We consider here a dynamic model for a gas in which a variable number of particles \(N \in \mathbb {N}_0:= \mathbb {N} \cup \{0\}\) can be located at a site. The dynamics are played by the shift acting on \(\Omega := \mathcal {A}^\mathbb {N}\) , where \(\mathcal {A}:= \{1,2,\ldots ,r\}\) . Introducing the chemical potential \(\mu\) , we adapt the concept of grand-canonical partition sum of thermodynamics of gases, considering a certain family of potentials \((A_N)_{N \in \mathbb {N}_0}\) , \(A_N:\Omega \rightarrow \mathbb {R}\) . Extending classical thermodynamic formalism, we introduce the grand-canonical-Ruelle operator: \(\mathcal {L}_{\beta , \mu }(f)=g\) , when, \(\beta >0,\mu <0,\) where \(\begin{aligned} g(x)= \mathcal {L}_{\beta , \mu }(f) (x) =\sum _{N \in \mathbb {N}_0} e^{\beta \, \mu \, N }\, \sum _{j \in \mathcal {A}} e^{- \,\beta \, A_N(jx)} f(jx). \end{aligned}\) We show the existence of the main eigenvalue, an associated eigenfunction, and an eigenprobability for \(\mathcal {L}_{\beta , \mu }^*\) . We also consider the concept of entropy for holonomic probabilities on \(\Omega \times \mathcal {A}^{\mathbb {N}_0}\) , relating these items with the problem of maximizing grand-canonical pressure. We briefly digress on a possible interpretation of the concept of topological pressure as related to the gas pressure of gas thermodynamics.
