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.

• The paper establishes explicit rates of convergence for soft constraints in underdamped Langevin dynamics, detailing both pathwise errors and invariant measure behavior.
• It demonstrates that phase-space penalization stabilizes positions and momenta uniformly in time, ensuring accurate realization of holonomic constraints.
• Results provide actionable insights for high-fidelity sampling in molecular simulations and stochastic optimization, clarifying transient dynamics and steady-state implications.

Realisation of Constraints in Underdamped Langevin Dynamics

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Introduction and Motivation

The study addresses the quantitative and pathwise realization of holonomic constraints in underdamped Langevin dynamics, a ubiquitous framework in molecular simulation, stochastic sampling, and statistical mechanics. The analysis is motivated by practical questions concerning the correct implementation and interpretation of soft (penalized) constraints, and how these relate, in the strong-confinement or singular-parameter limits, to the ideal dynamics on constrained submanifolds.

Contra the overdamped regime, rigorous pathwise error estimates and steady-state implications for various constraint mechanisms in underdamped Langevin systems are less developed. This work establishes explicit rates and modes of convergence for dynamically realized soft constraints and parameter regimes—large confinement, mass, and friction—clarifying both sampling properties and transient dynamics.

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Mathematical Setting and Constraint Realization Mechanisms

General Framework

Underdamped Langevin dynamics on $\mathbb{R}^{2d}$ is given by

$$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$

where $H(Q,P) = V(Q) + \frac12 |P|^2$, with $V$ a potential, $\gamma>0$, and $W_t$ standard Brownian motion. Constraints are imposed via a smooth function $\xi: \mathbb{R}^d \to \mathbb{R}^k$, with holonomic constraint manifold $\mathcal{M} = \xi^{-1}(0)$.

Mechanisms for Constraint Realization

The paper systematically treats several constraint realization protocols:

1. Strong Confinement (Soft Constraint Limit): Approximates the constraint using penalization: adding a stiff potential $\tfrac{1}{2\varepsilon} |\xi(Q)|^2$ to $V(Q)$ and analyzing the $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$0 limit.
2. Phase-Space Confinement: In contrast to spatial-only penalties, incorporates both $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$1 and $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$2 penalizations, targeting both positions and velocities. This suppresses unphysical oscillations in constrained momenta.
3. Physical Parameter Limits:

Constraints are approximated by limits such as - Zero-mass limit: mass matrix in constrained DOFs is scaled to zero, - Infinite-mass limit: mass in constrained DOFs diverges, - Large-friction limit: friction in constrained DOFs diverges, with or without matching the noise (fluctuation-dissipation).

The analysis is mainly carried out for linear (coordinate projection) constraints, but the structure outlined extends to more general holonomic settings.

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Main Results

Pathwise and Quantitative Convergence

Spatial Confinement:

If initial data are well-prepared ($$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$3 for projection constraint), $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$4 as $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$5, with explicit rates. However, the constrained momenta $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$6 exhibit persistent oscillations—they do not converge pathwise but vanish in a time-integrated sense only. Lack of uniform dissipativity in the fast subspace prevents strong stabilization of $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$7.

Phase-Space Confinement:

Penalizing both position and hidden velocity constraints yields \textbf{uniform-in-time explicit rates} for $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$8 as $$\begin{aligned} dQ_t &= \nabla_p H(Q_t, P_t)dt \ dP_t &= -\nabla_q H(Q_t, P_t)dt - \gamma \nabla_p H(Q_t, P_t)dt + \sqrt{2\gamma\beta^{-1}} dW_t \end{aligned}$$9, for arbitrary initial data, reflecting genuine stabilization to the constraint manifold in the phase space. The unconstrained variables $H(Q,P) = V(Q) + \frac12 |P|^2$0 converge pathwise to effective Langevin dynamics on the constraint.

