Analysis and Geometry on Configuration Spaces: Theory and Applications Invited Lectures in Lublin

Abstract

Abstract. Many fundamental phenomena in nature are described by usually nonlinear partial differential equations derived in a rather phenomenological way. Perhaps the most known ones are the Bolzmann equation, the Navier-Stokes equation, the reaction-diffusion equation, Fisher’s equation, Vlasov’s equation, etc. These equations describe a substance, e.g. gas or liquid, as a whole, without explicit considering its microscopic structure. They are fairly deterministic and operate with such macroscopic notions as pressure, fluid velocity, viscosity, and so on. On the other hand, the systems of large number of interacting microscopic agents (particles) are described by infinite chains of linear differential equations. For example, the systems of interacting gas molecules are described by the Bogoliubov hierarchy of linear differential equations, also called BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy. In this case, the motion of the particles and the inter-particle interactions are described explicitly and – due to the huge number of particle – in a probabilistic way. This means that the solutions give the time evolution of the probability distributions on the system’s phase space. Since the very appearance of these methods, the problem of deriving the macroscopic description of interacting particle systems from their microscopic statistical mechanical description was considered as a challenging mathematical task. The proposed cycle of lectures presents a number of examples where such derivation can be performed, including also the description on the intermediate (mesoscopic) level. Along with the statistical mechanical models, there will be considered models used in plant ecology, genetics, oceanology, economic and social sciences. In these models, the particles can die, be born, diffuse, perform jumps, etc. The microscopic description is performed in terms of the Markov evolution of states of infinite systems of interacting particles located in the space Rd. Then the mesocopic description, which leads to nonlocal nonlinear differential equations, is obtained by means of a procedure called scaling. In this way, a number of known phenomenological equations are obtained and studied.

Cite this paper

@inproceedings{Kondratiev2010AnalysisAG, title={Analysis and Geometry on Configuration Spaces: Theory and Applications Invited Lectures in Lublin}, author={Yuri Kondratiev}, year={2010} }