Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics (2510.21340v1)

Published 24 Oct 2025 in cond-mat.stat-mech

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 stochastic resonance and bifurcation. For instance, our analysis of stochastic 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.

Summary

The paper introduces a Koopman mode decomposition framework, establishing a method to quantify the contribution of oscillatory modes to thermodynamic entropy production in nonlinear systems.
It employs a geometric decomposition of entropy production into housekeeping and excess components, linking modal frequencies and intensities to dissipation rates.
The approach is validated on the noisy FitzHugh–Nagumo model, demonstrating improved reconstruction of dissipation and insights into bifurcation and stochastic resonance.

Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics

Introduction

The paper presents a rigorous framework for decomposing thermodynamic dissipation in nonlinear Langevin systems using Koopman mode decomposition. Oscillatory phenomena in nonequilibrium systems—such as neural firing, circadian rhythms, and chemical oscillations—require sustained energy dissipation to counteract noise. Quantifying how oscillatory features (frequency, amplitude, coherence) contribute to entropy production has been challenging, especially in nonlinear regimes. The authors address this by leveraging the Koopman operator formalism, which linearizes nonlinear dynamics in function space, enabling a modal decomposition of dissipation.

Geometric Decomposition of Entropy Production

The entropy production rate $\sigma_t$ in overdamped Langevin dynamics is defined as the $L^2$ norm of the local mean velocity field $\bm{\nu}_t$ under the metric $D_t^{-1}p_t(\bm{x})$ . The geometric decomposition splits $\sigma_t$ into housekeeping ( $\sigma_t^\mathrm{hk}$ ) and excess ( $\sigma_t^\mathrm{ex}$ ) components, where the former quantifies dissipation due to nonconservative forces and the latter arises from conservative dynamics. The housekeeping velocity field $\bm{\nu}_t^\mathrm{hk}$ is orthogonal to the excess part and does not contribute to the evolution of the probability density.

Figure 1: Schematic illustration of the geometric decomposition of the entropy production rate $\sigma_t$ into the housekeeping part $\sigma_t^\mathrm{hk}$ and the excess part $\sigma_t^\mathrm{ex}$ .

Koopman Mode Decomposition: Theory and Main Result

The central technical advance is the application of Koopman mode decomposition to the virtual deterministic dynamics driven by $\bm{\nu}_t^\mathrm{hk}$ :

$d\bm{x}_s = \bm{\nu}_t^\mathrm{hk}(\bm{x}_s) ds$

The Koopman generator $\mathcal{K}$ acts on observables $g(\bm{x})$ as $\mathcal{K}g = \nabla g^\top \bm{\nu}_t^\mathrm{hk}$ , yielding a linear evolution in function space. The eigenfunctions $\phi_k$ and eigenvalues $\lambda_k$ of $\mathcal{K}$ allow the state to be expanded as a sum of oscillatory modes:

$\bm{x}_{s+\Delta s} = \sum_k e^{\lambda_k \Delta s} \phi_k(\bm{x}_s) \bm{v}_k$

where $\bm{v}_k$ are Koopman modes and $\chi_k = \lambda_k/(2\pi i)$ is the frequency.

The main result is a modal decomposition of the housekeeping entropy production rate:

$\sigma_t^\mathrm{hk} = \sum_k (2\pi)^2 \chi_k^2 J_k$

where $J_k = \langle (\phi_k\bm{v}_k)^* D_t^{-1} (\phi_k\bm{v}_k) \rangle_t$ is the intensity of the $k$ -th mode. Thus, each mode's contribution to dissipation is proportional to its frequency squared and intensity.

Figure 2: (a) Koopman mode decomposition. The virtual dynamics are decomposed into oscillatory modes; (b) Each mode's contribution to dissipation is the product of frequency squared and intensity.

