H-Theorems from Autonomous Equations

Abstract

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.

13 pages; v1 -> v2: Sec. 1-2 considerably rewritten, minor corrections in Sec. 3-4

References in corpus (3)

• On the (Boltzmann) Entropy of Nonequilibrium Systems
• The Boltzmann Entropy for Dense Fluids Not in Local Equilibrium
• Time-Reversal and Entropy

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