Numerator: mathcal P_mathrm F[Gamma]

The passage around this formula

Under broad conditions, total stochastic entropy production can be represented as a log-likelihood ratio, $\Delta s_{\mathrm{tot}}[\Gamma] = k_{\mathrm B}\ln \frac{\mathcal P_{\mathrm F}[\Gamma]} {\mathcal P_{\mathrm R}[\Gamma^\dagger]}$. 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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