  We introduce a nonequilibrium grand-canonical ensemble defined by considering
the stationary state of a driven system of particles put in contact with a
nonequilibrium particle reservoir. At odds with its equilibrium counterpart, or
with purely formal constructions of a grand-canonical ensemble, this
physically-motivated construction yields a grand-canonical distribution that
depends on the details of the contact dynamics between the system and the
reservoir. For non-interacting driven particles, a grand-canonical chemical
potential can still be defined, although this chemical potential now differs
from that of the reservoir. However, in the general case, the usual exponential
factor (in the particle number) defining the grand-canonical chemical
potential, is replaced by the exponential of a non-linear function of the
density, this function being proportional to the volume. This case is
illustrated explicitly on a one-dimensional lattice model. Although a
grand-canonical chemical potential can no longer be defined in this case, it is
possible for a subclass of contact dynamics to generalize the equilibrium
fluctuation-response relation by introducing a small external potential
difference between the system and the reservoir.
