Course syllabus


TIF106 / FIM786 Non-equilibrium processes in physics, chemistry and biology lp4 VT21 (7.5 hp)

Course is offered by the department of Physics and the department of Microtechnology and Nanoscience - MC2.

This year the course will be given online using Zoom at the dates and times advertised in TimeEdit.

The lectures will be broadcasted. The links below will be valid at the time of the lecture . You are all welcome!


Home-reexam August 2021:

- pick up the sheet here in Canvas Monday August 23 at 8:00 (the link is at the bottom of this page and under Assignments)

- hand in your solutions here in Canvas Wednesday August 25 at 23:59 (earlier misprint corrected)

Note: if you have signed up for this exam in Ladok but decide to not take it anyway, please notify the teachers asap!


Home-exam May 2021:
- pick up the sheet here in Canvas Saturday May 29 at 8:00
- hand in your solutions here in Canvas Monday May 31 at 23:59

Oral exam: Thursday-Friday June 3-4. Check your time slot in the choodle: 

Note: enter Tomas' zoom room, see link and password below. You will at first be in the waiting room and you will be let in when it is your turn.

Contact details

First part of the course and second part, track IIa:

Second part of the course in track IIb:

Course aim

The great majority of physical, chemical, and biological processes occur outside the thermodynamic equilibrium. How do we describe many-particle system driven away from equilibrium, or evolving towards the equilibrium due to an interaction with an environment? In contrast to the universality of the thermodynamics, the non-equilibrium evolution is system specific and requires individual approach. The purpose of the course is to introduce basic concepts of kinetic theory and stochastic processes, and to study practical tools to investigate non-equilibrium states. We will discuss the origin of irreversible evolution and dissipation, hierarchy of relaxation processes, transport phenomena and noise, Brownian motion. The course includes a selection of applications to quantum solid state systems, chemical reaction kinetics, and soft matter like colloidal dispersions, polymers, gels, glasses and biological systems.


Lecture schedule 2021 (note that it can be adapted and changed during the course)


Course literature

Literature for parts I and IIa
The lectures in part I and part IIa to a large extent follow the line of Vitaly Shumeiko's lecture notes: Compendium.pdf  

L. Reichl, A modern course in Statistical Physics, Wiley, 1999.
L.D. Landau and E.M. Lifshits, Statistical Physics, part I, (Theoretical Physics vol. 5).
E.M. Lifshits and L.P. Pitayevsky, Physical Kinetics, (Theoretical Physics vol. 10).

The following compendium gives a slightly different point of view on a few of the topics covered in the course: link

Literature for part IIb
B. Mehlig, Lecture notes
Lennart Sjögren, Lecture notes stochastic processes (Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9,10)
N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Third Edition, North-Holland
H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) 284 [pdf]
P. L. Krapivsky, S. Redner & E. Ben-Naim, A kinetic view of statistical physics, Cambridge University Press (2010)
R. D. Astumian & P. Hänggi, Brownian motors, Physics Today, November 2002, p. 33
Previous exam questions for part IIb  [pdf]

Course design

The course is organized into two parts.

Part I, March 23 - May 5: Introduction to classical non-equilibrium processes.
Part II, May 5 - May 28:
  (a): Quantum kinetics.
  (b): Stochastic processes in chemistry, biology, and in the atmospheric sciences
Part I is obligatory for all students. However, one may follow either part IIa or IIb.

The course is based on a series of lectures and exercises. The second half of the course can be organized as a project work.

Plan for part IIb

For the schedule, please see TimeEdit.

Lecture 1: Repetition. Ornstein-Uhlenbeck process.  Fokker-Planck equation. Thermodynamic equilibrium. Maxwell-Boltzmann distribution. Fluctuation-dissipation theorem.
Recording part I
Recording part II

Lecture 2: Kramers problem and first passage times. Arrhenius law, transition-state theory, Kramers problem: overdamped limit: solution of Fokker-Planck equation with integrating factor. Saddle-point approximation. 
Recording part I
Recording part II

Lecture 3: Correlated random walks. Random walks vs. correlated random walks. Path-coalescence transition. Order parameter: Lyapunov exponent. Computation of Lyapunov exponent.
Recording part I
Recording part II

Lecture 4: Turbulent aerosols. Stokes equation. Relation to correlated random walks in the overdamped limit. Solution in the white-noise limit: mapping onto Kramers escape problem. Exact solution. WKB approximation (large-deviation theory).
Recording part I
Recording part II

Lecture 5: Turbulent aerosols continued. Algebraic perturbation theory for Lyapunov exponent: mapping to perturbed quantum harmonic oscillator. Perturbation theory.
Recording part I
Recording part II

Lecture 6: Population dynamics. Master equation for one-step processes: Poisson process, random walk. Steady state of one-step processes. Non-linear birth-death processes. Probability of extinction, time to extinction. Large-deviation theory for time to extinction. Time-dependent WKB approximation for Poisson process.
Recording part I
Recording part II

Lecture 7: Reaction kinetics. Branching-annihilation reaction, moment equations. Reaction rate for diffusive absorption. Diffusion-limited annihilation reactions. Failure of mean-field theory. Critical dimension. Self-avoiding random walk. Role of fluctuations: single-species reactions in one dimension.

Lecture 8: Brownian motors and ratchets. Life at low Reynolds number. Role of diffusion. Example: Brownian particle in asymmetric periodic potential. Modulation of potential or temperature as a function of time: out-of-equilbrium transport.
Recording part I
Recording part II

Lecture 9
Recording part I
Recording part II

Lecture 10

Learning objectives and syllabus

Learning objectives:

After this course, the student should have acquired a general knowledge of stochastic processes and their use to describe the time evolution of systems in nature. More specifically, the student should be able to:
* describe basic concepts of the kinetic theory for classical and quantum many particle systems
* describe the origin of irreversible evolution of physical, chemical, and biological systems, and the hierarchy of relaxation processes
* practically solve transport problems, as well as analyze dissipative and fluctuation phenomena, and the effects of an environment.

Link to the syllabus on Studieportalen:

Study plan Chalmers

Study plan GU

Note that the course is identical at Chalmers (tif106) and GU (fim786).

Examination form

  • Home assignments
  • Take-home written exam
  • Follow-up oral exam

Weights towards final grade:

  • Home assignments: 30%
  • Exam: 70%

Course summary:

Date Details Due