Recently, it has undergone a new set of developments when adiabatic systems were identified as an important area where MaZe dynamics can provide an original formal approach and an effective integration algorithm . The method of mass-zero constraints was originally introduced in the early 1980s to study the rotational–translational coupling in diatomic molecules . In practical implementations, use of the SHAKE algorithm enables symplectic and efficient numerical integration of the extended dynamical system. From these, the original parametric dynamics for the physical dofs is rigorously recovered by taking the limit of zero mass for the auxiliary variables. The coupled evolution equations for the overall constrained system are then conveniently obtained in the Lagrangian formalism. The method considers an extended system in which the parameters appear as (auxiliary) dynamical variables together with the original dofs and the conditions are interpreted as constraints. MaZe is a general simulation approach to study the motion of a set of physical degrees of freedom (dofs) whose evolution depends on parameters subject to given conditions. In this paper, we discuss, focusing on recent developments, the Mass-Zero (MaZe) constrained dynamics and further extend it to a simple but interesting model of classical polarizable systems in constant external magnetic field. The new development is presented in the second part of this paper and illustrated via a proof-of-principle calculation of the charge transport properties of a simple classical polarizable model of NaCl. Such conditions occur, for example, when describing systems in external magnetic field and they require to adapt MaZe to integrate semiholonomic constraints. We then generalize the approach to the case of conditions on the auxiliary variables that linearly involve their velocities. We begin by presenting MaZe for typical minimization problems where the imposed constraints are holonomic and summarizing its key formal properties, notably the exact Born–Oppenheimer dynamics followed by the physical variables and the exact sampling of the corresponding physical probability density. The method is formulated in the Lagrangian framework, enabling the properties of the approach to emerge naturally from a fully consistent dynamical and statistical viewpoint. In MaZe, the minimum condition is imposed as a constraint on the auxiliary variables treated as degrees of freedom of zero inertia driven by the physical system. In this paper, we discuss a recent algorithm proposed to efficiently and rigorously simulate this type of systems: the Mass-Zero (MaZe) Constrained Dynamics. In several domains of physics, including first principle simulations and classical models for polarizable systems, the minimization of an energy function with respect to a set of auxiliary variables must be performed to define the dynamics of physical degrees of freedom.
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