  We numerically study a billiard system with a time-dependent force, and our
results suggest the existence of a limitation on possible transitions between
steady states in Hamiltonian chaos, in analogy to the limitation on transitions
between equilibrium states described by the second law of thermodynamics. This
limitation is expressed in terms of irreversible information loss, which is
defined for each trajectory through Lyapunov analysis. As a key step in the
study, we demonstrate a non-equilibrium equality which means that the average
of the inverse exponential of the irreversible information loss is unity, where
the average is taken over initial conditions sampled from the microcanonical
ensemble.
