Equation 21 · The Statistical Mechanics of Irreversibility at Molecular Scale

What does this equation mean?

\langle\Sigma_\tau\rangle=\langle\Delta s_{\mathrm{tot}}\rangle/k_{\mathrm B}

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Inputs and operationslangleΔ s_totrangle/k_mathrm B

Result or conditionlangleSigma_τrangle

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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\Sigma_\tau

Symbol Sigma_τ

Sigma_τ is part of the quantity the equation computes from the expression on the right.

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\Delta s_{\mathrm{tot}}

Symbol Δ s_tot

Δ $s_t$ ot is an input to the expression that computes the quantity on the left.

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k_{\mathrm B}

Symbol k_mathrm B

$k_m$ athrm B is an input to the expression that computes the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

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

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