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Quantum hydrodynamic limit for a model of hard rods.Mathematical Physics & Probability
|Speaker: ||Yu. Suhov, University of Cambridge|
|Location: ||693 Kerr|
|Start time: ||Tue, May 18 1999, 4:10PM|
The question is how to obtain hydrodynamic equations (Euler,
Navier-Stokes) from Liouville-type equations of Hamiltonian mechanics
(classical or quantum). The original idea was due to Ch. Morrey (1956)
who introduced a concept of a hydrodynamic limit and was able to formally
derive an Euler equation from the classical Liouville equations (more
precisely, from the corresponding BBGKY hierarchy). However, Morrey had
to make some assumptions about the long-term behavior of the motion,
and this included a statement on ergodicity, in the sense that all
`reasonable' first integrals are functions of the energy, linear momentum
and the number of particles. Since then the idea of hydrodynamic limit
became very popular in the literature and has been successfully applied
to a variety of models of (mostly stochastic) dynamics relating them
to non-linear equations. However, in the original problem there was no
substantial progress until the work by S. Olla, S.R.S. Varadhan and T. Yau
(1992) where Morrey's assumptions were replaced by introducing a small
noise into the Hamiltonian (which effectively kills other integrals
of motion), and a classical Euler equation was correctly derived. The
talk will be on a quantum model for which the hydrodynamic limit can
be rigorously performed. The resulting Euler-type equation is similar
to the one that arises for the classical counterpart of the same model.
This suggests that perhaps classical and quantum hydrodynamic equations
must look sismilar if they are written for local densities of `canonical'
conserved quantities (the density of mass, linear momentum and energy).
The work is in collaboration with C.Boldrighini (University of Rome-1)
and A. Pellegrinotti (University of Rome-3).