Conserved quantities and ensemble measure for Martyna--Tobias--Klein barostats with restricted cell degrees of freedom

We derive the conserved energy-like quantity and ensemble measure for Martyna–Tobias–Klein (MTK) barostats in which only a restricted subset of the cell degrees of freedom are active. In the standard fully anisotropic MTK formulation, the number of barostat degrees of freedom is $d^2$, where $d$ is the spatial dimension. When only $n_c$ axes of the cell matrix are allowed to fluctuate, the conserved energy-like quantity retains the same functional form but with $d^2$ replaced by $n_c$ in every term that counts barostat degrees of freedom. The derivation builds on the generalized Liouville framework for non-Hamiltonian systems and the existing MTK integration machinery. We verify that this quantity is exactly conserved, show that the resulting dynamics samples the isothermal–isobaric ensemble restricted to the submanifold of cell shapes in which inactive components are held fixed, and provide a complete Liouville-operator-based integration scheme for the masked MTK variant.