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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.12165 |
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| _version_ | 1866917644097028096 |
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| author | Weidenmüller, Hans A. |
| author_facet | Weidenmüller, Hans A. |
| contents | An isolated quantum system is said to thermalize if ${\rm Tr} (A ρ(t)) \to {\rm Tr} (A ρ_{\rm eq})$ for time $t \to \infty$. Here $ρ(t)$ is the time-dependent density matrix of the system, $ρ_{\rm eq}$ is the time-independent density matrix that describes statistical equilibrium, and $A$ is a Hermitean operator standing for an observable. We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension $N$), all functions ${\rm Tr} (A ρ(t))$ in the ensemble thermalize: For $N \to \infty$ every such function tends to the value ${\rm Tr} (A ρ_{\rm eq}(\infty)) + {\rm Tr} (A ρ(0)) g^2(t)$. Here $ρ_{\rm
eq}(\infty)$ is the equilibrium density matrix at infinite temperature. The oscillatory function $g(t)$ is the Fourier transform of the average GOE level density and falls off as $1 / |t|$ for large $t$. With $g(t) = g(-t)$, thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble (GUE) of random matrices. Comparison with the ``eigenstate thermalization hypothesis'' of Ref.~\cite{Sre99} shows overall agreement but raises significant questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_12165 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Random-Matrix Model for Thermalization Weidenmüller, Hans A. Quantum Physics An isolated quantum system is said to thermalize if ${\rm Tr} (A ρ(t)) \to {\rm Tr} (A ρ_{\rm eq})$ for time $t \to \infty$. Here $ρ(t)$ is the time-dependent density matrix of the system, $ρ_{\rm eq}$ is the time-independent density matrix that describes statistical equilibrium, and $A$ is a Hermitean operator standing for an observable. We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension $N$), all functions ${\rm Tr} (A ρ(t))$ in the ensemble thermalize: For $N \to \infty$ every such function tends to the value ${\rm Tr} (A ρ_{\rm eq}(\infty)) + {\rm Tr} (A ρ(0)) g^2(t)$. Here $ρ_{\rm eq}(\infty)$ is the equilibrium density matrix at infinite temperature. The oscillatory function $g(t)$ is the Fourier transform of the average GOE level density and falls off as $1 / |t|$ for large $t$. With $g(t) = g(-t)$, thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble (GUE) of random matrices. Comparison with the ``eigenstate thermalization hypothesis'' of Ref.~\cite{Sre99} shows overall agreement but raises significant questions. |
| title | Random-Matrix Model for Thermalization |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2211.12165 |