Numerical Implementation and Application: Noisy FitzHugh–Nagumo Model

The framework is applied to the noisy FitzHugh–Nagumo model, a canonical excitable system. The authors compute the steady-state housekeeping velocity field via discretization and solve the virtual dynamics using high-order Runge–Kutta integration. Koopman modes and eigenfunctions are extracted from simulated trajectories using Hankel DMD and physics-informed DMD (PiDMD), ensuring the generator is skew-adjoint and eigenvalues are purely imaginary.

Figure 3: Application to the noisy FitzHugh–Nagumo model. (a) Trajectories of the original and virtual dynamics; (b) Reconstruction via Koopman modes; (c) Oscillatory eigenfunctions; (d,e) Modal contributions to dissipation.

The decomposition accurately reconstructs the total housekeeping entropy production rate, outperforming previous linear-only approaches. Varying the time constant $\tau$ shifts the dominant frequencies, and the reduction in dissipation is attributed to lower oscillation frequencies rather than amplitude changes.

Analysis of Nonlinear Phenomena: Bifurcation and Stochastic Resonance

Bifurcation

Near bifurcation points, the modal decomposition reveals that a broad spectrum of frequencies contributes to dissipation, but as the system transitions to a stable fixed point, modal contributions drop out intermittently, and eventually a single mode dominates.

Figure 4: Decomposition across bifurcation regimes. (a) Trajectories for varying input $I$ ; (b) Modal contributions to dissipation; (c) Frequency-resolved dropout near bifurcation.

Stochastic Resonance

In stochastic resonance, the total dissipation exhibits an inverted-U dependence on noise intensity. The modal decomposition shows that at optimal noise, dissipation is distributed across a broad frequency spectrum, while at suboptimal noise, a single mode dominates.

Figure 5: Decomposition in stochastic resonance. (a) Trajectories for varying noise $T$ ; (b) Correlation times; (c) Modal contributions to dissipation; (d) Frequency-resolved spectrum.

Theoretical and Practical Implications

The modal decomposition provides a quantitative link between oscillatory dynamics and thermodynamic cost in nonlinear, noisy systems. It generalizes previous results for linear systems and is distinct from approaches based on Markov rate matrices or cycle decompositions. The framework is robust to finite-dimensional approximations due to the skew-adjointness of the Koopman generator in the virtual dynamics. However, in systems with continuous spectra (e.g., chaos), the finite-mode approximation may break down, suggesting future work should incorporate DMD extensions for continuous spectra.

Practically, the approach enables frequency-resolved analysis of dissipation in experimental data, contingent on reliable estimation of the housekeeping velocity field and Koopman modes. Advances in data-driven Koopman analysis and optimal transport-based velocity estimation are promising for real-world applications.

Conclusion

The paper establishes a rigorous, interpretable framework for decomposing thermodynamic dissipation in nonlinear Langevin systems via Koopman mode analysis. Each oscillatory mode's contribution to dissipation is quantified by its frequency squared and intensity, enabling detailed analysis of complex nonlinear phenomena such as bifurcation and stochastic resonance. The approach provides a foundation for future studies of thermodynamic efficiency, design principles, and functional organization in biological and engineered oscillatory systems.

PDF Markdown

Paper Prompts

Top Community Prompts

Explain it Like I'm 14

off on

Knowledge Gaps

off on

Practical Applications

off on

Glossary

off on

Conceptual Simplification

off on

Explain it Like I'm 14

Overview

This paper looks at how “oscillations” (regular repeating patterns like heartbeats, brain waves, or daily sleep cycles) spend energy in noisy environments. The authors focus on a special kind of energy use called the housekeeping entropy production rate. They build a new way to break down complex, wiggly motion into simpler parts, so we can see which rhythms cost the most energy and why.

Key questions

The paper asks simple but important questions:

When a system shows rhythmic behavior, which features of the rhythm—like how fast it repeats (frequency) and how big the swings are (amplitude)—make it use more energy?
Can we separate the total energy cost into clear pieces tied to each rhythm, even if the system is nonlinear (not a simple straight line relationship) and noisy?
How does this work in real examples, like a simple model of nerve cells that can switch between quiet and active states?

