This seminar will investigate the relation between "exact" microscopic models that govern large particle systems' evolution and a specific type of approximate models known in Statistical Mechanics as "mean-field equations". Roughly speaking, a mean-field equation is a model that describes the evolution of a typical particle subject to the collective interaction created by a large number of other, identical particles. The most famous example of mean-field equation is the Vlasov-Poisson equation.
The mathematical justification of the mean-field limit involves very different ideas and methods. This course aims to give an introductory description of the classical approaches to the problem of the mean-field limit in mathematical analysis.
In particular, the intent is to learn essential tools and techniques for studying Partial Differential Equations while applying them to Vlasov equations.
Content of the course:To obtain the credits for the seminar each student must:
The topics and associated parts of the Lecture notes will be assigned to student on the first lecture.
| Date | Content | Speaker | Reference |
|---|---|---|---|
| 28.02 | Introduction to the seminar | Prof. Dr. Mikaela Iacobelli | |
| 07.03 | Tools for PDE's | Antoine Gagnebin | |
| 14.03 | Break | ||
| 21.03 | Transport Equations with Constant Coefficients Transport Equations with Variable Coefficients | Isabella Brovelli | Chap 2 Sect 2.1 & 2.2 [FG2013] |
| 28.03 | Conservative Transport and Weak Solutions | Adrian Dawid | Chap 2 Sect 2.3 [FG2013 ] |
| 04.04 | General formalism in classical mechanics Mean field characteristic flow | David Lenze | Chap 3 Sect 3.1 & 3.2 [FG2013] (Stop before Theorem 3.2.2) |
| 11.04 | Mean field characteristic flow The Monge-Kantorovich distance Dobrushin’s estimate | Fatime Rasiti | Chap 3 Sect 3.2, 3.3.1 & 3.3.2 [FG2013] (Start with Theorem 3.2.2) |
| 18.04 | Easter Monday | ||
| 25.04 | Sechseläuten afternoon off | ||
| 02.05 | The mean field limit On the choice of the initial data | Yuxiu Zhang | Chap 3 Sect 3.3.3 & 3.3.4 [FG2013] |
| 09.05 | |||
| 16.05 | |||
| 23.05 | |||
| 30.05 |
