Minimal dissipation per coherent oscillation of a stochastic clock
In plain words
Biochemical clocks, such as circadian oscillators in cells, keep time for only a limited number of cycles before noise scrambles them, and keeping them coherent costs free energy. A proposed universal minimum cost was disproved in 2026, and the correct universal bound is open.
Precise statement
For a finite Markov jump network with stationary entropy production rate $\sigma$ and second eigenvalue $-\lambda_R + i \lambda_I$ of the rate matrix, the conjecture $\sigma \ge \lambda_I^{2}/\lambda_R$ (arXiv:2112.01607) fails (counterexample arXiv:2609.34352), while a version weighted by an eigenvector-uniformity factor is proven (arXiv:2606.05498). Find the tightest bound on entropy produced per oscillation period in terms of the coherence of an observable correlation function $C(t)$ (number of oscillations before decay), and a model that saturates it.
What would settle it
A proof of a bound in terms of observable correlation functions together with a saturating model.
Status in the literature
Unverified note
Shiraishi disproved the spectral conjecture in September 2026 (arXiv:2609.34352); Gu proved a weaker bound with a mode-uniformity factor that reduces to the original for translation-invariant rings (Phys. Rev. E 113, 064130, 2026).