Role of Ergodicity in the Transient Fluctuation Relation and a New Relation for a Dissipative Non-Chaotic Map

Deterministic dynamical systems such as the baker maps are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the Transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics seems to be a necessary condition for the Transient Fluctuation Relation to hold. This investigation then leads to the numerical verification of a kind of transient relation which, differently from the usual Transient Fluctuation Relation (FR), holds only asymptotically. At the same time, it is not a steady state fluctuation relation, because no fluctuations are present in the steady state.