Transport Theory (Intro)


Introduction to Transport Theory

Dates

The specific times for the lectures and exercise sessions will be scheduled together with the participants during the week of 11 April. A specific time will be announced soon.

If you are interesting in taking the course but are not free at this time, please get in touch with the instructor.

The usual second half of this course, Advanced Topics in Transport Theory, will not be offered this year.

Information

The lecture is in English. Course materials and further information will be found in L2P. There are 5CP to earn with an oral examination.

Contents

Kinetic descriptions play an important role in a variety of physical, biological, and even social applications. Examples are the description of gases, radiation, bacteria or traffic. Typically, these systems are described locally not by a finite set of variables but instead by a probability density describing the distribution of a microscopic state. Its evolution is typically given by an integro-differential equation. Unfortunately, the large phase space associated with the kinetic description has made simulations impractical in most settings in the past. However, recent advances in computer resources, reduced-order modeling and numerical algorithms are making accurate approximations of kinetic models more tractable, and this trend is expected to continue in the future. On the theoretical mathematical side, two rather recent Fields medals (Pierre-Louis Lions 1994, Cédric Villani 2010) also indicate the continuing interest in this field, which has already been the subject of Hilbert's sixth out of the 23 problems presented at the World Congress of Mathematicians in 1900.

This course gives an introduction to kinetic or transport theory (by which we mean linear kinetic equations). Our purpose is to discuss the mathematical passage from a microscopic description of a system of particles, via a probabilistic description to a macroscopic view.

A broad range of mathematical techniques is used in this course. Besides mathematical modeling, we make use of statistics and probability theory, ordinary differential equations, hyperbolic partial differential equations, integral equations (and thus functional analysis) and infinite-dimensional optimization. Among the astonishing discoveries of kinetic theory are the statistical interpretation of the Second Law of Thermodynamics, induced by the Boltzmann-Grad limit, and the result that the macroscopic equations describing fluid motion (namely the Euler and Navier-Stokes equations) can be inferred from abstract geometrical properties of integral scattering operators.

Students who have completed this course should be able to

  • read and derive a kinetic equation for a particle system,
  • understand the passage from a microscopic description to a probabilistic one, the mathematical process involved, and the reason for emergence of irreversibility in form of the H Theorem,
  • analyze scattering operators, and to connect this information to conserved quantities and other invariants,
  • derive macroscopic equations from kinetic descriptions by using either of the two main techniques (asymptotic analysis and method of moments).
Literature
  • H. Babovsky, Die Boltzmann-Gleichung, Teubner, 1998.
  • F. Golse, Recent Results on the Periodic Lorentz Gas, in Nonlinear Partial Differential Equations, Springer, 2012.
  • B. Lapeyre, E. Pardoux, R. Sentis, Introduction to Monte-Carlo Methods for Transport and Diffusion Equations, Oxford University Press, 2003.
  • L. Saint-Raymond, Hydrodynamic limits of the Boltzmann Equation, Springer, 2009.
Last modified: 2016/03/16 10:57