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Maximum entropy distributions of wave functions at thermal equilibrium

Jacob T. Willson*, Henrik J. Heelweg, and Adam P. Willard†

  • *Also at Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachuetts, USA.
  • †Contact author: awillard@mit.edu
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Phys. Rev. E 114, 014145 – Published 27 July, 2026

DOI: https://doi.org/10.1103/yk5x-cnzv

Abstract

Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wave functions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wave function ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wave functions because they lack a definite energy. In this paper we present a maximum entropy principle for the quantum wave function ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the Rényi divergence of the ensemble with respect to the Gibbs state, indicating that the Rényi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.

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Outline

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Phys. Rev. E 114, 014145– Published 27 July, 2026
  • Received 25 March 2026
  • Revised 4 June 2026
  • Accepted 10 June 2026
Article part of CHORUS
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