Role of the nature of noise in the thermal conductance of mechanical systems

Focusing on a paradigmatic small system consisting of two coupled damped oscillators, we survey the role of the Levy-Ito nature of the noise in the thermal conductance. For white noises, we prove that the Levy-Ito composition (Lebesgue measure) of the noise is irrelevant for the thermal conductance of a nonequilibrium linearly coupled chain, which signals the independence of mechanical and thermodynamical properties. In contrast, for the nonlinearly coupled case, the two types of properties mix and the explicit definition of the noise plays a central role.