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 ΔH(ρ_m,ρ)+(2)F|M ⩾ 0, where ΔH(ρ_m,ρ) is the change in the cross entropy between the original phase-space probability distribution ρ and the measurement-updated distribution ρ_m and (2)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 integral fluctuation theorems. We demonstrate the formalism using simple analytical and numerical examples.