Entropy methods in hydrodynamic scaling. (English) Zbl 0872.60081

Varadhan, S. R. S.

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 196-208 (1995).

Hydrodynamic scaling is a procedure that attempts rigorously to derive large scale behavior of complex interacting systems from laws governing its evolution that are specified at a smaller scale. The procedure involves statistical averaging over the small scales and can be viewed as part of nonequilibrium statistical mechanics. The basic example is the classical problem of starting with a Hamiltonian system of interacting particles and deriving from it, after rescaling, the Euler equations of compressible gas dynamics.…Our goal is to establish some rigorous connection between the Hamiltonian equations on one hand and the Euler equations on the other. Randomness is important because some averaging has to be done with respect to small scales and one needs some information as to how the particles will arrange themselves locally in phase space if we only know their local density, local average velocity, and local temperature. One expects the arrangement to be given by an appropriate Maxwell-Gibbs distribution and formally the equations are derived under that assumption. To justify it, at the least, one needs a reasonable ergodic theory and for that more noise is better.
For the entire collection see [Zbl 0829.00014].

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C22	Interacting particle systems in time-dependent statistical mechanics

Keywords:

hydrodynamic scaling; nonequilibrium statistical mechanics; interacting particles; connection between the Hamiltonian equations on one hand and the Euler equations on the other; ergodic theory