← Mathematical compendium

Published equation contexts

Var⁡(Jτ)⟨Jτ⟩2≥2⟨Στ⟩\frac{\operatorname{Var}(J_\tau)}{\langle J_\tau\rangle^2} \ge \frac{2}{\langle\Sigma_\tau\rangle}

Why this formula appears here

A molecular clock or motor is useful not only when its average current is nonzero but when that current is sufficiently precise. Let JτJ_\tau be an integrated current observed for time τ\tau , such as net molecular steps or transported ions. Define dimensionless mean entropy production ⟨\langleΣτ\Sigma_\tau⟩\rangle=⟨\langleΔ\Delta stots_{\mathrm{tot}}⟩\rangle/kBk_{\mathrm B} . A canonical steady-state thermodynamic uncertainty relation has the form Var⁡(Jτ)⟨Jτ⟩2≥2⟨Στ⟩\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…

Read the full article-specific guide →

Read the representative guide

JτJ_\tau

Symbol J_τ

an integrated current observed for time τ\tau , such as net molecular steps or transported ions.

Read this term in its guide →
Στ\Sigma_\tau

Symbol Sigma_τ

Sigma_τ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Read this term in its guide →
⟨Jτ⟩2\langle J_\tau\rangle^2

Denominator: langle J_τrangle^2

The complete quantity below the fraction bar; it must be nonzero for this division.

Read this term in its guide →
⟨Στ⟩\langle\Sigma_\tau\rangle

Denominator: langleSigma_τrangle

The complete quantity below the fraction bar; it must be nonzero for this division.

Read this term in its guide →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

Var⁡(Jτ)⟨Jτ⟩2≥2⟨Στ⟩.\frac{\operatorname{Var}(J_\tau)}{\langle J_\tau\rangle^2} \ge \frac{2}{\langle\Sigma_\tau\rangle}.

Equation 22 · Physics

The Statistical Mechanics of Irreversibility at Molecular Scale

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

A molecular clock or motor is useful not only when its average current is nonzero but when that current is sufficiently precise. Let JτJ_\tau be an integrated current observed for time τ\tau , such as net molecular steps or transported ions. Define dimensionless mean entropy production ⟨\langleΣτ\Sigma_\tau⟩\rangle=⟨\langleΔ\Delta stots_{\mathrm{tot}}⟩\rangle/kBk_{\mathrm B} . A canonical steady-state thermodynamic uncertainty relation has the form Var⁡(Jτ)⟨Jτ⟩2≥2⟨Στ⟩\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…

Meanings in this article

  • JτJ_\tau: an integrated current observed for time τ\tau , such as net molecular steps or transported ions.
Equation guide → · Article →