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Thermal Equilibrium of a Macroscopic Quantum System in a Pure State

Sheldon Goldstein, David A. Huse, Joel L. Lebowitz, and Roderich Tumulka
Phys. Rev. Lett. 115, 100402 – Published 4 September 2015
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Abstract

We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may, however, not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. We note, however, that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.

  • Received 7 July 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.100402

© 2015 American Physical Society

Authors & Affiliations

Sheldon Goldstein1,*, David A. Huse2,†, Joel L. Lebowitz3,‡, and Roderich Tumulka4,§

  • 1Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA
  • 2Department of Physics, Princeton University, Jadwin Hall, Washington Road, Princeton, New Jersey 08544-0708, USA
  • 3Departments of Mathematics and Physics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA
  • 4Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA

  • *oldstein@math.rutgers.edu
  • huse@princeton.edu
  • lebowitz@math.rutgers.edu
  • §tumulka@math.rutgers.edu

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Issue

Vol. 115, Iss. 10 — 4 September 2015

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