Numerator: Var(J_τ)

The passage around this formula

A molecular clock or motor is useful not only when its average current is nonzero but when that current is sufficiently precise. Let $J_\tau$ be an integrated current observed for time $\tau$ , such as net molecular steps or transported ions. Define dimensionless mean entropy production $\langle$$\Sigma_\tau$$\rangle$=$\langle$$\Delta$ $s_{\mathrm{tot}}$$\rangle$/$k_{\mathrm B}$ . A canonical steady-state thermodynamic uncertainty relation has the form $\frac{\operatorname{Var}(J_\tau)}{\langle J_\tau\rangle^2} \ge \frac{2}{\langle\Sigma_\tau\rangle}$. The relation says that suppressing relative current fluctuations requires dissipation. Barato and Seifert introduced this precision–cost connection for biomolecular processes [ 5 ] , and later large-deviation results established broad bounds…

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