Stochastic thermodynamics based on incomplete information: Generalized Jarzynski equality with measurement errors with or without feedback

Abstract

In the derivation of fluctuation relations, and in stochastic thermodynamics in general, it is tacitly assumed that we can measure the system perfectly, i.e., without measurement errors. We here demonstrate for a driven system immersed in a single heat bath, for which the classic Jarzynski equality $⟨e^{-\beta (W-\Delta F)}⟩=1$ holds, how to relax this assumption. Based on a general measurement model akin to Bayesian inference we derive a general expression for the fluctuation relation of the measured work and we study the case of an overdamped Brownian particle and of a two-level system in particular. We then generalize our results further and incorporate feedback in our description. We show and argue that, if measurement errors are fully taken into account by the agent who controls and observes the system, the standard Jarzynski-Sagawa-Ueda relation should be formulated differently. We again explicitly demonstrate this for an overdamped Brownian particle and a two-level system where the fluctuation relation of the measured work differs significantly from the efficacy parameter introduced by Sagawa and Ueda. Instead, the generalized fluctuation relation under feedback control, $⟨e^{-\beta (W-\Delta F)-I}⟩=1$, holds only for a superobserver having perfect access to both the system and detector degrees of freedom, independently of whether or not the detector yields a noisy measurement record and whether or not we perform feedback.

21 pages, 5 figures

References in corpus (10)

• Experimental Test of Quantum Jarzynski Equality with a Trapped Ion System
• Ensemble and Trajectory Thermodynamics: A Brief Introduction
• Experimental study of mutual information in a Maxwell Demon
• Thermodynamics of a physical model implementing a Maxwell demon
• Nonequilibrium Detailed Fluctuation Theorem for Repeated Discrete Feedback
• Entropy production and coarse-graining in Markov processes
• Incomplete measurement of work in a dissipative two level system
• General achievable bound of extractable work under feedback control
• Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles
• Using a Szilard engine to illustrate the validity of the modified Jarzynski equality in presence of measurement errors