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arxiv: 2509.06421 · v3 · submitted 2025-09-08 · ❄️ cond-mat.stat-mech

Duality between dissipation-coherence trade-off and thermodynamic speed limit based on thermodynamic uncertainty relation for stochastic limit cycles

Ryuna Nagayama , Sosuke Ito This is my paper

Pith reviewed 2026-05-18 18:40 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords stochastic limit cyclesthermodynamic uncertainty relationdissipation-coherence trade-offthermodynamic speed limitweak-noise limitentropy productionphase reductionnoisy oscillators
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The pith

A duality derived from the thermodynamic uncertainty relation shows the dissipation-coherence trade-off is the dual of the thermodynamic speed limit for stochastic limit cycles.

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

The paper derives two trade-offs that apply to any stochastic limit cycle in the weak-noise limit. One bounds the entropy production per oscillation period by the number of cycles before correlations decay, while the other bounds entropy production by the geometric length of the cycle trajectory. Both bounds follow from inserting a pair of observables that are dual to each other, chosen from the linear stability analysis of the cycle, into the thermodynamic uncertainty relation. This substitution makes the two trade-offs dual versions of the same underlying inequality. The result unifies separate constraints on the cost and timing of noisy periodic motion and extends to chemical reaction networks even when some diffusion eigenvalues vanish.

Core claim

For general stochastic limit cycles in the weak-noise limit, the thermodynamic uncertainty relation applied to mutually dual observables constructed from the cycle's linear stability analysis produces both the dissipation-coherence trade-off (bounding entropy production by the duration of steady-state correlations) and the thermodynamic speed limit (bounding entropy production by the Euclidean length of the limit cycle), thereby establishing that the dissipation-coherence trade-off is the dual of the thermodynamic speed limit.

What carries the argument

Substitution of mutually dual observables derived from the linear stability analysis of the limit cycle into the thermodynamic uncertainty relation.

If this is right

  • The dissipation-coherence trade-off holds for general stochastic limit cycles without needing the extra assumptions used in prior work.
  • The thermodynamic speed limit constrains entropy production directly by the Euclidean length of the limit-cycle orbit.
  • Both trade-offs remain usable for stochastic chemical systems even when the diffusion matrix has zero eigenvalues.
  • The dissipation-coherence bound can be saturated exactly by suitably modifying the diffusion matrix via phase reduction.

Where Pith is reading between the lines

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

  • The same duality construction may extend to other nonequilibrium periodic systems where a thermodynamic uncertainty relation can be formulated.
  • Numerical checks on the noisy Rössler attractor already illustrate the bounds; similar tests could be performed on biological oscillators such as circadian clocks.
  • If the duality is robust, optimizing entropy production for one observable would automatically tighten the bound for its dual without separate calculation.

Load-bearing premise

The thermodynamic uncertainty relation remains valid when applied to the pair of mutually dual observables obtained from the linear stability analysis in the weak-noise limit.

What would settle it

A concrete counterexample would be a stochastic limit cycle in the weak-noise regime where measured entropy production per period falls below the lower bound set by either the correlation lifetime or the cycle length after the dual observables are correctly identified.

Figures

Figures reproduced from arXiv: 2509.06421 by Ryuna Nagayama, Sosuke Ito.

Figure 1
Figure 1. Figure 1: FIG. 1. Duality of the dissipation-coherence trade-off and the TSL. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical demonstration of R [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Numerical demonstration of the chemical oscillator with [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

We derive two fundamental trade-offs for general stochastic limit cycles in the weak-noise limit. The first is the dissipation-coherence trade-off, which was discovered and proved under additional assumptions by Santolin and Falasco [Phys. Rev. Lett. 135, 057101 (2025)]. This trade-off bounds the entropy production required for one oscillatory period using the number of oscillations that occur before steady-state correlations are disrupted. The second is the thermodynamic speed limit, which bounds the entropy production by the Euclidean length of the limit cycle. These trade-offs are obtained by substituting mutually dual observables, derived from the stability of the limit cycle, into the thermodynamic uncertainty relation. This fact allows us to regard the dissipation-coherence trade-off as the dual of the thermodynamic speed limit. We numerically demonstrate these trade-offs using the noisy R\"{o}ssler model. We also apply the trade-offs to stochastic chemical systems, where the diffusion coefficient matrix may contain zero eigenvalues. Furthermore, we show that the dissipation-coherence trade-off is always achievable by appropriately modifying the diffusion coefficient matrix based on the phase reduction.

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 derives two fundamental trade-offs for general stochastic limit cycles in the weak-noise limit: a dissipation-coherence trade-off (bounding entropy production per period by the number of oscillations before steady-state correlations decay) and a thermodynamic speed limit (bounding entropy production by the Euclidean length of the limit cycle). Both are obtained by substituting a pair of mutually dual observables—constructed from the linear stability matrix of the deterministic limit cycle—into the thermodynamic uncertainty relation (TUR). This substitution is used to establish a duality between the two trade-offs. Results are illustrated numerically on the noisy Rössler oscillator and extended to stochastic chemical reaction networks (including cases with zero eigenvalues in the diffusion matrix); an additional claim is that the dissipation-coherence bound is always achievable by a suitable phase-reduction-based modification of the diffusion matrix.

