Elementary derivation of the dissipation-coherence bound for stochastic oscillators

The dissipation-coherence bound is a conjectured trade-off between entropy production and the quality of stochastic oscillations. We show that this bound can be derived by combining the higher-order "thermodynamic uncertainty relation" with a simple condition on phase-current fluctuations. In one-dimensional cyclic systems, our proposed condition is shown to be equivalent to the dissipation-coherence bound itself. Our approach yields an elementary proof in the weak-noise Gaussian regime and extends naturally to some non-Gaussian systems, as we illustrate with a run-and-tumble particle. Finally, we contrast current-based and spectral formulations of the dissipation-coherence bound.