The Bayesian Second Law of Thermodynamics

We derive a generalization of the Second Law of Thermodynamics that uses Bayesian updates to explicitly incorporate the effects of a measurement of a system at some point in its evolution. By allowing an experimenter’s knowledge to be updated by the measurement process, this formulation resolves a tension between the fact that the entropy of a statistical system can sometimes fluctuate downward and the information-theoretic idea that knowledge of a stochastically-evolving system degrades over time. The Bayesian Second Law can be written as ∆H(ρm, ρ) + 〈Q〉F |m ≥ 0, where ∆H(ρm, ρ) is the change in the cross entropy between the original phase-space probability distribution ρ and the measurement-updated distribution ρm, and 〈Q〉F |m is the expectation value of a generalized heat flow out of the system. We also derive refined versions of the Second Law that bound the entropy increase from below by a non-negative number, as well as Bayesian versions of the Jarzynski equality. We demonstrate the formalism using simple analytical and numerical examples. ar X iv :1 50 8. 02 42 1v 1 [ co nd -m at .s ta tm ec h] 1 0 A ug 2 01 5