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Equation 7 · Part 5 · The Statistical Mechanics of Irreversibility at Molecular Scale

Symbol P_mathrm R

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]}.
PRP_{\mathrm R}

What this part means

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

Its job in the formula

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

The passage around this formula

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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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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