Vlasov billiards

A particle gas in a smooth bounded region interacts with the table in such a way that the total momentum of gas and table is conserved.
If the gas consists of finitely many particles, then the table gets a kick each time, a particle hits the boundary.
Already for two particles in a circular table, the motion of the coupled system is not integrable.
If the particle gas is given by a smooth density in the phase space, the table moves continuously. Even with a smooth interparticle interaction, there is an existence theorem for such infinite-dimensional Hamiltonian systems, which extends the existence theorem of Dobrushin-Brown-Hepp for Vlasov dynamics.
The system is then a coupled system of an infinite-dimensional Hamiltonian system coupled to the finite-dimensional boundary motion by the law of total momentum conservation.
For such dynamical systems, we expect:
  • Convergence to equilibrium: while for finitely many particles, Poincare recurrence holds, this is no more the case, when the particle density is continuous. We expect the amplitude of the table motion to decrase to zero as time is increased. For a chaotic billiard table like the Bunimovich stadium, we expect the phase space density P(t) to converge weakly to a time-independent measure.
  • Convergence of energy: We expect the energy of the boundary averaged over a time interval [t,t+1) to converge to a nonzero value. This means that the wall continues to oscillate with smaller and smaller amplitude but in such a way that the velocity p(t) of the wall (which stays bounded due to energy conservation), does not go to zero.
  • Growth of fluctuations: we expect that some particles will get accelerated to larger and larger speed, leading to an indefinit growth of d/dt p(t). In other words, the time-independent limiting phase space density P will have no more compact support.
  • Lyapunov exponent: we expect the Lyapunov exponent of the gas (which is defined as the exponential growth rate of ||DX(t)||, where (x(t),y(t))=X(t)(x(0),y(0)) describes the motion of a single particle), is the Lyapunov exponent of the billiard flow, when the table is kept at rest.

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© 2000, Oliver Knill , dynamical-systems.org