Bartolotta, Anthony and Carroll, Sean M. and Leichenauer, Stefan and Pollack, Jason
(2015)
*Bayesian second law of thermodynamics.*
Physical Review E, 94
(2).
Art. No. 022102.
ISSN 2470-0045.
https://resolver.caltech.edu/CaltechAUTHORS:20150812-142230808

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## 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)+(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.

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Alternate Title: | The Bayesian Second Law of Thermodynamics | ||||||||||||||||||

Additional Information: | © 2016 American Physical Society. (Received 13 August 2015; revised manuscript received 20 June 2016; published 1 August 2016). It is a pleasure to thank Gavin Crooks, Christopher Jarzynski, Timothy Maxwell, and Nicole Yunger Halpern for helpful conversations. This research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech, by DOE Grant No. DE-SC0011632, and by the Gordon and Betty Moore Foundation through Grant No. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics and by the John Simon Guggenheim Memorial Foundation. The work of S.L. is supported in part by the Berkeley Center for Theoretical Physics, by the National Science Foundation (Awards No. 1214644 and No. 1316783), by fqxi Grant No. RFP3-1323, and by the US Department of Energy under Contract No. DE-AC02-05CH11231. | ||||||||||||||||||

Group: | Walter Burke Institute for Theoretical Physics, Moore Center for Theoretical Cosmology and Physics | ||||||||||||||||||

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Issue or Number: | 2 | ||||||||||||||||||

Record Number: | CaltechAUTHORS:20150812-142230808 | ||||||||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20150812-142230808 | ||||||||||||||||||

Official Citation: | Bayesian second law of thermodynamics Anthony Bartolotta, Sean M. Carroll, Stefan Leichenauer, and Jason Pollack Phys. Rev. E 94, 022102 | ||||||||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||||

ID Code: | 59452 | ||||||||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||||||||

Deposited By: | Joy Painter | ||||||||||||||||||

Deposited On: | 12 Aug 2015 22:31 | ||||||||||||||||||

Last Modified: | 03 Oct 2019 08:47 |

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