How did they paper it? Methods explained simply

To tackle messy, nonlinear, and noisy motion, the authors use three main ideas:

1) Noisy motion (Langevin dynamics)

Think of a tiny bead in water. It’s pushed by forces (like a toy motor) and jiggled by random kicks from water molecules (noise). This kind of motion is called Langevin dynamics. The paper studies the “overdamped” case, meaning the bead doesn’t coast—it moves only as forces and noise push it.

2) Housekeeping entropy production (energy cost of nonconservative forces)

Entropy production rate is a measure of how much energy a system spends to keep going. The “housekeeping” part is the portion of that energy cost caused by nonconservative forces—the push that doesn’t come from a simple downhill potential and is needed to keep cycles and flows going in steady state. You can think of it as the energy needed to maintain rhythmic activity against noise.

3) Koopman mode decomposition (turning messy motion into clean waves)

This is the key trick. Koopman decomposition takes a nonlinear motion and represents it as a sum of simple, clean oscillations (like combining musical notes to make a song). Each “note” is a mode with:

a frequency (how fast it cycles), and
an intensity (how strong that mode is in the motion).

To make this decomposition work, they consider a “virtual dynamics”—a shadow system that moves according to the housekeeping part of the motion. This virtual system has the same overall probability distribution as the original noisy system, but it’s easier to analyze because it’s deterministic (no random kicks appear explicitly). Then, they use data-driven tools (Dynamic Mode Decomposition, or DMD) to find the modes and their frequencies from time series.

Main findings

1) A clean formula: energy cost per rhythm

They prove that the housekeeping entropy production rate splits neatly into positive pieces from each oscillatory mode. For each mode k, the contribution looks like:

$\sigma^{\mathrm{hk},(k)} = (2\pi)^2 \,\chi_k^2 \, J_k,$

where:

$\chi_k$ is the mode’s frequency, and
$J_k$ is the mode’s intensity (how “big” or “strong” that oscillation is).

In everyday language: faster rhythms cost more—quadratically more—and stronger rhythms cost more. The total cost is the sum over all modes.

2) It works in real nonlinear examples

They apply this to the noisy FitzHugh–Nagumo model, a classic simple model of a neuron. They paper two nonlinear phenomena:

Bifurcation: When changing a parameter, the system’s behavior shifts from large loops (big oscillations) to small loops (tiny oscillations near a stable point). Their decomposition shows that, as the system approaches this shift, energy contributions from many frequencies drop out intermittently—eventually one main frequency dominates near the stable point.
Stochastic resonance: This is when noise surprisingly helps produce more regular behavior. As they increase noise:
- At low noise, one slow mode dominates (the system stays near one stable state).
- At optimal noise, many frequencies contribute strongly—energy cost peaks and rhythms are most organized.
- At high noise, contributions fade—energy cost drops and behavior gets disordered.

3) Why this is better than linear methods

If you pretend the system is linear, you miss key nonlinear features. The authors show that their nonlinear decomposition matches the true energy cost, while a linear-only method fails in these nonlinear cases.

Why this matters

This work offers a new, understandable “mode-by-mode” picture of how complex rhythms spend energy in noisy settings. That’s important because many living systems—brains, hearts, biological clocks—use oscillations to function, and energy is limited. With this approach:

Engineers and scientists can see which rhythms are expensive and why.
It’s possible to design or tune systems to be more energy-efficient by managing their dominant modes.
The method connects the shape of oscillations (frequency and intensity) to thermodynamic cost in a clear way.

In short, the paper gives a powerful lens: think of complex oscillations as a set of clean waves, and measure how each wave contributes to the energy bill.

View Paper Prompt View All Prompts

Knowledge Gaps

Below is a concise list of knowledge gaps, limitations, and open questions that remain unresolved and could guide future research.

