Thermodynamic Vs Statistical Entropy Production in Thermostatted Hamiltonian Dynamics

Sufficient conditions were recently established for the rates of thermodynamic entropy production and phase space volume contraction to be identically equal in thermostatted Hamiltonian systems in non-equilibrium steady states (Ramshaw 2017 Phys. Rev. E 96 052122). This equality has previously been interpreted as a statistical analogue of the second law of thermodynamics (SLT). However, that interpretation is unjustified because the volume contraction rate represents the production rate of the statistical entropy of the system and thermostat together, which differs in general from that of the Hamiltonian system itself. This logical lacuna is remedied by identifying two further conditions which combine with the previous conditions to imply a proper statistical analogue of the SLT. The first of these new conditions can simply be imposed a priori, whereas the second, like ergodicity and mixing, involves the asymptotic statistical properties of the dynamics as t → ∞. The latter condition seems likely to be analytically intractable in general, and therefore to require numerical investigation and confirmation in systems of practical interest