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In a recent paper [M. Colangeli \textit{et al.}, J.\ Stat.\ Mech.\ P04021,(2011)] it was argued that the Fluctuation Relation for the phase spacecontraction rate $\Lambda$ could suitably be extended to non-reversibledissipative systems. We strengthen here those arguments, providing analyticaland numerical evidence based on the properties of a simple irreversiblenonequilibrium baker model. We also consider the problem of response, showingthat the transport coefficients are not affected by the irreversibility of themicroscopic dynamics. In addition, we prove that a form of \textit{detailedbalance}, hence of equilibrium, holds in the space of relevant variables,despite the irreversibility of the phase space dynamics. This corroborates theidea that the same stochastic description, which arises from a projection ontoa subspace of relevant coordinates, is compatible with quite differentunderlying deterministic dynamics. In other words, the details of themicroscopic dynamics are largely irrelevant, for what concerns properties suchas those concerning the Fluctuation Relations, the equilibrium behaviour andthe response to perturbations.

Equilibrium, fluctuation relations and transport for irreversible deterministic dynamics