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arXiv:2303.12330 · Journal of Statistical Physics

Boltzmann Entropy of a Freely Expanding Quantum Ideal Gas

Saurav Pandey, Junaid Majeed Bhat, Abhishek Dhar, Sheldon Goldstein, David A. Huse, Manas Kulkarni, Anupam Kundu, Joel L. Lebowitz

Published Aug 15, 2023• 14 citations
arXivHTMLPDF
DOI
10.1007/s10955-023-03154-y

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A recent work considered the Page curve for entanglement entropy of a freely expanding one-dimensional Fermi gas starting from a finite box (the system) in a zero-temperature initial state that then expands in to a semi-infinite vacuum (the environment). For the case with a single defect at the system-environment junction, it was shown that the Yang-Yang entropy of the system computed from semiclassical hydrodynamics agrees with the microscopically computed entanglement entropy, yielding the Page curve. On the other hand, the environment Yang-Yang entropy showed a monotonic increase. Here, we propose a hydrodynamic description to obtain the eigenvalues of the microscopic correlation matrix for a nonzero temperature freely-expanding Fermi gas. For the system, the phase space density directly gives the correlation matrix eigenvalues and thus, via the Yang-Yang formula, yields the von-Neumann entropy. For the reservoir, a modification of the phase space density is required in the Yang-Yang formula, to incorporate coherences, in order to reproduce accurately the entropy. The hydrodynamic description of correlation matrix eigenvalues also leads to a Levitov-Lesovik type formula for the full-counting-statistics for a finite system in the transient regime. We provide a heuristic derivation of the correlation matrix eigenvalues based on a semiclassical wave-packet picture.