Thermostats,constants of motion and heat in nonequilibrium statistical mechanics

Thermostats are a contrivance designed to keep temperature constant in regions with which a system subject to nonconservative forces is in contact. In nonequilibrium statistical mechanics thermostat models can be Hamiltonian and infinite or artificial and finite. The problem examined is the equivalence between some Hamiltonian and artificial model: the artificial models considered are not the stochastic thermostats but the iso-ernergetic or isokinetic thermostats which introduce artificial forces to maintain the constrained temperatures: such thermostats are interesting because they are employed in simulations leading to recent developments in nonequilibrium theory. Connection heat conduction and Fourier law might find a place in the talk if time permits.