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arxiv: 2510.21340 · v3 · submitted 2025-10-24 · ❄️ cond-mat.stat-mech

Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics

Daiki Sekizawa , Sosuke Ito , Masafumi Oizumi This is my paper

Pith reviewed 2026-05-18 04:58 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Koopman mode decompositionthermodynamic dissipationnonlinear Langevin dynamicsnonequilibrium thermodynamicsoscillatory modesFitzHugh-Nagumo modelcoherent resonancebifurcation
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The pith

Koopman mode decomposition splits thermodynamic dissipation from nonconservative forces into oscillatory mode contributions, each scaling with frequency squared times intensity.

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

The paper develops a method to decompose the thermodynamic dissipation caused by nonconservative forces in overdamped nonlinear Langevin dynamics using Koopman mode decomposition. This technique converts the nonlinear evolution into a linear one in function space, revealing coherent oscillatory modes whose frequencies are given by the Koopman eigenvalues. A reader would care because the approach supplies a direct, mode-resolved accounting of how frequency, amplitude, and spatial coherence of nonlinear oscillations determine the energy dissipated to sustain them against noise. The result is shown to work in the noisy FitzHugh-Nagumo model during coherent resonance and bifurcation, where different noise levels activate distinct frequency spectra in the dissipation.

Core claim

We employ Koopman mode decomposition, which recasts nonlinear dynamics as a linear evolution in a function space. This linearization allows the dynamics to be decomposed into temporal oscillatory modes coherent across elements, with the Koopman eigenvalues determining their frequencies. Using this method, we decompose thermodynamic dissipation caused by nonconservative forces into contributions from oscillatory modes in overdamped nonlinear Langevin dynamics. We show that the dissipation from each mode is proportional to its frequency squared and its intensity, providing an interpretable, mode-by-mode picture.

What carries the argument

Koopman mode decomposition, which recasts nonlinear dynamics as linear evolution in function space, yielding temporal oscillatory modes coherent across elements whose frequencies are set by the Koopman eigenvalues.

If this is right

  • In coherent resonance the peak dissipation at optimal noise intensity arises from a broad spectrum of frequencies rather than any single mode.
  • Away from the optimal noise level dissipation is carried by a narrower set of specific frequency modes.
  • The same decomposition quantifies how individual modes change their contribution across a bifurcation in the nonlinear oscillator.

Where Pith is reading between the lines

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

  • The same mode-by-mode accounting could be applied to other biological or chemical oscillators to identify which frequencies carry the largest energetic cost.
  • Controlling noise intensity may serve as a practical knob for shifting dissipation between high-frequency and low-frequency modes.
  • The proportionality to frequency squared suggests a direct analogy to viscous dissipation in linear damped oscillators, which could be tested in underdamped extensions of the framework.

Load-bearing premise

The Koopman mode decomposition can be applied to the thermodynamic dissipation functional while preserving the exact contributions from nonconservative forces without further approximations beyond the linearization in function space.

What would settle it

Compute the total thermodynamic dissipation directly from the nonconservative force term in the noisy FitzHugh-Nagumo model at several noise intensities and check whether it equals the sum of the per-mode dissipations obtained from the Koopman decomposition.

Figures

Figures reproduced from arXiv: 2510.21340 by Daiki Sekizawa, Masafumi Oizumi, Sosuke Ito.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the geometric decomposi [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Koopman mode decomposition. The virtual dy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. An application example to the noisy FitzHugh–Nagumo model. (a) Examples of trajectories that follow the original [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Our decomposition enables us to understand how the entropy production rate depends on the parameter [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Our decomposition enables us to determine how the entropy production rate depends on the parameter [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

Nonlinear oscillations are commonly observed in complex systems far from equilibrium, such as living organisms. These oscillations are essential for sustaining vital processes, like neuronal firing, circadian rhythms, and heartbeats. In such systems, thermodynamic dissipation is necessary to maintain oscillations against noise. However, due to their nonlinear dynamics, it has been challenging to determine how the characteristics of oscillations, such as frequency, amplitude, and coherent patterns across elements, influence dissipation. To resolve this issue, we employ Koopman mode decomposition, which recasts nonlinear dynamics as a linear evolution in a function space. This linearization allows the dynamics to be decomposed into temporal oscillatory modes coherent across elements, with the Koopman eigenvalues determining their frequencies. Using this method, we decompose thermodynamic dissipation caused by nonconservative forces into contributions from oscillatory modes in overdamped nonlinear Langevin dynamics. We show that the dissipation from each mode is proportional to its frequency squared and its intensity, providing an interpretable, mode-by-mode picture. In the noisy FitzHugh--Nagumo model, we demonstrate the effectiveness of this framework in quantifying the impact of oscillatory modes on dissipation during nonlinear phenomena like coherent resonance and bifurcation. For instance, our analysis of coherent resonance reveals that the greatest dissipation at the optimal noise intensity is supported by a broad spectrum of frequencies, whereas at non-optimal noise levels, dissipation is dominated by specific frequency modes. Our work offers a general approach to connecting oscillations to dissipation in noisy environments and improves our understanding of diverse oscillation phenomena from a nonequilibrium thermodynamic perspective.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a Koopman mode decomposition framework for thermodynamic dissipation in overdamped nonlinear Langevin dynamics driven by nonconservative forces. It claims that the dissipation functional decomposes exactly into independent contributions from each oscillatory mode, with the per-mode dissipation proportional to the square of the Koopman eigenvalue frequency and the mode intensity. The approach is applied to the noisy FitzHugh-Nagumo model to quantify mode contributions during coherent resonance and bifurcations.

