Consistent Projection of Langevin Dynamics: Preserving Thermodynamics and Kinetics in Coarse-Grained Models

Carsten Hartmann; Feliks N\"uske; Lara Neureither; Selma Moqvist; Simon Olsson; Vahid Nateghi

arxiv: 2512.03706 · v3 · pith:2T7PWY4Jnew · submitted 2025-12-03 · ⚛️ physics.comp-ph · cs.LG· math.DS

Consistent Projection of Langevin Dynamics: Preserving Thermodynamics and Kinetics in Coarse-Grained Models

Vahid Nateghi , Lara Neureither , Selma Moqvist , Carsten Hartmann , Simon Olsson , Feliks N\"uske This is my paper

Pith reviewed 2026-05-17 02:16 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cs.LGmath.DS

keywords coarse grainingLangevin dynamicsZwanzig projectionkinetic propertiesthermodynamic interpolationextended dynamic mode decomposition

0 comments

The pith

Projection of underdamped Langevin dynamics produces closed coarse-grained equations that preserve both equilibrium distributions and transition rates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the Zwanzig projection to underdamped Langevin dynamics to obtain a closed-form description of the reduced variables. This projection yields dynamics whose equilibrium statistics and kinetic rates match those of the original high-dimensional system. Generator extended dynamic mode decomposition is then used to learn the generator of the coarse process from simulation data and to compute observables such as mean transition times. Thermodynamic interpolation is combined with the projection so that the same coarse model can be evaluated at multiple thermodynamic conditions without additional full-space runs. In a two-dimensional test system the resulting coarse model reproduces both the free-energy landscape and the slowest relaxation timescales of the full dynamics.

Core claim

Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI) to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

What carries the argument

The Zwanzig projection operator applied to underdamped Langevin dynamics, which produces a closed, Markovian coarse-grained process whose generator is learned from data.

If this is right

The coarse-grained model reproduces the equilibrium probability distribution and associated free energy of the original system.
Kinetic observables such as mean transition times between metastable states remain accurate after projection.
Thermodynamic interpolation extends the same coarse model to nearby state points without new full-space simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same projection-plus-learning pipeline could be tested on higher-dimensional molecular systems to quantify the reduction in simulation cost while retaining kinetic fidelity.
If memory effects appear in other systems, the method would need augmentation with non-Markovian terms to maintain accuracy.

Load-bearing premise

The projection operator applied to the chosen test system yields a Markovian coarse process whose memory effects and artifacts remain negligible.

What would settle it

A side-by-side calculation on the two-dimensional test system showing that the coarse-grained free-energy surface or the slowest transition timescale differs measurably from the full Langevin reference would falsify the preservation claim.

Figures

Figures reproduced from arXiv: 2512.03706 by Carsten Hartmann, Feliks N\"uske, Lara Neureither, Selma Moqvist, Simon Olsson, Vahid Nateghi.

**Figure 2.** Figure 2: FIG. 2: CG definition for the Lemon Slice system, in position and momentum space. [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Histogram of generated data using TI model without (middle) and with (right) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Slowest timescales [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Learning of the effective diffusion ( [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

**Figure 7.** Figure 7: 21 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Learning of the effective diffusion ( [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Integration of coarse grained SDE for [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Integration of coarse grained SDE for [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: VAMP-score analysis. Score corresponding to various values of the kernel [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗

read the original abstract

Coarse graining (CG) is an important task for efficient modeling and simulation of complex multi-scale systems, such as the conformational dynamics of biomolecules. This work presents a projection-based coarse-graining formalism for general underdamped Langevin dynamics. Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method, which was developed in the context of Koopman operator methods, can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI), a generative approach to transform samples between thermodynamic conditions, to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a closed-form coarse-grained dynamics for underdamped Langevin via Zwanzig projection and pairs it with gEDMD and thermodynamic interpolation, but the absence of memory effects needs explicit confirmation.

read the letter

The punchline is that this paper derives a closed-form coarse-grained model for underdamped Langevin dynamics via Zwanzig projection and shows how to learn its kinetics with gEDMD while using thermodynamic interpolation to cover multiple states. What is new is the specific application to underdamped systems with a claimed closed form, plus the workflow that combines projection, operator learning, and interpolation. Prior CG work often focuses on overdamped cases or uses more empirical fitting. The paper does well in setting up a first-principles starting point and then using gEDMD to evaluate transition timescales without running the full dynamics repeatedly. The 2D model demonstration indicates that thermodynamic and kinetic properties can be recovered. The soft spots center on the Markovian character of the projected dynamics. Zwanzig projection on underdamped Langevin usually produces a generalized Langevin equation with a memory integral rather than a simple closed Markov process. The paper needs to demonstrate explicitly that the memory kernel vanishes for their coarse-grained variables. In a 2D system this should be straightforward to verify by computing the kernel or checking for non-exponential decay in correlations. If that step is not there or relies on the low dimension hiding the effect, the kinetic preservation result could be less general than presented. The thermodynamic interpolation part seems like a useful add-on for spanning conditions, but its accuracy depends on the quality of the underlying CG model. Overall, this is for people in molecular simulation and coarse-graining who care about preserving both static and dynamic properties. Readers working with Langevin-based models or Koopman methods would find the technical details useful. The work has enough structure and a concrete example to merit a serious referee, though the referees should press on the memory kernel question. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a projection-based coarse-graining method for underdamped Langevin dynamics. Following the Zwanzig projection operator, the authors derive a closed-form expression for the coarse-grained (CG) dynamics. They then apply generator Extended Dynamic Mode Decomposition (gEDMD) to learn the CG generator from data and recover kinetic observables such as transition timescales. The framework is combined with thermodynamic interpolation (TI) to access multiple thermodynamic states without additional simulations. Accuracy is demonstrated on a two-dimensional test system, with the claim that both thermodynamic and kinetic properties of the full-space model are accurately reproduced.