Significance. If the central substitution argument holds exactly, the work supplies a compact duality perspective that unifies dissipation-coherence and speed-limit bounds for stochastic oscillations via the TUR and linear stability analysis. The numerical checks on the Rössler model and the explicit treatment of singular diffusion matrices in chemical networks are concrete strengths; the phase-reduction achievability result, if rigorously justified, would further enhance applicability to non-equilibrium chemical systems.

major comments (2)
  1. [Abstract and derivation of trade-offs via TUR substitution] Abstract (substitution step) and the corresponding derivation section: the claim that the two trade-offs follow directly from substituting the dual observables (one tied to decorrelation time, one to cycle length) into the standard TUR assumes these observables inherit the precise fluctuation-dissipation form required by the TUR without residual corrections. In the weak-noise limit this requires that higher-order phase-diffusion terms, periodic modulation of the stability eigenvalues along the orbit, and projection onto the tangent space when the diffusion matrix is singular all vanish identically; the manuscript must supply an explicit error estimate or proof that these residuals are absent, because they are load-bearing for the exactness of both bounds.
  2. [Achievability via diffusion-matrix modification] Abstract (final sentence) and the section on achievability: the assertion that the dissipation-coherence trade-off is always achievable by modifying the diffusion coefficient matrix according to phase reduction must demonstrate that the modification leaves the underlying stochastic dynamics, the limit-cycle stability, and the entropy-production rate unchanged except for the intended adjustment; without this, the achievability statement risks altering the very quantities the bound is meant to constrain.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'mutually dual observables, derived from the stability of the limit cycle' would benefit from a one-sentence definition or cross-reference to the explicit construction (e.g., via the Floquet exponents or the phase-response curve) to aid readers who have not yet reached the methods section.
  2. [Numerical checks on Rössler model] Numerical demonstration section: the Rössler-model plots should include statistical error bars on the measured entropy production and coherence times, together with the number of independent trajectories used, so that the tightness of the reported bounds can be assessed quantitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comments point by point below. We agree that additional justifications are needed for rigor and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and derivation of trade-offs via TUR substitution] Abstract (substitution step) and the corresponding derivation section: the claim that the two trade-offs follow directly from substituting the dual observables (one tied to decorrelation time, one to cycle length) into the standard TUR assumes these observables inherit the precise fluctuation-dissipation form required by the TUR without residual corrections. In the weak-noise limit this requires that higher-order phase-diffusion terms, periodic modulation of the stability eigenvalues along the orbit, and projection onto the tangent space when the diffusion matrix is singular all vanish identically; the manuscript must supply an explicit error estimate or proof that these residuals are absent, because they are load-bearing for the exactness of both bounds.

    Authors: We thank the referee for highlighting this important point regarding the validity of the substitution in the weak-noise limit. In our derivation, the dual observables are defined using the linear stability matrix of the deterministic limit cycle, and in the weak-noise expansion, the leading-order contributions to the TUR indeed satisfy the required fluctuation-dissipation relation exactly. Higher-order terms, including phase-diffusion corrections and periodic modulations, contribute at orders that vanish as the noise strength approaches zero. For the singular diffusion case, the projection is handled by restricting to the tangent space of the cycle, which is justified by the chemical reaction network structure. To strengthen the manuscript, we will include an explicit error estimate showing that the residuals are of higher order O(ε), where ε denotes the noise intensity, thereby confirming the exactness of the bounds in the weak-noise limit. revision: yes

  2. Referee: [Achievability via diffusion-matrix modification] Abstract (final sentence) and the section on achievability: the assertion that the dissipation-coherence trade-off is always achievable by modifying the diffusion coefficient matrix according to phase reduction must demonstrate that the modification leaves the underlying stochastic dynamics, the limit-cycle stability, and the entropy-production rate unchanged except for the intended adjustment; without this, the achievability statement risks altering the very quantities the bound is meant to constrain.

    Authors: We agree that a careful justification is required for the achievability claim. The phase-reduction modification adjusts only the diffusion matrix in a manner consistent with the phase dynamics, while keeping the deterministic drift term (and thus the limit cycle and its stability) identical. The entropy production rate is modified solely through the diffusion adjustment, which is the intended effect to achieve the bound. This construction does not alter the underlying deterministic dynamics or the stability properties. In the revised version, we will provide a more detailed proof that the modified system retains the same limit cycle and that the entropy production change is precisely the one needed for achievability, without unintended side effects on the stochastic trajectories beyond the diffusion scaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central derivation substitutes observables constructed independently from the linear stability analysis of the deterministic limit cycle into the established thermodynamic uncertainty relation (TUR). The duality between the dissipation-coherence trade-off and the thermodynamic speed limit emerges as a consequence of this substitution rather than being presupposed by definition or by a self-citation chain. The cited prior result on the dissipation-coherence trade-off (Santolin and Falasco) has non-overlapping authors and is invoked only for context, not as load-bearing justification for the present substitution step. No fitted parameters are renamed as predictions, no ansatz is smuggled via self-citation, and the weak-noise limit is used only to justify the construction of the observables, not to force the final bounds by construction. The derivation remains self-contained against external benchmarks such as the standard TUR and linear stability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Derivation rests on the thermodynamic uncertainty relation applied to stability-derived observables and on the weak-noise limit; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Thermodynamic uncertainty relation holds for the dual observables constructed from limit-cycle stability
    Used directly to obtain both trade-offs by substitution.
  • domain assumption Weak-noise limit is valid for the stochastic limit cycles under study
    Stated as the regime in which the two trade-offs are derived.

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Cited by 1 Pith paper

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