Mathematical guarantees for the finite-dimensional approximation: provide conditions under which the Koopman generator of the ν^hk-driven virtual dynamics is diagonalizable and admits a discrete spectrum, and derive convergence/error bounds when truncating to r modes.
Applicability to systems with continuous spectra: extend the decomposition to chaotic, quasi-periodic, or mixing dynamics where the Koopman operator has continuous or mixed spectra (e.g., via spectral measures instead of discrete sums).
Under-damped and odd-parity variables: generalize the framework to inertial Langevin dynamics with momentum (odd parity) and clarify how parity affects ν^hk, skew-adjointness, and the modal dissipation formula.
Time-dependent (non-steady-state) regimes: develop a decomposition for transient entropy production when p_t is time-varying (including driven or nonautonomous systems), with time-dependent Koopman operators and mode tracking.
Multiplicative or state-dependent noise: analyze how D(x,t) (non-constant diffusion) affects the construction of ν^hk, skew-adjointness, eigenvalue properties, and the mode-wise dissipation formula.
Robust computation of ν^hk and U(x): design scalable algorithms to solve for the potential U and ν^hk in high dimensions, with guarantees on accuracy, stability, and computational complexity.
Density estimation at scale: establish sample complexity and robust, high-dimensional estimators for p_t(x) needed to compute ν^hk, including strategies that mitigate bias/variance and boundary effects.
Error quantification for DMD-based Koopman learning: provide statistical and numerical error bars for eigenvalues, eigenfunctions, and modes; characterize sensitivity to sampling rate, trajectory length, noise, and hyperparameters.
Mode selection and degeneracy handling: develop principled criteria for selecting r, resolving near-degenerate eigenvalues/mode mixing, and testing the significance of identified modes (including cross-validation and bootstrapping).
Uniqueness and invariance of the decomposition: clarify how the modal contributions depend on the choice of observables (e.g., identity function), coordinates, and metric; characterize invariance under reparameterizations or linear transformations.
Mapping virtual to physical time: formalize the relation between frequencies in virtual time s and physically observed time t, and determine when and how modal contributions translate to experimentally measured spectra.
Excess entropy production: extend the Koopman mode decomposition to the excess part σ_t^ex and characterize whether similar frequency–intensity factorization holds.
Bounds and thermodynamic constraints: connect modal contributions to thermodynamic uncertainty relations or coherence–dissipation trade-offs; derive bounds on per-mode dissipation given spectral or amplitude constraints.
Partial observability and measurement noise: devise methods to infer ν^hk and Koopman modes from incomplete or noisy observations (e.g., delay embeddings, latent-state reconstruction), with guarantees on identifiability.
Data-driven inference without force models: establish procedures to recover ν^hk and modal dissipation solely from time-series data when the force field and diffusion are unknown, including practical requirements for experimental datasets.
Scalability to high-dimensional, networked, or spatially extended systems: test and optimize the framework on large systems to assess computational feasibility and interpretability of mode-wise dissipation across many elements.
Analytical characterization near bifurcations: link modal spectra and contributions to normal-form theory and stochastic bifurcation analysis, predicting when and how “dropouts” or mode reorganizations occur.
Validation across diverse nonlinear phenomena: assess generality by applying the method to limit cycles, relaxation oscillators, multistable systems, reaction–diffusion/Turing patterns, and systems exhibiting canard dynamics.
Treatment of boundary conditions and non-compact state spaces: specify conditions ensuring skew-adjointness and purely imaginary spectra when state spaces are unbounded or have complex boundaries.
Physical interpretability of mode intensities J_k: test whether J_k correlates with measurable amplitudes/energies and establish experimental protocols to validate the frequency-squared scaling of dissipation.
Coarse-graining and relation to cycle decompositions: explore how modal dissipation relates to network cycle decompositions and whether modes can be mapped to physical cycles or mesoscopic mechanisms.
Control and design: formulate optimization problems to redistribute dissipation across modes (e.g., lowering high-frequency costs) and explore control strategies that shape modal spectra under constraints.
Generalization to discrete-time and non-Markovian dynamics: adapt the decomposition to maps and memoryful processes, clarifying how ν^hk and the modal framework change.
Rigorous comparison with linear approaches: quantify when the nonlinear Koopman-based decomposition materially deviates from linear eigenmode analyses and identify regimes where linearization suffices.
Empirical reproducibility: define standardized pipelines (estimation of p_t, ν^hk, DMD setup, uncertainty quantification) and benchmark datasets to ensure reproducible modal dissipation analyses.

