Counterexamples to the conjectured ordering between the waiting-time bound and the thermodynamic uncertainty bound on entropy production

TL;DR

The paper investigates whether the waiting-time distribution bound $\sigma_{\mathcal{L}}$ universally dominates the thermodynamic uncertainty relation bound $\sigma_{\mathrm{TUR}}$ for entropy production under partial observation. Using a formal CTMC framework with a single observed bidirectional link, it derives the TUR bound $\sigma_{\mathrm{TUR}}=j^2/D$ and the waiting-time bound $\sigma_{\mathcal{L}}=\sigma_{\ell}+\sigma_t$ based on the observed sequence of transitions and waiting times. Through a large-scale numerical search in 4-state networks, the authors exhibit counterexamples where $\sigma_{\mathcal{L}}<\sigma_{\mathrm{TUR}}$, notably for a specific matrix with $\sigma_{\mathrm{TUR}} \approx 12.353$ and $\sigma_{\mathcal{L}} \approx 12.204$, while the full entropy production is $\sigma \approx 40.991$. This result shows that no universal ordering exists between the two inference bounds under partial observation and suggests the need to identify subclasses of networks or observation schemes where a bound ordering can be established.

Abstract

Two widely used model-free lower bounds on the steady-state entropy production rate of a continuous-time Markov jump process are the thermodynamic uncertainty relation (TUR) bound $σ_\text{TUR}$, derived from the mean and variance of a current, and the waiting-time distribution (WTD) bound $σ_\mathcal{L}$, derived from the irreversibility of partially observed transition sequences together with their waiting times. It has been conjectured that $σ_{\mathcal L}$ is always at least as tight as $σ_{\mathrm{TUR}}$ when both are constructed from the same partially observed link. Here we provide four-state counterexamples in a nonequilibrium steady state where $σ_{\mathcal L}<σ_{\mathrm{TUR}}$. This result shows that no universal ordering exists between these two inference bounds under partial observation.

Figures (1)

• Figure 1: Numerical search over randomly generated $4$-state Markov jump processes (all-to-all positive rates drawn from a lognormal distribution) with a single observed link $1\leftrightarrow 4$. The scatter plot shows (a) $\sigma_{\mathrm{TUR}}/\sigma_{\mathcal{L}}$ and (b) $\sigma_{\mathrm{TUR}}/\sigma_{\ell}$ versus the index of valid sampled generators. Points above the horizontal line at $1$ (represented by red crosses) indicate counterexamples with $\sigma_{\mathcal{L}}<\sigma_{\mathrm{TUR}}$ or $\sigma_{\mathcal{L}}<\sigma_{\ell}$.