Information-theoretic equilibrium and observable thermalization

Abstract

To understand under which conditions thermodynamics emerges from the microscopic dynamics is the ultimate goal of statistical mechanics. Despite the fact that the theory is more than 100 years old, we are still discussing its foundations and its regime of applicability. A point of crucial importance is the definition of the notion of thermal equilibrium, which is given as the state that maximises the von Neumann entropy. Here we argue that it is necessary to propose a new way of describing thermal equilibrium, focused on observables rather than on the full state of the quantum system. We characterise the notion of thermal equilibrium, for a given observable, via the maximisation of its Shannon entropy and highlight the thermal properties that such a principle heralds. The relation with Gibbs ensembles is brought to light. Furthermore, we apply such a notion of equilibrium to a closed quantum systems and prove that there is always a class of observables which exhibits thermal equilibrium properties and we give a recipe to explicitly construct them. Eventually, we bring to light an intimate connection of such a principle with the Eigenstate Thermalisation Hypothesis.

arXiv commentAccepted by Scientific Report

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• Quantum operator entropies under unitary evolution
• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
• Generalized Wigner-von Neumann entropy and its typicality
• Relaxation of Shannon entropy for trapped interacting bosons with dipolar interactions
• Quantifying Information Extraction using Generalized Quantum Measurements
• The Boltzmann distribution and the quantum-classical correspondence
• New equilibrium ensembles for isolated quantum systems
• Logarithmic growth of local entropy and total correlations in many-body localized dynamics
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