View Paper Prompt View All Prompts

Practical Applications

Immediate Applications

Below are concrete ways the paper’s Koopman-based decomposition of housekeeping entropy production (EPR) can be used now, given current data-driven tooling (e.g., DMD/EDMD, PyDMD), typical lab- and industry-grade time-series, and steady-state or near–steady-state operating regimes:

Frequency-resolved thermodynamic auditing of oscillators (academia, synthetic biology, chemical engineering, soft matter, MEMS/NEMS)
- What: Quantify the thermodynamic cost of each oscillatory mode in experiments (chemical oscillators, gene circuits, micromechanical resonators, active-matter flows).
- Tools/workflow: Data acquisition → estimate diffusion/noise matrix and steady-state density → compute housekeeping velocity field via geometric decomposition → virtual-dynamics simulation or direct Koopman operator learning → DMD/EDMD to get modes → compute modal intensities J_k and frequencies χ_k → produce “thermodynamic spectrograms.”
- Dependencies/assumptions: Overdamped Langevin approximation; even-parity states; adequate sampling to estimate p(x) and D; finite-mode Koopman approximation; approximately stationary statistics.
Early-warning indicators for regime shifts via modal dissipation signatures (manufacturing/process control, power electronics, labs; academia)
- What: Detect impending bifurcations or qualitative regime changes by monitoring redistribution/dropout/broadening of modal contributions to EPR (as demonstrated near Hopf-like transitions).
- Tools: Real-time sliding-window DMD + modal-EPR dashboard; alerting on shifts in σ_t^{hk,(k)} spectra.
- Dependencies: Sufficient sampling rate and windowing; robust density/current estimation; stationarity over the window.
Noise tuning for stochastic resonance to optimize performance-cost trade-offs (sensors, neuromorphic devices, biochemical assays)
- What: Identify noise levels that maximize functional coherence while understanding which frequency bands carry the dissipation burden.
- Tools: Controlled noise injection/dither + modal EPR readout; “noise advisor” routine to pick operating point.
- Dependencies: Ability to modulate noise; measurable state; steady-state or cyclo-stationary operation.
Mode-targeted control to reduce energy cost or improve coherence (robotics/CPGs, haptics, precision mechatronics, oscillatory process units)
- What: Use mode-by-mode cost maps (σ_t^{hk,(k)} ∝ χ_k² J_k) to damp high-cost modes or amplify low-cost functional modes.
- Tools: Modal feedback/filters, selective actuation or parameter tuning informed by Koopman modal analysis.
- Dependencies: Actuation authority over modal content; partial observability handled via time-delay embeddings or EDMD.
Neuroscience time-series analysis with thermodynamic markers (academia; translational research)
- What: Relate EEG/LFP/MEA band activity to mode-specific dissipation to paper excitability, seizures, and resonance-like phenomena.
- Tools: Preprocessing + EDMD on neural state embeddings; compute frequency-resolved σ_t^{hk,(k)} during tasks or pathology.
- Dependencies: Approximate stationarity over analysis windows; surrogate-state embeddings when full state unobserved.
Quality assurance and comparative benchmarking of oscillator designs (R&D; synthetic biology, chemical oscillators, MEMS/NEMS)
- What: Compare competing designs by their modal dissipation profile at target frequencies/amplitudes.
- Tools: Standardized “oscillation cost maps”; batch pipelines for datasets.
- Dependencies: Consistent measurement protocols; known or well-estimated noise covariance.
Software/tooling deliverables (academia, industry R&D)
- What: Open-source Python toolkit that unifies housekeeping-velocity estimation, virtual-dynamics KMD, and modal EPR computation; “Thermodynamic Spectrogram” visualization.
- Tools: PyDMD/EDMD integration; density/score estimation (KDE, diffusion maps, score matching).
- Dependencies: Documentation and validated examples; dataset-specific hyperparameter tuning.

