On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems

Abstract:We consider a closed macroscopic quantum system in a pure state $\psi$ evolving unitarily with Hamiltonian $H$ in a finite-dimensional Hilbert space $\mathscr{H}$. Following von Neumann, we assume a decomposition of $\mathscr{H}$ into mutually orthogonal subspaces $\mathscr{H}_{\nu}$, where each $\nu$ represents a macro state, and $\mathscr{H}_\nu$ contains the pure states compatible with $\nu$. We define the quantum Boltzmann entropy for a macro state $\nu$ as $S_\mathrm{qB}(\nu) = k_{\mathrm{B}} \log \dim \mathscr{H}_{\nu}$, where $k_\mathrm{B}$ 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 $S_\mathrm{qB}$ 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 $\mathscr{H}_\nu$.

arXiv:2610.18553 [quant-ph]

(or arXiv:2610.18553v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.18553 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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