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
- 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
describes the motion of a single particle), is the Lyapunov exponent of
the billiard flow, when the table is kept at rest.