Nonequilibrium Stationary States (Deterministic)

Nonequilibrium stationary states (NESS) describe the state of a mechanical system driven and maintained out of equilibrium by external forces. The external forces can be mechanical or thermodynamical. Examples of mechanical forces are external non-Hamiltonian forces imposed on the system or boundary forcing for example in a shear flow. Thermodynamical forces occur when a system is coupled to several reservoirs which have different thermodynamical parameters such as temperatures, chemical potentials, or others potentials. The main characteristic of a NESS is that it sustains sustain steady flows, for example energy flows (heat conduction), particles flow (like in heat or electrical conduction), momentum flows (like in shear flows). The occurence of these various flows in NESS can summarized by saying the NESS exhibits a positive entropy production. To give a formal definition of a NESS, let M be the state space of the system and let Φt : M → M be a group of invertible transformations which describe the dynamics of the system. We assume that a probability measure μ is given. The measure μ serves as a reference measure and is given a-priori. In applications it depends on the physical parameters of the system. We emphasize that, in general, μ is not an invariant measure for the dynamics φt.