Classical dynamical coarse-grained entropy and comparison with the quantum version

Abstract

We develop the framework of classical Observational entropy, which is a mathematically rigorous and precise framework for non-equilibrium thermodynamics, explicitly defined in terms of a set of observables. Observational entropy can be seen as a generalization of Boltzmann entropy to systems with indeterminate initial conditions, and describes the knowledge achievable about the system by a macroscopic observer with limited measurement capabilities; it becomes Gibbs entropy in the limit of perfectly fine-grained measurements. This quantity, while previously mentioned in the literature, has been investigated in detail only in the quantum case. We describe this framework reasonably pedagogically, then show that in this framework, certain choices of coarse-graining lead to an entropy that is well-defined out of equilibrium, additive on independent systems, and that grows towards thermodynamic entropy as the system reaches equilibrium, even for systems that are genuinely isolated. Choosing certain macroscopic regions, this dynamical thermodynamic entropy measures how close these regions are to thermal equilibrium. We also show that in the given formalism, the correspondence between classical entropy (defined on classical phase space) and quantum entropy (defined on Hilbert space) becomes surprisingly direct and transparent, while manifesting differences stemming from non-commutativity of coarse-grainings and from non-existence of a direct classical analogue of quantum energy eigenstates.

arXiv comment15+6 pages, 6 figures, 1 table. v2: literature review extended, added a discussion of Observational entropy with local energy coarse-graining (FOE) and its properties, added physical/operational interpretations of both FOE and S_xE. Comments and questions are very much welcome

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• First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy
• A brief introduction to observational entropy
• Quantum correlation entropy
• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
• Boltzmann entropy of a freely expanding quantum ideal gas
• Witnessing quantum chaos using observational entropy
• Continuity bounds on observational entropy and measured relative entropies
• Observational entropic study of Anderson localization
• Multi-temperature atomic ensemble: nonequilibrium evolution after ultrafast electronic excitation
• On the generic increase of observational entropy in isolated systems
• Entropy production as change in observational entropy
• A Schmidt decomposition approach to quantum thermodynamics
• Quantifying Information Extraction using Generalized Quantum Measurements
• Comparative Microscopic Study of Entropies and their Production
• Observational entropy with general quantum priors
• Non-conjugate quantum subsystems
• Boltzmann's entropy during free expansion of an interacting ideal gas
• Entropic partial orderings of quantum measurements
• Coarse-graining and symplectic non-squeezing
• Quantum information in Riemannian spaces
• Basics of observational entropy
• Macroscopicity and observational deficit in states, operations, and correlations
• Entropy of Quantum Measurements