On the conservation of specific energy and entropy in infinite anharmonic systems

We work with infinite, closed, translation-invariant, finite-range lattice systems with “unbounded classical spins”, also known as anharmonic crystals, under assumptions close to those used by Lanford, Lebowitz and Lieb (J. Stat. Phys., 1977); among other conditions, the pinning dominates the interaction. In this context, we prove conservation of the specific energy and specific entropy under the time evolution, and we discuss their relation to approach to thermal equilibrium, paralleling known results in the theory of quantum spin systems, where noncommutativity, as opposed to lack of compactness, is the main source of difficulties.