# Functional Analysis

## Scaling limits for particle systems

Summer Term 2016

Graduate Seminar on Analysis (S4B1)

Prof. Dr. J.J.L. Velázquez, Dr. A. Nota

Seminar sessions Fridays 10–12, room N0.008

**Preliminary Meeting 10.02.2016 at 12 c.t., room 2.025**

** Second Preliminary Meeting 18.03.2016 at 13 c.t., room 2.025**

### Synopsis

Many interesting systems in physics are constituted by a large number of identical components so that they are difficult to analyze from a mathematical point of view. At the same time we are not interested in a detailed description of the system but rather in its collective behavior. Therefore it is necessary to look for all the procedures leading to simplified models which preserve all the interesting physical informations of the original system, cutting away redundant information. This is the methodology of the Kinetic Theory.

We want to outline the limiting procedures which lead from the microscopic description based on the fundamental laws of mechanics to a kinetic picture described by nonlinear partial differential equations depending on a small number of degrees of freedom. The exact derivation of these macroscopic evolution equations is one of the central problems in mathematical physics.

We will focus on a simple microscopic model, the Lorentz gas, which is a gas of non-interacting particles in a random configuration of scatterers. This model is paradigmatic since it provides a rare source of exact results in kinetic theory. Indeed one can prove, under suitable scaling limits, a rigorous validation of linear kinetic equations.

### Prerequisites

Basic knowledge of PDEs and Functional Analysis is essential.

### Organization

The seminar sessions are on Fridays 10–12 in room N0.008.

Topics will be distributed in a preliminary meeting on 10.02.16 at 12 c.t. in room 2.025. Interested students are asked to conatct us by email (A. Nota or J. Velázquez) in advance.

### Literature

- Desvillettes L., Pulvirenti M.: The linear Boltzmann equation for long-range forces: a derivation from particle systems. Models Methods Appl. Sci. 9, 1123-1145 (1999) [article]
- Desvillettes L., Ricci V.: A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions. J. Stat. Phys. 104, 1173-1189 (2001) [article]
- Gallavotti G.: Grad-Boltzmann limit and Lorentz’s Gas. In: Statistical Mechanics. A short treatise. Appendix 1.A2. Springer, Berlin (1999) [article]
- Desvillettes L., Ricci V.:The Boltzmann-Grad limit of a stochastic Lorentz Gas in a force field. Bull. Inst. Math. Acad. Sin. (New Ser.)
**2**(2), 637-648 (2007) [article] - Braun W., Hepp K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Comm. Math. Phys. 56 (2), 101-113 (1977) [article]