  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.
