  For non-equilibrium systems of interacting particles and for interacting
diffusions in d dimensions, a novel fluctuation relation is derived. The
theorem establishes a quantitative relation between the probabilities of
observing two current values in different spatial directions. The result is a
consequence of spatial symmetries of the microscopic dynamics, generalizing in
this way the Gallavotti-Cohen fluctuation theorem related to the time-reversal
symmetry. This new perspective opens up the possibility of direct experimental
measurements of fluctuation relations of vectorial observables.
