Title: Statistical mechanics, 2-D fluids, and structure-preserving numerics

Abstract:

In a 1949 paper, the Norwegian American genius Lars Onsager demonstrated that Boltzmann’s theory of statistical mechanics provides insights to the long-time behavior of 2-D fluids described by Euler's equations. The key notion is to “approximate” the fluid by a finite number of interacting point vortices. Since these are governed by Hamiltonian dynamics, the framework of statistical mechanics provides a characterization of the long-time asymptotic behavior. The approach successfully predicts clustering of equally signed vortices, which is part of Onsager’s legacy. Nevertheless, and as Onsager proclaimed in his paper, the point vortex approximation is highly non-regular and might therefore be poor. Furthermore, statistical mechanics assumes ergodicity, which fails for 2-D Euler. In this talk, I will present another finite-dimensional approximation of 2-D Euler due to Vladimir Zeitlin. Unlike point vortices, Zeitlin’s model gives smooth approximations. It thereby renders a refinement of Onsager’s statistical approach, and, together with a symplectic numerical time-integration scheme, provides a structure preserving numerical method. Simulations using this method suggest compelling connections between 2-D Euler and integrability of low-dimensional Hamiltonian systems with symmetry. The work is joint with Milo Viviani.

Il colloquio sarà preceduto come da consueto da un rinfresco a partire dalle ore 17:30. Siete tutti invitati a partecipare. Saluti,

Dario Trevisan

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Dipartimento di Matematica
Universita di Pisa
Largo Bruno Pontecorvo 5
56127 Pisa PI Italy

http://people.dm.unipi.it/trevisan