Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics

TL;DR

This work introduces a mode-resolved thermodynamic framework by applying Koopman mode decomposition to the housekeeping entropy production rate in nonlinear Langevin systems. By constructing a virtual deterministic dynamics driven by the non-conservative (housekeeping) component, the authors diagonalize the dynamics in function space and express the dissipation as a sum over oscillatory Koopman modes, each contributing proportional to the square of its frequency and its intensity. The approach is demonstrated on the noisy FitzHugh–Nagumo model, revealing how bifurcations and stochastic resonance reorganize mode contributions to dissipation, including cases where many frequencies participate or a single mode dominates. The framework provides a transparent, mode-by-mode link between oscillatory structure and thermodynamic cost, with potential extensions to data-driven analyses and more complex nonlinear systems.

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.

Figures (5)

• 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 $\sigma_t^\mathrm{ex}$Dechant2022-gtyoshimura2023housekeeping. The excess part $\bm{\nu}_t^\mathrm{ex}$ means the velocity field given by the conservative force that provides the same time evolution as the original velocity field $\bm{\nu}_t$. The remainder housekeeping part $\bm{\nu}_t^\mathrm{hk}$ corresponds to the non-conservative force and does not contribute to the time evolution of $p_t(\bm{x})$. The parts of the entropy production rates associated with the respective parts of the local mean velocities are $\sigma_t^\mathrm{ex} = \langle (\bm{\nu}_t^\mathrm{hk})^\top \mathit{D}_t^{-1} \bm{\nu}_t^\mathrm{hk}\rangle_t$ and $\sigma_t^\mathrm{hk} = \langle (\bm{\nu}_t^\mathrm{hk})^\top \mathit{D}_t^{-1} \bm{\nu}_t^\mathrm{hk}\rangle_t$. We use here the noisy FitzHugh--Nagumo model to describe these schematics.
• Figure 2: (a) Koopman mode decomposition. The virtual dynamics in Eq. \ref{['eq:Langevin_hk']} are decomposed into a sum of the oscillatory modes using the Koopman mode decomposition. The Koopman generator $\mathcal{K}$ transforms a nonlinear dynamical system into the linear dynamics on a function space. Using the eigenvalues $\{\lambda_k\}_{k=1}^r$ and the eigenfunctions $\{\phi_k\}_{k=1}^r$ of the Koopman generator $\mathcal{K}$, the time variation of $\bm{x}_s$ in virtual dynamics can be expressed as a sum of modes. (b) Our main result. The housekeeping entropy production rate is decomposed into a sum of contributions from oscillatory modes. Each mode's contribution is the product of the square of its frequency and its oscillation intensity.
• Figure 3: An application example to the noisy FitzHugh--Nagumo model. (a) Examples of trajectories that follow the original Langevin process in Eq. \ref{['eq:Langevin_FN']} and the virtual deterministic process in Eq. \ref{['eq:Langevin_hk']} (b) An example of a trajectory following the virtual dynamics of the noisy FitzHugh--Nagumo model driven by the housekeeping part of the local mean velocity [Eq. \ref{['eq:Langevin_hk']}] (colored lines) and the dynamics reconstructed from the Koopman mode decomposition [Eq. \ref{['eq:oscillation_x']}] (black dotted line). (c) Koopman eigenfunctions $\phi_k(\bm{x})$ along an example trajectory. Top: The value of $\text{Re}(\phi_k(\bm{x}))$ at each point along the trajectory. The color represents the time $s$ modulo the period of the slowest oscillation $1/\chi_1$, and is consistent with the color used in the bottom panel. Bottom: The temporal evolution of $\text{Re}(\phi_k(\bm{x}))$ along the trajectory. (d) The contribution of each oscillatory mode to the housekeeping entropy production rate $\sigma_t^{\mathrm{hk}, (k)}$. Each dot represents the contribution $\sigma_t^{\mathrm{hk}, (k)}$. The vertical dashed lines at 0.05 Hz and 0.12 Hz are provided to facilitate comparison with (e). Top: Result from a single trajectory. Bottom: Results from 1000 trajectories, computed using a moving window over frequency. (e) The sum of the contributions from (d) almost equals the total value of the housekeeping entropy production rate. Left: a stacked bar plot of $\sigma_t^{\mathrm{hk}, (k)}$. The colors represent the frequencies of the oscillatory modes. The error bars indicate 95% confidence intervals of the sum of the contributions. Middle: a stacked bar plot of $\sigma_t^{\mathrm{hk}, (k)}$ under the assumption of the linear dynamics, calculated using the methods in Ref. sekizawa2024decomposing. The colors represent the frequencies of the oscillatory modes. Right: the true housekeeping entropy production rate. (f) The stacked bar plots to show our decomposition for different values of the time constant $\tau$ in Eq. \ref{['eq:Langevin_FN']}. The stacked bar plot shows the sum of the contributions from the oscillatory modes. The colors represent the frequencies of