H-theorems from macroscopic autonomous equations

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 rigourously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, the macrostate evolves au-tonomously, 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 rec-oncile 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 semigroup property for the macroscopic evolution. KEY WORDS: H-theorem, entropy, irreversible equations. 1