the exact statement of how far a process can run against the second law, and how rarely. Crooks: $P(+W)/P(-W) = e^{(W-\Delta F)/k_BT}$ — the odds of a fluctuation that consumes rather than produces entropy fall exponentially with its size. Jarzynski: $\langle e^{-W/k_BT}\rangle = e^{-\Delta F/k_BT}$ — an equality, not an inequality, that recovers free energy differences from irreversible pulls
verified in experiment: RNA hairpin unfolding (Liphardt 2002, Collin 2005), colloidal particles in optical traps, single-electron boxes. the second law is thereby not a prohibition but a statistic
the information form is the one cyber uses. Kawai–Parrondo–Van den Broeck: dissipated work equals a relative entropy between the forward process and its time reverse,
$$W_{\text{diss}} = k_BT\, D_{\mathrm{KL}}\big(P_{\rightarrow} \,\|\, P_{\leftarrow}\big)$$
so dissipation is measured surprise. this is the physics under Bayesian Truth Serum: an honest report is the quasi-static limit with zero dissipation, a lie is irreversible work, and the zero-sum that moves stake from liars to truth-tellers is the second law written on a ledger
see entropy · thermodynamics · landauer limit · physical analogies
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