= Integral fluctuation theorem
{title2=$\langle e^{-\Sigma_F}\rangle_F=1$}
Normalization of the reverse law gives an exact finite identity:
$$
\left\langle e^{-\Sigma_F}\right\rangle_F
=\sum_\gamma P_F[\gamma]\frac{P_R[\Theta\gamma]}{P_F[\gamma]}=1.
$$
The sum notation describes discrete paths; the corresponding path <integral> has the same change-of-variable argument. <Jensen's inequality> for the <exponential function> then gives $e^{-\langle\Sigma_F\rangle_F}\le1$, hence $\langle\Sigma_F\rangle_F\ge0$. Equivalently this mean is the <Kullback-Leibler divergence> between the forward path law and the reverse path law pulled back by $\Theta$. Equality requires those two laws to agree. <Entropy production along a stochastic trajectory and an integral fluctuation theorem by Udo Seifert>[Seifert, 2005, Eqs. (17)–(18): integral relation and its mean inequality]{ref}
Rare negative <entropy production> contributes disproportionately to this <expectation> through $e^{-\Sigma_F}$. Missing such events in a finite sample can spoil an estimate of the identity. Matching <probability support> is also essential: reverse paths absent from the forward law cannot be recovered by cancelling forward weights. This <Second law of thermodynamics> statement concerns an <ensemble average>; an individual negative value is compatible with it.
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