bardarson, Room A4:1049, Email: bardarson@kth.se.
Course description
Nonequilibrium situations are far more common in nature than equilibrium ones. This course gives an introduction to the common ideas and different approaches for studying systems in statistical mechanics that are not in equilibrium, i.e., with a time dependence in the description of the system. We begin with a review of the origin of irreversibility and the second law of thermodynamics, which are at the foundations of equilibrium statistical mechanics. Then various different techniques for studying nonequilibrium situations follows, which treat the problem on different levels of detail. The main part of the course considers effective descriptions in terms of stochastic processes, closely related to simple random walk problems. We also discuss the Boltzmann equation, which provides a microscopic framework for studying transport in dilute systems, and leads up to coarsegrained hydrodynamic descriptions on longer length scales. We further discuss the linear regime close to equilibrium, where it is possible to obtain the linear response of the system from its equilibrium fluctuations, via the fluctuationdissipation theorem. Finally, we discuss thermalisation in quantum mechanics via the eigenstate thermalisation hypothesis.
Objectives
After the course you shall

have a broad overview of concepts, methods and approaches within nonequilibrium statistical mechanics.

be able to model new physical situations using the methods exemplified in the course.

be able to generalize and apply the methods to new problems.

have gained insights into more advanced methods which touch upon modern research.
More specifically you shall

be able to model various physical processes using stochastic differential equations and Master equations.

be able to solve stochastic differential equations, e.g., the Langevin equation (analytically or numerically).

be able to solve (simple) Master equations using generating functions.

be able to describe the principles behind the Boltzmann equation, its approximations, and its consequences.

be able to solve simple transport problems using the Boltzmann equation.

be able to explain the relation between fluctuations and dissipation.

be able to describe the importance of and the consequences of microscopic time reversebility and causality.

be able to use linear response theory to calculate susceptibilities and transport coefficients in physical systems.
 be able to explain the basic tenets of the eigenstate thermalisation hypothesis.
Examination
The examination consists of three or more sets of home assignments. In addition, PhD students who take the course will have an oral examination.
The problems should be solved individually, but you are allowed (and encouraged!) to discuss them with each other.
When solving the problems it is important that you clearly motivate all the steps in your calculations. Check if your results are reasonable! Also, please take a look at the objectives above – they will be used when evaluating your performance. For the highest grades your solutions should be easy to follow with clear reasoning and all steps motivated. Your solutions should show that you understand what you have done. Furthermore, you should be able to discuss and explain your solutions to the teacher.
Contents

Irreversibility and the second law

Brownian motion: Random walks, Langevin equation, FokkerPlanck equation.

Stochastic processes in physics: Master equations, Genereating functions.

The Boltzmann equation: The Htheorem and irreversibility. Conservation laws and hydrodynamics.

Linear response theory: Kubo formula, Fluctuationdissipation theorem, Onsager relations.
 Quantum quenches and eigenstate thermalisation hypothesis.
Course material
The course material consists of lecture notes, which are made available online at the course homepage.
Literature
There is no single book which covers all of the course content. Here are a couple of recommendations for further reading.

R. Zwanzig, Nonequilibrium statistical mechanics (Oxford, 2001), Ch. 17.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGrawHill, 1965), Ch. 1315.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (NorthHolland, 1981).

M. Le Bellac, F. Mortessagne, and G. G. Batrouni, Equilibrium and NonEquilibrium Statistical Thermodynamics (Cambridge, 2004), Ch. 6, 89.
Teachers
 Jens Bardarson Examiner, Course responsible, Teacher
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