## Functional Analysis

## Derivation of macroscopic evolution models from deterministic microscopic equations

Winter 2012/2013

Lecture course "Advanced Topics in PDE and Mathematical Models" V5B3

Prof. Dr. J. J. L. Velázquez

Lectures Monday 12-14 and Friday 10-12, Room N0.003 (Endenicher Allee 60)

### Synopsis

In Mathematical Physics and Applied Mathematics there are many problems of systems composed of many particles, which interact by means of deterministic microscopic rules but for which the dynamics of some macroscopic particle densities can be defined by means of a system of Partial Differential Equations or some kinetic model. In some cases it is possible to use Mean-Field Approximations in which the dynamics of individual particles can be approximated by the sum of the effects of many small contributions of many other particles (cf. for instance [2,6]). In other cases a given particle interacts only with a small number of neighbouring particles, but – either because the microscopic dynamics drives the system very fast towards "local equilibrium'' configurations or because the interactions between particles are rare – it is possible to derive an effective set of equations for a set of macroscopic quantities which describe some important properties of the system. One of the most relevant examples is the Boltzmann Equation that was rigorously obtained as the limit of a many particle system in [4].

A common feature of most of the systems considered in this course is that their microscopic dynamics are described by a system of Ordinary Differential Equations, i. e. no stochastic behaviour is present in the microscopic evolution. In many interesting examples, the microscopic equations describing the evolution of the system are reversible in time, but the effective macroscopic equations which describe the collective behaviour of the particle system have dissipative properties, increasing entropies and similar properties. The solutions of the microscopic equations which exhibit such type of irreversible behaviour must be chosen usually according to suitable probability distributions that ensure that the solutions of the microscopic system mimic the behaviour of a random system. Such randomness of the distribution of particles must be preserved along the evolution. One of the main goals is to study the conditions on the microscopic distributions of particles in the currently available rigorous results which guarantee the dissipative behaviour of the macroscopic problem.

The course addresses students interested in Differential Equations with some background in basic Probability Theory. At the beginning, some examples of rigorous derivations of Mean Field Theories will be described in detail and some open problems will be discussed. The second part of the course will consist of the detailed study of the derivation by O. E. Lanford of the Boltzmann Equation, taking as starting point an uncorrelated system of particles.

### References

- L. A. Buniwovich and Y. G. Sinai. Statistical Properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78, 479–497, 1980/81.
- W. Braun and K. Hepp. The Vlasov Dynamics and Its Fluctuations in the 1/N Limit of Interacting Classical Particles. Comm. Math. Phys. 56, 125–146, 1977.
- F. Golse. The Mean-Field Limit for the Dynamics of Large Particle Systems. Journées Équations aux dérivées partielles, Forges-les-Eaux, 2-6 juin 2003, GDR 2434 (CNRS).
- O. E. Lanford III. Time evolution of large classical systems. Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1–111. Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975.
- B. Niethammer and J. J. L. Velázquez. Homogenization in coarsening systems II. Stochastic case. Math. Methods Mod. Appl. Sci. 14(8), 1401–1424, 2004.
- K. Oelschläger. Large systems of interacting particles and the porous medium equation. Journal of Differential Equations, 88(2), 294–346, 1990.
- H. Spohn. Kinetic equations from Hamiltonian dynamics: Markovian limits. Review of Modern Physics 53, 569–615, 1980.