Parameter Limits:

• Zero-mass: $H(Q,P) = V(Q) + \frac12 |P|^2$1 and $H(Q,P) = V(Q) + \frac12 |P|^2$2 follows overdamped effective dynamics.
• Infinite-mass/infinite-friction: $H(Q,P) = V(Q) + \frac12 |P|^2$3 is frozen at its initial value, while $H(Q,P) = V(Q) + \frac12 |P|^2$4 may become undefined or behave as an uncontrolled process (depending on the scaling/noise).
• Fluctuation-dissipation preservation: Stationary distributions are non-singular in $H(Q,P) = V(Q) + \frac12 |P|^2$5, otherwise, marginals can develop degeneracies (e.g., delta functions in momentum).

All main results are explicit in the rate of convergence with respect to the singular parameter, the time horizon, and the initial data norm, and handle both transient and steady-state regimes.

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Invariant Measures and Sampling Properties

The paper gives a rigorous characterization of the relationship between pre-limit and limit invariant measures for each constraint realization mechanism. For instance,

• Spatial confinement leads, as $H(Q,P) = V(Q) + \frac12 |P|^2$6, to the Gibbs measure on $H(Q,P) = V(Q) + \frac12 |P|^2$7, but with marginal in $H(Q,P) = V(Q) + \frac12 |P|^2$8 possibly oscillatory.
• Phase-space confinement yields the conditional Gibbs measure both in $H(Q,P) = V(Q) + \frac12 |P|^2$9 and $V$0, aligning with what would be obtained via geometric/Lagrange-multiplier projection.
• Physical limits can produce steady states that do not coincide with the Gibbs measure on the constraint; the order of limit $V$1 and parameter scaling can fail to commute, reflecting subtle non-ergodic effects.

A strong claim is that the steady-state of the constrained Langevin system achieved via phase-space penalization coincides with the reduction (conditioning) of the unconstrained Gibbs measure on the full space, while other mechanisms can yield singular or otherwise "incorrect" measures.

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Technical Approach

The methods combine:

• Pathwise analysis via explicit variation-of-constants solutions to linearized (or affine) SDEs for the constrained blocks;
• Uniform a priori estimates leveraging the separation of timescales induced by stiff penalties or singular parameters;
• Gronwall-type arguments for explicit, uniform-in-time control of errors;
• Distributional convergence analysis for invariant measures, identifying the limiting conditional laws and their precise dependence on the realization protocol;
• Extension to random or unprepared initial data, with care to distinguish when initial energy/incompatibility with the constraint injects persistent error terms.

A significant technical contribution is demonstrating that for quadratic confining potentials (i.e., linear constraints), the uniform-in-time $V$2 error bounds hold for the resolved (unconstrained) DOFs, a property that fails for general nonlinear constraints.

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Implications and Future Directions

The findings clarify which constraint approximation schemes faithfully reproduce both the pathwise and invariant measure structure of the ideal constrained underdamped Langevin dynamics—and which do not.

• For high-fidelity sampling (e.g., constrained MCMC, computation of conditional expectations, free energy on submanifolds), phase-space penalization should be preferred over spatial-only methods to avoid spurious momentum artifacts and nonphysical invariant marginals.
• Insights into the interplay between thermostatting, time-step stability (e.g., the effect of stiff oscillatory fast modes), and geometric structure preservation (e.g., symmetry and ergodicity) are directly relevant to contemporary statistical mechanics and MCMC on manifolds [projection_diffusion].
• The work provides rigorous evidence supporting soft penalty regularization approaches in the training of constrained neural networks, where analogues of underdamped Langevin dynamics underpin stochastic optimization with hard regularizers [leimkuhler-constraint-regularization-nn, leimkuhler2022better].

Future theoretical developments should address general nonlinear (possibly non-holonomic) constraints, interacting particle systems, and devise high-order numerical schemes respecting the stiffness-induced homogenized dynamics and their sampling properties.

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Conclusion

This article provides a comprehensive mathematical analysis of multiple mechanisms for the realization of holonomic constraints in underdamped Langevin systems. The quantitative, pathwise, and invariant measure results establish precise conditions on the appropriateness of various soft-constraint approaches, the role of initial data, and explain the mechanism-dependent transition between correct and pathological sampling in both the transient and asymptotic regimes. The results directly inform best practices in computational physics, molecular simulation, and machine learning contexts where geometric constraint enforcement and accurate statistical sampling are critical (2604.02129).