Stationary nonequilibrium statistical mechanics

Systems in stationary nonequilibrium are mechanical systems subject to nonconservative external forces and to thermostat forces which forbid indefinite increase of the energy and allow reaching statistically stationary states. A system Σ is described by the positions and velocities of its n particles X, Ẋ, with the particles positions confined to a finite volume container C0. If X = (x1, . . ., xn) are the particles positions in a Cartesian inertial system of coordinates, the equations of motion are determined by their masses mi > 0, i = 1, . . . , n, by the potential energy of interaction V (x1, . . . ,xn) ≡ V (X), by the external nonconservative forces Fi(X,Φ) and by the thermostat forces −θi as