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 $\mathscr{H}$. Following von Neumann, we assume a decomposition of $\mathscr{H}$ into mutually orthogonal subspaces $\mathscr{H}<em>ν$, where each $ν$ represents a macro state, and $\mathscr{H}</em>ν$ contains the pure states compatible with $ν$. We define the quantum Boltzmann entropy for a macro state $ν$ as $$S_\mathrm{qB}(ν) = k_{\mathrm{B}} \log \dim \mathscr{H}<em>ν$$, where $k</em>\mathrm{B}$ 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 $\mathscr{H}_ν$.