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Realisation of constraints in underdamped Langevin dynamics

Carsten Hartmann, Lara Neureither, Upanshu Sharma

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.

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.

Paper Structure

This paper contains 63 sections, 15 theorems, 254 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Relevant previous work
  3. Contributions of this article
  4. Guiding example
  5. Outline
  6. Main results
  7. Strong confinement of Langevin dynamics
  8. Set-up: coordinate projection constraints and physical parameter limits
  9. Confinement via stiff potential
  10. Spatial confinement.
  11. Phase-space confinement.
  12. Steady states.
  13. Constraining via physical parameter limits
  14. Zero mass limit
  15. Infinite mass limit
  16. ...and 48 more sections

Key Result

Proposition 3.1

Given $\varepsilon>0$, let $(Q^1_t,P^1_t)$ for $i=1,2$, be the solution of eq:intro-Spat-Cons-Fast with corresponding initial datum $(Q^1_0,P^1_0)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Two coupled point masses subject to friction and noise

Theorems & Definitions (39)

  • Remark 1.1
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.3: Convergence in probability and pointwise in time bounds
  • proof
  • Remark 3.4: Uniform in time estimate
  • Remark 3.5: Comparison to related literature
  • Remark 3.6
  • Proposition 3.7: Phase-space constrained limits
  • Remark 3.8
  • ...and 29 more