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

TL;DR

This work studies the thermodynamic constraints on noisy oscillations in stochastic limit cycles by applying dual observables to the thermodynamic uncertainty relation (TUR). In the weak-noise limit, it proves a general dissipation-coherence trade-off and derives a dual thermodynamic speed limit (TSL) that connect entropy production to coherent oscillations and to the geometric length of the limit cycle, respectively. The dissipation-coherence bound is shown to hold beyond previous special cases (identity diffusion or Hopf-bifurcation proximity), and the duality between the two bounds is established on the TUR level; the results are demonstrated in a noisy Rössler system and extended to stochastic chemical networks with conservation laws by a positive-definite reduced Langevin description. The findings tie deterministic limit-cycle metrics (cycle length, Euclidean length) to stochastic thermodynamics, offering a framework to classify bifurcations and analyze thermodynamic efficiency in complex oscillatory systems.

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 numerically conjectured and partially proved 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 with 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 Rössler model. We also apply the trade-offs to stochastic chemical systems, where the diffusion coefficient matrix may contain zero eigenvalues.

Figures (3)

• Figure 1: Duality of the dissipation-coherence trade-off and the TSL. We derive these trade-offs by substituting dual observables into the (short-time) TUR.
• Figure 2: Numerical demonstration of Rössler model. (a) $c$-dependence of the period $\tau_{\mathrm{p}}$. (b) Typical stable limit cycles in each phase of the period-doubling bifurcation and corresponding $c$. (c) $c$-dependence of the tightness of the dissipation-coherence trade-off. (d) $c$-dependence of the tightness of the TSL. In (a), (c), and (d), we color the range of $c$ corresponding to each phase of the period-doubling bifurcation in red, green, and blue.
• Figure 3: Numerical demonstration of the chemical oscillator with a conservation law. (a) The stable limit cycles on $x_2$-$x_1$ plane for several $\kappa$. The number written below each limit cycle corresponds to $\kappa$. (b) $\kappa$-dependence of $\eta^{\mathrm{RE}}_{\mathrm{DCT}}$ (red line) and $\eta^{\mathrm{RE}}_{\mathrm{TSL}}$ (blue line). (c) $\kappa$-dependence of $\eta^{\mathrm{ME}}_{\mathrm{DCT}}$ (red line) and $\eta^{\mathrm{ME}}_{\mathrm{TSL}}$ (blue line). (d) The EPs multiplied by $\epsilon$ and their lower bounds. We plot the value in $\epsilon\to0$.