Incorporate the finite-time improvement factor into the refined iTUR

Establish whether combining the local dissipation-based approach with the proof of the inverse thermodynamic uncertainty relation for overdamped Langevin dynamics yields the refined inverse thermodynamic uncertainty relation with the factor $g(\bar\lambda_{\mathrm{gap}}t)$ included, thereby sharpening the variance bound at finite observation times when the maximal dissipation rate satisfies $\Sigma_\infty<\infty$.

Background

The paper derives a refined inverse thermodynamic uncertainty relation that bounds the variance of a generalized current from above using locally observed dissipation, the observable’s intrinsic fluctuations, and the spectral gap. It compares this result with an earlier inverse thermodynamic uncertainty relation for overdamped Langevin dynamics, which contains an additional finite-time factor $g(\bar\lambda_{\mathrm{gap}}t)\leq 1$.

The authors observe that this factor can sharpen the bound for finite times when the global maximal dissipation rate is finite, but they do not establish that it can be incorporated into their locally refined inequality. They therefore formulate the combination of the two proof approaches as a conjecture.