the oscillatory modes. The gray line indicates the sum of the contributions from the oscillatory modes, with error bars representing 95% confidence intervals. The black dashed line shows the true housekeeping entropy production rate. The insets represent examples of the trajectories. (g) The contribution of each oscillatory mode to the housekeeping entropy production rate $\sigma_t^{\mathrm{hk}, (k)}$ for different time constants $\tau$, computed using a moving window over frequency. The vertical dashed lines make it easier to compare peak positions; the red and orange lines respectively indicate the peaks for $\tau = 7.5$ and $\tau = 12.5$. (h) The intensities of the oscillatory modes. The vertical dashed lines also make it easier to compare peak positions; the red and orange lines respectively indicate the peaks for $\tau = 7.5$ and $\tau = 12.5$.
• Figure 4: Our decomposition enables us to understand how the entropy production rate depends on the parameter $I$ near the bifurcation point of the noisy FitzHugh–Nagumo model. (a) Examples of the trajectory of the virtual dynamics in Eq. \ref{['eq:Langevin_hk']} for the noisy FitzHugh--Nagumo model with different input values of $I$ in Eq. \ref{['eq:Langevin_hk']}. The black dashed line represents the nullclines of the noisy FitzHugh--Nagumo model, which were calculated from the original Langevin dynamics in Eq. \ref{['eq:Langevin_FN']} by ignoring the noise term. The blue and red crosses represent the unstable and stable fixed points, respectively. For $I < 1.5$, the trajectory forms a large loop around the unstable fixed point. However, near $I = 1.5$, a stable fixed point emerges, and the trajectory transitions to a smaller loop around this stable point. (b) The parameter-dependent behavior of the housekeeping entropy production rate and its decomposition. The stacked bar plot shows the sum of the contributions from the oscillatory modes. The colors represent the frequencies of the oscillatory modes. The gray line indicates the sum of the contributions from the oscillatory modes, with error bars representing 95% confidence intervals. The black dashed line shows the true housekeeping entropy production rate. As the trajectory transitions from the large loop to the small loop, the housekeeping entropy production rate $\sigma_t^\mathrm{hk}$ significantly decreases. (c) The contribution of each oscillatory mode to the housekeeping entropy production rate $\sigma_t^{\mathrm{hk}, (k)}$ for different input values of $I$. For $I<2.4$, a variety of frequencies contribute to the housekeeping entropy production rate $\sigma_t^\mathrm{hk}$. As $I$ approaches $2.4$, the contributions from frequencies undergo intermittent dropout. At $I=2.4$, almost a single frequency predominantly contributes to the housekeeping entropy production rate $\sigma_t^\mathrm{hk}$.
• Figure 5: Our decomposition enables us to determine how the entropy production rate depends on the parameter $T$ in the context of the stochastic resonance in the noisy FitzHugh–Nagumo model. (a) Examples of the trajectory of the virtual dynamics in Eq. \ref{['eq:Langevin_hk']} for the noisy FitzHugh--Nagumo model with different input values of $I$ in Eq. \ref{['eq:Langevin_hk']}. The black dashed line represents the nullclines of the noisy FitzHugh--Nagumo model, which were calculated from the original Langevin dynamics in Eq. \ref{['eq:Langevin_FN']} by ignoring the noise term. The blue and red crosses represent the unstable and stable fixed points, respectively. For $T \leq 10^{-3.5}$, the trajectories are trapped in the small loop near the two fixed points. However, for $10^{-3.5}<T$, the the trajectories transition between the two stable points, forming large loops in phase space. (b) The correlation times $\tau_\textbf{corr}$, required to detect stochastic resonance. The shaded areas represent 95% confidence intervals. (c) The parameter-dependent behavior of the housekeeping entropy production rate and its decomposition. The stacked bar plot shows the sum of the contributions from the oscillatory modes. The colors represent the frequencies of the oscillatory modes. The gray line indicates the sum of the contributions from the oscillatory modes, with error bars representing 95% confidence intervals. The black dashed line shows the true housekeeping entropy production rate. This curve exhibits an inverted U shape, which is characteristic of stochastic resonance. (d) The contribution of each oscillatory mode to the housekeeping entropy production rate $\sigma_t^{\mathrm{hk}, (k)}$ for different $T$. When $T$ is small, only one frequency mode significantly contributes to the housekeeping entropy production rate$\sigma_t^\mathrm{hk}$. As $T$ increases a broader range of frequency modes begins to contribute. As the total entropy production rate begins to decrease at higher noise intensities, contributions from oscillatory modes gradually decrease.