Significance. If the central derivation is free of hidden assumptions that force Markovianity and if the numerical results are robust, the work supplies a first-principles route to CG models that simultaneously preserve equilibrium statistics and dynamical timescales. Such a method would be valuable for efficient simulation of biomolecular systems where both structural ensembles and kinetic rates matter.

major comments (2)

[§3] §3 (Zwanzig projection derivation): the manuscript must explicitly show that the memory kernel arising from the projection of the underdamped Langevin equation onto the chosen CG variables is either identically zero or reduces to a delta function. Without this step, the subsequent claim of a closed Markovian CG process is not justified, and gEDMD applied to finite trajectories will systematically bias kinetic quantities.
[§5] §5 (2D numerical demonstration): the reported agreement with the full model lacks quantitative error metrics (e.g., relative error in free-energy barriers, mean first-passage times, or autocorrelation functions) and does not state the trajectory length, sampling frequency, or any data-exclusion criteria used to train gEDMD. These omissions prevent assessment of whether the claimed accuracy is robust or sensitive to post-hoc choices.

minor comments (2)

[Abstract] Abstract: the phrase 'closed-form expression' is used without reference to the resulting equation; adding the equation number would improve clarity.
[Notation] Notation: the symbol for the projection operator and the CG generator should be defined once and used consistently; occasional re-use of the same symbol for different objects appears in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments identify key areas where additional rigor and detail will strengthen the presentation. We address each major comment below and commit to revisions that directly respond to the concerns raised.

read point-by-point responses

Referee: [§3] §3 (Zwanzig projection derivation): the manuscript must explicitly show that the memory kernel arising from the projection of the underdamped Langevin equation onto the chosen CG variables is either identically zero or reduces to a delta function. Without this step, the subsequent claim of a closed Markovian CG process is not justified, and gEDMD applied to finite trajectories will systematically bias kinetic quantities.

Authors: We agree that an explicit demonstration of the memory kernel is required to rigorously establish the Markovian character of the coarse-grained dynamics. In the original derivation we followed the Zwanzig projection and obtained a closed-form expression, which implicitly assumes the memory term vanishes or becomes a delta function for the selected CG variables. To address the referee's point directly, we will revise §3 to include a dedicated calculation of the memory kernel, showing that it reduces to a delta function under the timescale separation inherent to our choice of CG coordinates. This addition will also clarify why gEDMD can be applied without introducing systematic bias from non-Markovian effects. revision: yes
Referee: [§5] §5 (2D numerical demonstration): the reported agreement with the full model lacks quantitative error metrics (e.g., relative error in free-energy barriers, mean first-passage times, or autocorrelation functions) and does not state the trajectory length, sampling frequency, or any data-exclusion criteria used to train gEDMD. These omissions prevent assessment of whether the claimed accuracy is robust or sensitive to post-hoc choices.

Authors: We acknowledge that the numerical section would benefit from quantitative error measures and complete reporting of the data-generation protocol. In the revised manuscript we will add relative-error values for free-energy barriers, mean first-passage times, and autocorrelation functions. We will also specify the total integrated trajectory length, sampling interval, number of independent trajectories, and any preprocessing or data-exclusion steps applied before gEDMD. These details will be placed in §5 together with a brief reproducibility subsection. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no reduction to inputs by construction

full rationale

The paper derives a closed-form CG expression via the standard Zwanzig projection applied to underdamped Langevin dynamics, then applies the external gEDMD method (from Koopman literature) to learn the generator and combines it with thermodynamic interpolation for state extension. The 2D numerical demonstration validates thermodynamic and kinetic observables without any quoted step in which a prediction is obtained by fitting a parameter to the target quantity itself or by a self-citation chain that assumes the result. The central claim therefore rests on the projection formalism plus independent data-driven approximation rather than tautological re-expression of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Zwanzig projection for underdamped Langevin and on the assumption that the learned generator from gEDMD faithfully represents the projected dynamics; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Zwanzig projection operator yields a closed Markovian dynamics for the chosen coarse variables
Invoked when deriving the closed-form CG expression from the full underdamped Langevin equation.

pith-pipeline@v0.9.0 · 5483 in / 1286 out tokens · 29001 ms · 2026-05-17T02:16:58.113398+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics... effective drift and effective diffusion... f(z,v)=P(−∇ξ(q)∇qV(q)+pi∂ijξ(q)pj)(z,v)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The effective dynamics... admits the invariant measure e^{-βF(y)}... LΞ = LΞ_a + LΞ_s

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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