Quantum Macrostates, Equivalence of Ensembles and an H-Theorem

Abstract

Before the thermodynamic limit, macroscopic averages need not commute for a quantum system. As a consequence, aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis, when dealing with several observables. We propose an implementation of ideas that go back to John von Neumann's writing about the macroscopic measurement. We apply our scheme to the relation between macroscopic autonomy and an H-theorem, and to the problem of equivalence of ensembles. In particular, we show how the latter is related to the asymptotic equipartition theorem. The main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate, as the size of the system gets larger, on the macroscopic values for the different macroscopic observables. Deviations from that law are governed by the entropy.

16 pages; v1 -> v2: Sec. 3 slightly rewritten, 2 references added

References in corpus (3)

• A quantum version of Sanov's theorem
• Large deviations for quantum spin systems
• Large deviations in quantum lattice systems: one-phase region

Cited by in corpus (14)

• Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
• Normal Typicality and von Neumann's Quantum Ergodic Theorem
• Thermalization and canonical typicality in translation-invariant quantum lattice systems
• Thermal equilibrium of a macroscopic quantum system in a pure state
• Macroscopic and Microscopic Thermal Equilibrium
• Eigenstate thermalization from the clustering property of correlation
• Typical support and Sanov large deviations of correlated states
• Nonequivalence of ensembles in the Curie-Weiss anisotropic quantum Heisenberg model
• Boltzmann entropy of a freely expanding quantum ideal gas
• On the foundations of statistical mechanics
• Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement
• The Shannon-McMillan Theorem for AF $C^{\ast }$-systems
• Boltzmann's entropy during free expansion of an interacting ideal gas
• Large deviations in the quantum quasi-1D jellium