Title: From Hamiltonian Systems to Kenetic and Macroscopic Dynamics
Speaker: Professor Claude Bardos
Speaker Info: University of Paris 7, France
The same physical phenomena can be described by several type of equations which form a hierarchy of problems. Each of them is in some obtained (formally or rigourously) as the limit of the previous one and in fact turn ou to be valid when the previous one ceases to be computable. In this hierarchy the kinetic equations introduced by Maxwell and Boltzmann play a crucial role as a natural link between microcopic and macroscopic models. If the derivation of the different macroscopic models from kinetic equations is well established. Formal results are available and rigourous results are in reasonnably good agreement with the rigourous results already established for the macroscopic equations.Date: Saturday, March 11, 2000
The relation between microscopic and kinetic model is a harder problem and therefore in a much less complete form, however it shares with the previous one the following aspect. The scaling parameter which are used to understand the derivation are well understood. However, the challenges are that in most cases from the reversible Hamiltonian system one deduces an irreversible macroscopic equation. This has two consequences:
i) One the appearance of the irreversible phenomena is systematically related to some type of entropy and this leads to the introduction of this notion.
ii) Since at the limit quantities which are conserved at the level of the Hamiltonian system decay convergence cannot in general holds in a weak sense.
In many example some randomness is introduced in the dynamical system and it seems important to understand or to find out when this randomness is really compulsory for the process.
Eventually one should also consider direct derivations of the macroscopic equation (say the Euler equations) from the original Hamiltonian system. However at this level no proof are available and an entropy principle seems to play a crucial role whose mathematical content is yet not understood.