On the Gallavotti-Cohen symmetry for stochastic systems

Considering Langevin dynamics we derive the general form of the stochastic differential that satisfies the Gallavotti-Cohen symmetry. This extends the work previously done by Kurchan, and Lebowitz and Spohn on such systems, and we treat systems with and without inertia in a unified manner. We further shown that for systems with a time-reversal invariant steady state there exists a stochastic differential for which then the Gallavotti-Cohen symmetry, and all its consequences, are valid for finite times. For these systems the differential can be seen as the direct analogy of the Gibbs-entropy creation along paths in deterministic systems. It differs from previously studied differentials in that it identically zero for equilibrium systems while on average strictly positive for non-equilibrium system. When the steady state is not time-reversal invariant the Gallavotti-Cohen symmetry is asymptotically valid in the usual long time limit.