title: Bayesian second law of thermodynamics.
abstract: 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)+〈Q〉_{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 〈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 integral fluctuation theorems. We demonstrate the formalism using simple analytical and numerical examples.
authors.affiliation: Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA.
authors.forename: Anthony
authors.initials: A
authors.lastname: Bartolotta
authors.affiliation: Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA.
authors.forename: Sean M
authors.initials: SM
authors.lastname: Carroll
authors.affiliation: Center for Theoretical Physics and Department of Physics, University of California, Berkeley, California 94720, USA.
authors.forename: Stefan
authors.initials: S
authors.lastname: Leichenauer
authors.affiliation: Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA.
authors.forename: Jason
authors.initials: J
authors.lastname: Pollack