Realisation of constraints in underdamped Langevin dynamics

Abstract

This article deals with the realisation of constraints in underdamped Langevin dynamics via soft-constrained dynamics. Specifically, we study systems with a large (or small) parameter that controls the constraint mechanisms, e.g. the strength of confinement forces, mass or friction coefficients, and we derive quantitative convergence results for both the constrained variables and the softly constrained dynamics on the limiting subspace. The latter can be either a spatial or a momentum or velocity subspace, depending on the underlying soft constraint mechanism; in this paper we treat only holonomic constraints, i.e. all momentum- or velocity-level constraints are integrable. We explicitly include the initial conditions so that it is clear whether they must satisfy the constraint or not in order to realise the desired constrained dynamics. We discuss the implications of these results as well as questions related to the sampling of the corresponding conditional probability measures.