Long-Term Applications

The following opportunities are feasible with further theoretical extensions (e.g., underdamped/odd-parity variables), scalable estimation in higher dimensions, and domain validation:

Clinical thermodynamic biomarkers and closed-loop therapies (healthcare)
- What: Use mode-resolved dissipation in ECG/EEG to detect arrhythmia/seizure risk earlier and drive closed-loop stimulation that shifts modal costs to safer regimes.
- Potential products: ICU monitors with “modal dissipation” channels; neuromodulation controllers that modulate noise/stimulus.
- Dependencies: Validation on partial/latent physiological states; extension to underdamped dynamics; regulatory approval.
Energy-aware design principles for biological and synthetic oscillators (synthetic biology, chronotherapy)
- What: Optimize gene circuits/circadian interventions for required function at minimal σ_t^{hk}, allocating cost across modes intentionally.
- Potential workflows: Design–build–test cycles guided by modal EPR constraints; in silico screening with KMD-in-the-loop.
- Dependencies: Robust estimation under cell-to-cell variability; integration with CRN/SSA simulators; standardization of thermodynamic metrics.
Grid and infrastructure oscillation governance (energy, utilities)
- What: Frequency-resolved dissipation diagnostics for inter-area oscillations and damping controller design in power systems.
- Potential tools: Wide-area “thermodynamic oscillation monitors” for control rooms.
- Dependencies: Extension to underdamped, hybrid, and constraint-rich dynamics; multi-source telemetry fusion.
Energy-efficient locomotion and manipulation via modal cost shaping (robotics)
- What: Tune CPGs and feedback policies to minimize high-χ_k² J_k modes while preserving task performance.
- Potential products: Controller add-ons for legged robots and exoskeletons prioritizing “thermodynamic frugality.”
- Dependencies: Real-time high-dimensional Koopman/operator learning; partial observability; hardware-in-the-loop validation.
Process-intensified oscillatory reactors with cost-aware control (chemical industry)
- What: Run oscillatory reactors at resonance-like operating points with favorable cost/throughput ratios, informed by modal EPR spectra.
- Potential tools: Advanced control modules with noise/forcing co-design.
- Dependencies: Plant-scale identification; safe noise/forcing modulation; integration with APC/MPC stacks.
Active matter and swarm materials with minimal dissipation patterns (materials, soft robotics)
- What: Engineer collective oscillations with targeted modal cost distribution for function (transport, mixing, morphing).
- Potential tools: Agent-level parameter synthesis guided by macro-scale modal dissipation goals.
- Dependencies: High-dimensional Koopman models; handling continuous spectra/chaos; experimental tractability.
Finance and macroeconomics regime-shift indicators (finance/policy analytics)
- What: Use modal dissipation analogs as features for detecting market/macro regime transitions.
- Potential tools: Time-series analytics packages with “thermodynamic features.”
- Dependencies: Justifiable stochastic-Langevin embeddings; careful interpretation (thermodynamic meaning is metaphorical here).
Standards and policy for energy/dissipation reporting in bio- and nano-devices (policy, industry consortia)
- What: Develop reporting guidelines around steady-state σ_t and modal allocation for oscillatory technologies.
- Potential tools: Benchmarks, certification schemes, and regulatory frameworks for “thermodynamic efficiency” claims.
- Dependencies: Community consensus on estimators and protocols; cross-domain validation.

Cross-cutting assumptions and dependencies to watch

Modeling scope: Overdamped Langevin with even-parity states; steady state implies σ_t = σ_t^hk. Underdamped/odd-parity extensions remain open.
Estimation challenges: Accurate density/score and diffusion estimation in higher dimensions; sensitivity to window lengths and sampling; robustness to measurement noise and latent states.
Koopman approximation: Finite-mode diagonalizable approximation is assumed; continuous spectra/chaotic dynamics may require alternative spectral treatments.
Computation: Real-time deployment requires efficient EDMD/neural-Koopman implementations and edge inference strategies.
Data and actuation: Sufficient observability and the ability to modulate parameters/noise to act on identified modes.

