  Ergodicity is a fundamental requirement for a dynamical system to reach a
state of statistical equilibrium. On the other hand, it is known that in
slow-fast systems ergodicity of the fast sub- system impedes the equilibration
of the whole system due to the presence of adiabatic invariants. Here, we show
that the violation of ergodicity in the fast dynamics effectively drives the
whole system to equilibrium. To demonstrate this principle we investigate
dynamics of the so-called springy billiards. These consist of a point particle
of a small mass which bounces elastically in a billiard where one of the walls
can move - the wall is of a finite mass and is attached to a spring. We propose
a random process model for the slow wall dynamics and perform numerical
experiments with the springy billiards themselves and the model. The
experiments show that for such systems equilibration is always achieved; yet,
in the adiabatic limit, the system equilibrates with a positive exponential
rate only when the fast particle dynamics has more than one ergodic component
for certain wall positions.
