Statistisk mekanik för icke-jämviktssystem

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General Information

Course start: January 18, 2016, at 13:15 in Room FB51 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 2-4 problems. For PhD students there is in addition an oral examination.

Prerequisites: Introductory thermodynamics and statistical physics, and some quantum mechanics.

Lecturer: Jens H Bardarson, Room A4:1049, Email: bardarson

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 non-equilibrium 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 coarse-grained hydrodynamic descriptions on longer length scales. Finally, we 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 fluctuation-dissipation theorem. A brief discussion of fluctuation theorems, e.g., the Jarzinsky identity, valid arbitrary far from equilibrium is also included.


After the course you shall

  • have a broad overview of concepts, methods and approaches within non-equilibrium 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.


The examination consists of three sets of home assignments. In addition, PhD students who take the course will have an oral examination.

The problems should be solved individually (with one possible exception), but you are allowed (and encouraged!) to discuss them with each other.

The one exception is one of the problems in the first set where you will use a computer for a numerical solution. For this problem I encourage you to work in groups of two, but you will still have to hand in an individual solution.

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.

Important dates:

Voluntary deadline for the first problem set: February 10, 2016.

Final deadline for all three home assignments: March 23, 2016.

The first set of problems is treated differently compared to the other two: If you hand in the problems in time before the voluntary deadline, they will be promptly corrected and returned to you for further improvement, if necessary, before the final deadline. Use this opportunity to get some feedback on your solutions during the course! Note that this only applies to the first set of problems.


  • Irreversibility and the second law

  • Brownian motion: Random walks, Langevin equation, Fokker-Planck equation, Functional integrals.

  • Stochastic processes in physics: Master equations, Genereating functions, Doi formalism.

  • The Boltzmann equation: The H-theorem and irreversibility. Conservation laws and hydrodynamics.

  • Linear response theory: Kubo formula, Fluctuation-dissipation theorem, Onsager relations.

Course material

The course material consists of lecture notes, which are made available online at the course homepage.


There is no single book which covers all of the course content. Here are a couple of recommendations for further reading.

  • W. Ebeling, I.M. Sokolov, Statistical thermodynamics and stochastic theory of nonequilibrium systems (Singapore : World Scientific, 2005).

  • R. Zwanzig, Nonequilibrium statistical mechanics (Oxford, 2001), Ch. 1-7.

  • F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965), Ch. 13-15.

  • N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, 1981).

  • M. Le Bellac, F. Mortessagne, and G. G. Batrouni, Equilibrium and Non-Equilibrium Statistical Thermodynamics (Cambridge, 2004), Ch. 6, 8-9.


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