# FSI3075 Non-Equilibrium Statistical Mechanics 7.5 credits

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.

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Headings with content from the Course syllabus FSI3075 (Spring 2019–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

Irreversibility and the second law. Brownian motion: random walks, Langevin equation, Fokker-Planck equation, functional integrals. Stochastic processes in physics: master equations, generating 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.

### Intended learning outcomes

After completed course, the PhD student should be able to:

• 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.

### Course disposition

No information inserted

## Literature and preparations

### Specific prerequisites

Introductory thermodynamics and statistical physics, and some quantum mechanics.

### Recommended prerequisites

No information inserted

### Equipment

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### Literature

Lecture notes will be available.

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

P, F

### Examination

• INL1 - Assignments, 5,0 hp, betygsskala: P, F
• TEN1 - Oral exam, 2,5 hp, betygsskala: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

### Other requirements for final grade

Solution of exercises, which are to be discussed with the examiner. PhD students will have an oral exam.

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web FSI3075

SCI/Physics

### Main field of study

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Third cycle