Certified Zero-Frequency Response and a Fourth-Order Mechanism Test for Nonlinear Soft-Constrained Langevin Dynamics
- 1Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, MD, USA, 20742-4015
Abstract
Eliminating stiff internal coordinates can preserve equilibrium while changing kinetics, and a finite-time coarse-graining error does not by itself control a zero-frequency transport observable. We study a reversible overdamped Langevin model with constant microscopic mobility and nonlinear soft constraints, Uϵ(x, y; γ) = V (x− Fγ) + |y − g(x)|2 2ϵ , where V may be nonquadratic and nonconvex. An exact coordinate transformation factorizes the equilibrium measure but produces a nontrivial kinetic metric. For any mean-zero source depending only on x, we prove a direct zero-frequency comparison between the full microscopic inverse response Rϵ and the constrained geometric response R0: 0 ≤ R0 −Rϵ ≤ ϵmT 2 Eπ∥Dw0∥2F , where w0 is the source-specific corrector derived from the effective Poisson equation. The bound contains no inverse global spectral gap. For a one-dimensional translated potential, the conjugateforce friction closes explicitly and admits a certified first finite-stiffness correction with two-sided O(ϵ2) remainder bounds. We then derive a mechanism test: any autonomous hidden-history model for which two-time forces are conditionally standard Gaussian with history-dependent correlation must have a nonnegative connected fourth quantity K4(t). A fully reversible nonlinear constant-mobility Langevin model violates this condition: K4,ϵ(t) < 0 for a nonzero short-time interval at every finite ϵ, while its constrained limit satisfies K4(t) < 0 for every t > 0. The all-time sign is certified by a five-dimensional outward-rounded interval Schur test controlling the infinite even sector of the generator. Integer interval arithmetic, independent primal–dual calculations of the full two-dimensional process, and a reversible finite-volume discretization provide mutually distinct checks. These results are a controlled response and model-discrimination theory for a specified soft-constrained class; they do not constitute a general theory of glass formation or a bulk shear-viscosity prediction.
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BibTeX
@misc{yang2026certified,
title = {{Certified Zero-Frequency Response and a Fourth-Order Mechanism Test for Nonlinear Soft-Constrained Langevin Dynamics}},
author = {Yueer Yang},
howpublished = {ChemRxiv},
year = {2026},
month = sep,
doi = {10.26434/chemrxiv.15009185/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009185/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - Certified Zero-Frequency Response and a Fourth-Order Mechanism Test for Nonlinear Soft-Constrained Langevin Dynamics
AU - Yueer Yang
PY - 2026
DA - 2026/09/21
PB - ChemRxiv
DO - 10.26434/chemrxiv.15009185/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009185/v1
AB - Eliminating stiff internal coordinates can preserve equilibrium while changing kinetics, and a finite-time coarse-graining error does not by itself control a zero-frequency transport observable. We study a reversible overdamped Langevin model with constant microscopic mobility and nonlinear soft constraints, Uϵ(x, y; γ) = V (x− Fγ) + |y − g(x)|2 2ϵ , where V may be nonquadratic and nonconvex. An exact coordinate transformation factorizes the equilibrium measure but produces a nontrivial kinetic metric. For any mean-zero source depending only on x, we prove a direct zero-frequency comparison between the full microscopic inverse response Rϵ and the constrained geometric response R0: 0 ≤ R0 −Rϵ ≤ ϵmT 2 Eπ∥Dw0∥2F , where w0 is the source-specific corrector derived from the effective Poisson equation. The bound contains no inverse global spectral gap. For a one-dimensional translated potential, the conjugateforce friction closes explicitly and admits a certified first finite-stiffness correction with two-sided O(ϵ2) remainder bounds. We then derive a mechanism test: any autonomous hidden-history model for which two-time forces are conditionally standard Gaussian with history-dependent correlation must have a nonnegative connected fourth quantity K4(t). A fully reversible nonlinear constant-mobility Langevin model violates this condition: K4,ϵ(t) < 0 for a nonzero short-time interval at every finite ϵ, while its constrained limit satisfies K4(t) < 0 for every t > 0. The all-time sign is certified by a five-dimensional outward-rounded interval Schur test controlling the infinite even sector of the generator. Integer interval arithmetic, independent primal–dual calculations of the full two-dimensional process, and a reversible finite-volume discretization provide mutually distinct checks. These results are a controlled response and model-discrimination theory for a specified soft-constrained class; they do not constitute a general theory of glass formation or a bulk shear-viscosity prediction.
ER -