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

Abstract

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.