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Published equation contexts

Δstot[Γ]=kBln⁡PF[Γ]PR[Γ†]\Delta s_{\mathrm{tot}}[\Gamma] = k_{\mathrm B}\ln \frac{\mathcal P_{\mathrm F}[\Gamma]} {\mathcal P_{\mathrm R}[\Gamma^\dagger]}

Why this formula appears here

Under broad conditions, total stochastic entropy production can be represented as a log-likelihood ratio, Δstot[Γ]=kBln⁡PF[Γ]PR[Γ†]\Delta s_{\mathrm{tot}}[\Gamma] = k_{\mathrm B}\ln \frac{\mathcal P_{\mathrm F}[\Gamma]} {\mathcal P_{\mathrm R}[\Gamma^\dagger]}. This equation supplies a precise meaning for the arrow of time. If a path is far more probable under the forward process than its conjugate is under the reverse process, observing it provides evidence about temporal direction. If the probabilities are equal, the path itself carries no arrow-of-time information. Irreversibility is therefore related to statistical distinguishability, not to an absolute prohibition on reverse-looking motion.

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Δstot\Delta s_{\mathrm{tot}}

Symbol Δ s_tot

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

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kBk_{\mathrm B}

Symbol k_mathrm B

kmk_mathrm B is an input to the expression that computes the quantity on the left.

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PFP_{\mathrm F}

Symbol P_mathrm F

PmP_mathrm F occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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PRP_{\mathrm R}

Symbol P_mathrm R

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

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PF[Γ]\mathcal P_{\mathrm F}[\Gamma]

Numerator: mathcal P_mathrm F[Gamma]

The complete quantity above the fraction bar.

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PR[Γ†]\mathcal P_{\mathrm R}[\Gamma^\dagger]

Denominator: mathcal P_mathrm R[Gamma^dagger]

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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

Δstot[Γ]=kBln⁡PF[Γ]PR[Γ†].\Delta s_{\mathrm{tot}}[\Gamma] = k_{\mathrm B}\ln \frac{\mathcal P_{\mathrm F}[\Gamma]} {\mathcal P_{\mathrm R}[\Gamma^\dagger]}.

Equation 7 · Physics

The Statistical Mechanics of Irreversibility at Molecular Scale

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Under broad conditions, total stochastic entropy production can be represented as a log-likelihood ratio, Δstot[Γ]=kBln⁡PF[Γ]PR[Γ†]\Delta s_{\mathrm{tot}}[\Gamma] = k_{\mathrm B}\ln \frac{\mathcal P_{\mathrm F}[\Gamma]} {\mathcal P_{\mathrm R}[\Gamma^\dagger]}. This equation supplies a precise meaning for the arrow of time. If a path is far more probable under the forward process than its conjugate is under the reverse process, observing it provides evidence about temporal direction. If the probabilities are equal, the path itself carries no arrow-of-time information. Irreversibility is therefore related to statistical distinguishability, not to an absolute prohibition on reverse-looking motion.

Meanings in this article

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