Active Fluctuation Symmetries

In contrast with the understanding of fluctuation symmetries for entropy production, similar ideas applied to the time-symmetric fluctuation sector have been less explored. Here we give detailed derivations of time-symmetric fluctuation symmetries in boundary driven particle systems such as the open Kawasaki lattice gas and the zero range model. As a measure of time-symmetric dynamical activity we take the difference $(N_1 - N_L)/T$ in the number of particles entering or leaving the system at the left versus the right edge of the system over time $T$. We show that this quantity satisfies a fluctuation symmetry from which we derive a new Green-Kubo type relation. It will follow then that the system is more active at the edge connected to the particle reservoir with the largest chemical potential. We also apply these exact relations derived for stochastic particle models to a deterministic case, the spinning Lorentz gas, where the symmetry relation for the activity is checked numerically.