Bitemporal Dynamics Sinai Divergence, An Energetic Analog to Boltzmann's   Entropy?
  Sinai chaos is characterized by exponential divergence between neighboring trajectories of a point billiard. If the repulsive potential of the finite-diameter fixed particle in the middle of the table is made smooth, the Sinai divergence persists with finite measure. So it does if the smooth potential is made attractive. So it still does if the potential is in addition made time-dependent (periodic). Then a systematic decrease in energy of the moving particle can be predicted to occur in both time directions for a long time. If so, classical entropy acquires an analog in real space. 
Otto E. Rossler, Ramis Movassagh