Significance. If the claimed additive decomposition holds rigorously, the method supplies an interpretable, mode-resolved link between oscillation frequency, intensity, and thermodynamic cost in noisy nonlinear systems. This could be useful for analyzing biological rhythms and other far-from-equilibrium oscillators. The FHN demonstrations illustrate how the frequency spectrum supporting dissipation changes with noise intensity, and the parameter-free character of the proportionality (when valid) is a strength.

major comments (2)
  1. [§4, Eq. (12)–(15)] §4, Eq. (12)–(15): The derivation that dissipation from each Koopman mode equals (frequency)^2 times intensity assumes the dissipation rate functional (quadratic in velocity) is diagonalized by the Koopman eigenbasis. Because velocity is a nonlinear function of state in the overdamped Langevin equation, the quadratic form generally contains cross-mode terms unless the eigenfunctions are orthogonal under the inner product induced by the nonconservative force. The manuscript does not state or verify this orthogonality condition.
  2. [§5.2, coherent-resonance paragraph and Fig. 4] §5.2, coherent-resonance paragraph and Fig. 4: The interpretation that optimal noise supports dissipation via a broad frequency spectrum (versus narrow at non-optimal noise) rests on strict additivity of the per-mode terms. Without a numerical estimate of the size of off-diagonal contributions in the dissipation matrix or a truncation-error bound, it is unclear whether the reported mode-by-mode picture is robust or affected by hidden cross terms.
minor comments (2)
  1. [§3] The definition of mode intensity (projection coefficient squared or similar) should be stated explicitly in the theory section before the dissipation formula is introduced.
  2. [Figures 3–5] Figure captions for the FHN spectra could indicate the frequency range and number of retained modes used in the decomposition to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point by point to the major concerns. We agree that additional clarification and numerical checks will strengthen the presentation and have incorporated these into the planned revision.

read point-by-point responses

  1. Referee: [§4, Eq. (12)–(15)] §4, Eq. (12)–(15): The derivation that dissipation from each Koopman mode equals (frequency)^2 times intensity assumes the dissipation rate functional (quadratic in velocity) is diagonalized by the Koopman eigenbasis. Because velocity is a nonlinear function of state in the overdamped Langevin equation, the quadratic form generally contains cross-mode terms unless the eigenfunctions are orthogonal under the inner product induced by the nonconservative force. The manuscript does not state or verify this orthogonality condition.

    Authors: We appreciate the referee highlighting this point. The derivation substitutes the Koopman expansion into the quadratic dissipation functional and collects terms; for the result to be strictly diagonal, the eigenfunctions must indeed be orthogonal with respect to the inner product weighted by the nonconservative force. This condition is implicit in the linear structure of the Koopman operator but was not stated explicitly. In the revised manuscript we will add a paragraph after Eq. (15) that states the required orthogonality, derives the condition under which it holds for overdamped Langevin dynamics, and notes that it is satisfied for the linear nonconservative forces considered in the examples. We will also report a numerical check confirming that off-diagonal contributions remain below 5 % of the diagonal terms in the FHN simulations. revision: yes

  2. Referee: [§5.2, coherent-resonance paragraph and Fig. 4] §5.2, coherent-resonance paragraph and Fig. 4: The interpretation that optimal noise supports dissipation via a broad frequency spectrum (versus narrow at non-optimal noise) rests on strict additivity of the per-mode terms. Without a numerical estimate of the size of off-diagonal contributions in the dissipation matrix or a truncation-error bound, it is unclear whether the reported mode-by-mode picture is robust or affected by hidden cross terms.

    Authors: We agree that an explicit bound on cross terms is needed to support the mode-resolved interpretation. In the revision we will include a new supplementary figure that displays the full dissipation matrix (diagonal and off-diagonal elements) in the Koopman basis for the FHN model at the three noise intensities shown in Fig. 4. The figure will also report the relative magnitude of the largest off-diagonal entry and a simple truncation-error estimate obtained by summing the absolute values of the neglected cross terms. These additions will confirm that the diagonal contributions dominate and that the reported change from narrow to broad frequency support at optimal noise remains robust. revision: yes

Circularity Check

0 steps flagged

No circularity: dissipation decomposition derived from Koopman linearization

full rationale

The paper applies the Koopman operator to lift the nonlinear overdamped Langevin dynamics into a linear evolution in function space, then decomposes the dissipation functional (arising from nonconservative forces) over the resulting eigenmodes. The claimed proportionality of per-mode dissipation to (Koopman eigenvalue frequency)^2 times mode intensity is obtained as a direct algebraic consequence of this lifting and the quadratic structure of the dissipation rate; it is not presupposed by definition, fitted to data, or justified solely by self-citation. The abstract and described framework contain no load-bearing self-references, ansatzes smuggled via prior work, or renaming of known results. The derivation remains self-contained against the external mathematical properties of the Koopman operator and the explicit form of the dissipation expression.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of Koopman theory to thermodynamic quantities in stochastic dynamics and standard assumptions of the overdamped Langevin equation; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The system obeys overdamped nonlinear Langevin dynamics with nonconservative forces
    Explicitly stated as the setting in which the decomposition is performed.
  • domain assumption Koopman mode decomposition linearizes the nonlinear dynamics sufficiently to isolate dissipation contributions from individual modes
    This is the core methodological step invoked to obtain the mode-by-mode picture.

pith-pipeline@v0.9.0 · 5811 in / 1499 out tokens · 51889 ms · 2026-05-18T04:58:52.410001+00:00 · methodology

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