Rare interaction limit in a billiard system: standard pairs and recollisions
speaker: Peter Toth (Budapest University of Technology and Economics)abstract: We study a billiard system describing two particles, in the limit when the particles exchange energy rarely. This is motivated by a heat conduction model studied earlier by Bunimovich, Liverani, Pellegrinotti and Sukhov, and later by Gaspard and Gilbert.
When energy exchange is rare, we have a fast-slow system with the energies being the slow coordinates, and the fast subsystem mixing fast. In the rare interaction limit we show convergece of the energy process to a Markov jump process. The proof uses the method of standard pairs, which are measures on the phase space, concentrated on unstable curves. As time evolves, a standard pair evolves into a collection of standard pairs, supported on many unstable curves cut by the singularities of the dynamics. An essential part of the argument is to show that "fast recollisions" are unlikely, so that after an energy exchange, the fast subsystem has sufficient time to equilibrate before another energy exchange occurs.
In this talk I will explain what this exactly means, and how fast recollisions can be ruled out. The argument uses two key facts about the time evolution of standard pairs: one is fast convergence to the equilibrium measure (in the fast subsystem), and the other is a "growth lemma" saying that, as time evolves, most of the measure will always be on unstable curves that are not too short.
Joint work with Péter Bálint, Péter Nándori and Domokos Szász.
timetable:
Fri 7 Jun, 9:50 - 10:35, Aula Dini
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