Course syllabus

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TIF106 / FIM786 TIF106 / FIM786 Non-equilibrium processes in physics, chemistry and biology lp4 VT22 (7.5 hp)

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

Latest course material is under Modules.

Questions during exam preparation regarding part IIB? You can discuss them in "Discussion". Go to "Part IIB".

 

Note on exam: the exam is closed book, but we allow the following material: 3 handwritten pages A4. Physics Handbook.

Contact details

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

Second part of the course in track IIb:

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.

Schedule

TimeEdit

Lecture schedule

The teaching will take place on campus as listed in TimeEdit and will not be streamed since there is no equipment for that in the FL62-FL64 lecture halls.

Organization

The course is organized into two parts.

Part I: Introduction to classical non-equilibrium processes.
Part II:
  (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.

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.

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

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).

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

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.

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.

Course literature

Literature for part 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 (Chapter 1, 2, 3, 4, 5, 6, 7)
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]

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

Study plan GU

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

Examination form

  • Home assignments
  • Written exam

Weights towards final grade:

  • Home assignments: 30%
  • Exam: 70%

Course summary:

Date Details Due