The Bayesian Second Law of Thermodynamics

TL;DR

This paper introduces a Bayesian formulation of the Second Law of Thermodynamics that incorporates measurement updates, resolving apparent entropy fluctuations and linking information theory with thermodynamic principles.

Contribution

It presents a novel Bayesian generalization of the Second Law, including refined bounds and Bayesian Jarzynski equalities, with analytical and numerical demonstrations.

Findings

Bayesian Second Law accounts for measurement updates in entropy change.

Refined bounds on entropy increase are derived.

Bayesian Jarzynski equality is established.

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 $\Delta H(\rho_m, \rho) + \langle \mathcal{Q}\rangle_{F|m}\geq 0$, 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 $\langle \mathcal{Q}\rangle_{F|m}$ 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.