Role of ergodicity in the transient Fluctuation Relation and a new relation for a dissipative non-chaotic map

•We verify that ergodicity is a crucial condition for the validity of the transient Fluctuation Relation.•We numerically test the onset of a new Fluctuation Relation for a family of non-chaotic maps.•We analytically evaluate the Lyapunov exponents, the basins of attraction and the invariant sets of our dynamical system.

Toy model 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 (also known as “ergodic consistency”) seems to be a necessary condition for the transient Fluctuation Relation. This investigation then leads to the numerical verification of a kind of transient relation which, differently from the usual transient Fluctuation Relation, holds only asymptotically. At the same time, this relation is not a steady state Fluctuation Relation, because the steady state is a fixed point without fluctuations.