On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems

Abstract

We consider a closed macroscopic quantum system in a pure state ψ evolving unitarily with Hamiltonian H in a finite-dimensional Hilbert space ℋ. Following von Neumann, we assume a decomposition of ℋ into mutually orthogonal subspaces ℋ_ν, where each ν represents a macro state, and ℋ_ν contains the pure states compatible with ν. We define the quantum Boltzmann entropy for a macro state ν as SqB(ν) = k_B log dim ℋ_ν, where kB is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that SqB predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of H and ℋ_ν.

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BibTeX RIS

BibTeX

@misc{igelspacher2026increase,
  title = {{On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems}},
  author = {Cedric Igelspacher and Roderich Tumulka},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18553},
  eprint = {2610.18553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18553},
}

RIS

TY  - UNPB
TI  - On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems
AU  - Cedric Igelspacher
AU  - Roderich Tumulka
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.18553
UR  - https://arxiv.org/abs/2610.18553
AB  - We consider a closed macroscopic quantum system in a pure state ψ evolving unitarily with Hamiltonian H in a finite-dimensional Hilbert space ℋ. Following von Neumann, we assume a decomposition of ℋ into mutually orthogonal subspaces ℋ_ν, where each ν represents a macro state, and ℋ_ν contains the pure states compatible with ν. We define the quantum Boltzmann entropy for a macro state ν as S_qB(ν) = k_B log dim ℋ_ν, where k_B is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that S_qB predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of H and ℋ_ν.
ER  -

Abstract

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