Digressions on Irreversibility and Stochastic Systems

Abstract: We attempt to characterize irreversibility of a dynamical system from the existence of different forward and backward mathematical representations depending on the direction of the time arrow. Such different representations only exist for stochastic diffusion models so one has to face the justification of stochastic descriptions for physical systems which are inherently conservative. This representation can be shown to hold for conservative finite dimensional deterministic systems coupled to an infinite-dimensional conservative heat bath. We show that the heat bath acts on the finite-dimensional model by {\em state-feedback} and shifts its eigenvalues to make the system dissipative. Moreover, under a natural family of invariant measures the heat bath induces a white noise input acting on the system making it look like a true dissipative diffusion.