A non-equilibrium equality in Hamiltonian chaos
  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. 
Shin-ichi Sasa