Nonequilibrium grand-canonical ensemble built from a physical particle reservoir

We introduce a nonequilibrium grand-canonical ensemble defined by considering the stationary state of driven system particles put in contact with particle reservoir. When an additivity assumption holds for large deviation function density, chemical potential reservoir can be defined. The distribution then takes form similar to equilibrium one. At variance equilibrium, though, probability weight is ``renormalized'' contribution coming from contact, respect canonical isolated system. A formal introduced terms scaled cumulant generating function, as Legendre-Fenchel transform density. role Legendre parameter played, physically, when latter defined, or energy difference applied between and Static fluctuation-response relations naturally follow structure. Some results are illustrated on two different explicit examples, gas noninteracting active lattice model interacting particles.