  The H-theorem is an extension of the Second Law to a time-sequence of states
that need not be equilibrium ones. In this paper we review and we rigorously
establish the connection with macroscopic autonomy.
  If for a Hamiltonian dynamics for many particles, at all times the present
macrostate determines the future macrostate, then its entropy is non-decreasing
as a consequence of Liouville's theorem. That observation, made since long, is
here rigorously analyzed with special care to reconcile the application of
Liouville's theorem (for a finite number of particles) with the condition of
autonomous macroscopic evolution (sharp only in the limit of infinite scale
separation); and to evaluate the presumed necessity of a Markov property for
the macroscopic evolution.
