The 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 $\Delta H({\rho }_m,\rho )+⟨\mathcal{Q}{⟩}_{F|m}\ge 0$, where $\Delta H({\rho }_m,\rho )$ is the change in the cross entropy between the original phase-space probability distribution $\rho$ and the measurement-updated distribution ${\rho }_m$, and $⟨\mathcal{Q}{⟩}_{F|m}$ 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.

40 pages. Additional information and animations at http://preposterousuniverse.com/research/bsl/

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Cited by in corpus (4)

• Marginal and Conditional Second Laws of Thermodynamics
• Quantum State Reduction: Generalized Bipartitions from Algebras of Observables
• Nonequilibrium entropic bounds for Darwinian replicators
• Learning stochastic filtering