The Steady State Fluctuation Relation for the Dissipation Function

We give a proof of transient uctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient uctuation relation yields a steady state uctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and su cient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state uctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of decay of correlations in the dissipation. It does not require the steady state probability measure to enjoy special properties. Because of this generality the resulting steady state uctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state uctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are correctly described in the context of this work.