View Paper Prompt View All Prompts

Glossary

Bifurcation: A qualitative change in system dynamics as a parameter varies, often creating or destroying oscillations or altering their character. "undergoing bifurcation and stochastic resonance."
Brownian motion: Random motion of particles modeled as a continuous-time stochastic process; in SDEs, it drives the noise term. "The term $d\boldsymbol{B}_t$ denotes a standard $d$ -dimensional Brownian motion"
Correlation time: A measure of temporal coherence indicating how long a signal remains correlated with itself. "The correlation times $\tau_corr$ , required to detect stochastic resonance."
Cycle decomposition: A representation of nonequilibrium steady-state dissipation as sums over cyclic flows in a network. "this decomposition is similar to the cycle decomposition of housekeeping entropy production rates"
Detailed balance condition: A balance between forward and reverse transitions that characterizes equilibrium; its violation indicates nonequilibrium. "violate the detailed balance condition"
Diagonalizable: A property of a linear operator whereby it can be expressed in a basis of eigenvectors, simplifying analysis. "and is therefore diagonalizable under this finite-dimensional approximation."
Diffusion-induced force: An effective drift arising from spatial variations in probability density, given by $-D_t\nabla \ln p_t(\bm{x})$ . "the diffusion-induced force $-D_t\nabla \ln p_t(\bm{x})$ "
Dynamic Mode Decomposition (DMD): A data-driven method to approximate Koopman spectra and modes from time-series data. "using dynamic mode decomposition (DMD)"
Eigenfunction: A function that is scaled (but not altered in form) by an operator, satisfying $\mathcal{K}\phi=\lambda\phi$ . "the eigenfunctions $\{\phi_k\}_{k=1}^r$ of the Koopman generator $\mathcal{K}$ "
Eigenvalue: The scalar associated with an eigenfunction indicating its growth/oscillation rate under the operator. "the eigenvalues $\{\lambda_k\}_{k=1}^r$ "
Entropy production rate: A nonnegative measure of irreversibility/dissipation in nonequilibrium systems, defined via a norm of the local mean velocity. "The entropy production rate $\sigma_t$ is defined as the following $L^2$ norm"
Excess entropy production rate: The component of entropy production linked to conservative dynamics, complementing the housekeeping part. "the excess entropy production rate $\sigma\mathrm{ex}_t := \sigma_t - \sigma\mathrm{hk}_t"</li> <li>FitzHugh–Nagumo model: A canonical two-variable model of neuronal excitability exhibiting nonlinear oscillations. "the noisy FitzHughâNagumo model"</li> <li>Fokker–Planck equation: A PDE describing the time evolution of the probability density under drift and diffusion. "the following Fokker--Planck equation"</li> <li>Geometric decomposition: A splitting of entropy production into orthogonal housekeeping and excess parts via a generalized Pythagorean relation. "called the geometric decomposition"</li> <li>Hatano–Sasa decomposition: A framework separating housekeeping and excess contributions in nonequilibrium thermodynamics. "the housekeeping-excess decomposition by Hatano and Sasa"</li> <li>Hopf bifurcation: A transition where a fixed point loses stability and a limit cycle emerges. "this transition corresponds to a Hopf bifurcation"</li> <li>Housekeeping entropy production rate: The dissipation caused solely by nonconservative forces, independent of changes in probability density. "housekeeping entropy production rate $\sigma_t^\mathrm{hk}$"</li> <li>Invariant measure: A probability distribution that remains unchanged under the dynamics of a system. "becomes the invariant measure of the virtual dynamics."</li> <li>Koopman generator: A linear operator governing the evolution of observables in function space for a nonlinear system. "known as the Koopman generator"</li> <li>Koopman mode: The vector coefficient in the expansion of the identity observable associated with a Koopman eigenfunction. "with the weight $\bm{v}_k$ called Koopman mode"</li> <li>Koopman mode decomposition: A representation of nonlinear dynamics as a sum of oscillatory modes via the Koopman operator’s spectral properties. "we employ Koopman mode decomposition"</li> <li>Langevin equation: A stochastic differential equation describing dynamics under deterministic forces and Gaussian noise. "We consider the overdamped multidimensional Langevin equation"</li> <li>Limit cycle: A closed, isolated periodic orbit that attracts nearby trajectories in nonlinear systems. "such as limit cycles, bifurcations"</li> <li>Local mean velocity: The effective drift field in the Fokker–Planck description capturing average motion in state space. "The velocity field $\bm{\nu}_t(\bm{x})$ is called the local mean velocity."</li> <li>$L^2$ norm: The integral of the squared magnitude of a function, used here to quantify entropy production. "the following $L^2$ norm of the local mean velocity"</li> <li>Nonconservative forces: Forces not derivable from a potential, responsible for persistent currents and dissipation. "caused by nonconservative forces"</li> <li>Nonequilibrium thermodynamics: The paper of thermodynamic behavior in systems away from equilibrium. "from a nonequilibrium thermodynamic perspective."</li> <li>Nullclines: Curves in phase space where the time derivative of a state variable is zero. "The black dashed line represents the nullclines"</li> <li>Overdamped: A regime where inertial effects are negligible and motion is dominated by friction and noise. "overdamped nonlinear Langevin dynamics."</li> <li>Parity (even parity): A symmetry property under time reversal; even-parity variables remain unchanged. "have even parity"</li> <li>Schnakenberg's network theory: A graph-based theory for analyzing currents and entropy production in Markov networks. "based on Schnakenberg's network theory"</li> <li>Skew-adjoint: An operator $K $satisfying$ K^*=-K$, implying purely imaginary eigenvalues in suitable settings. "the Koopman generator $\mathcal{K}$ is skew-adjoint"</li> <li>Stochastic resonance: A phenomenon where noise enhances the system’s response or coherence, often yielding optimal performance at intermediate noise. "stochastic resonance"</li> <li>Stochastic thermodynamics: A framework quantifying thermodynamic quantities for stochastic processes and small systems. "Stochastic thermodynamics $|$ Langevin equation"
Steady state: A condition where the probability distribution does not change over time. "A system is in a steady state"
Virtual dynamics: An auxiliary deterministic system constructed to analyze housekeeping contributions. "the virtual dynamics in Eq.~\ref{eq:Langevin_hk}"
Wiener process: A continuous-time stochastic process with independent Gaussian increments; the mathematical model of Brownian motion. "which is a Wiener process"

