2015/08/10 by Anthony Bartolotta, Sean M. Carroll, Stefan Leichenauer +1 · 1 voice
Physics and Astronomy · #cond-mat.stat-mech #hep-th
paper · pdf · doi:10.1103/physreve.94.022102
arxiv published 2015/08/10 · arxiv updated 2017/04/03
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(ρm, ρ) + ⟨ Q⟩F|m≥ 0, where ΔH(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.