View Paper Prompt View All Prompts

Open Problems

Validity of Koopman mode decomposition under finite-dimensional approximation for continuous-spectrum systems

Continue Learning

Authors (3)

Collections

Tweets

This paper has been mentioned in 2 tweets and received 351 likes.

Upgrade to Pro to view all of the tweets about this paper:

Start a free 7-day Pro trial

Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics (2510.21340v1)

Summary

Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics

Introduction

Geometric Decomposition of Entropy Production

Koopman Mode Decomposition: Theory and Main Result

Numerical Implementation and Application: Noisy FitzHugh–Nagumo Model

Analysis of Nonlinear Phenomena: Bifurcation and Stochastic Resonance

Bifurcation

Stochastic Resonance

Theoretical and Practical Implications

Conclusion

Paper Prompts

Top Community Prompts

Explain it Like I'm 14

Overview

Key questions

How did they paper it? Methods explained simply

1) Noisy motion (Langevin dynamics)

2) Housekeeping entropy production (energy cost of nonconservative forces)

3) Koopman mode decomposition (turning messy motion into clean waves)

Main findings

1) A clean formula: energy cost per rhythm

2) It works in real nonlinear examples

3) Why this is better than linear methods

Why this matters

Knowledge Gaps

Practical Applications

Immediate Applications

Long-Term Applications

Cross-cutting assumptions and dependencies to watch

Glossary

Open Problems

Continue Learning

Related Papers

Authors (3)

Collections

Tweets