Estimation of equilibration time scales from nested fraction approximations
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914925156237312 |
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| author | Bartsch, Christian Dymarsky, Anatoly Lamann, Mats H. Wang, Jiaozi Steinigeweg, Robin Gemmer, Jochen |
| author_facet | Bartsch, Christian Dymarsky, Anatoly Lamann, Mats H. Wang, Jiaozi Steinigeweg, Robin Gemmer, Jochen |
| contents | We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients, or solving a system of coupled differential equations in the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function, and use it to estimate the equilibration time $τ$ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of $τ$, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a new practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_12822 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Estimation of equilibration time scales from nested fraction approximations Bartsch, Christian Dymarsky, Anatoly Lamann, Mats H. Wang, Jiaozi Steinigeweg, Robin Gemmer, Jochen Statistical Mechanics High Energy Physics - Theory We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients, or solving a system of coupled differential equations in the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function, and use it to estimate the equilibration time $τ$ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of $τ$, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a new practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult. |
| title | Estimation of equilibration time scales from nested fraction approximations |
| topic | Statistical Mechanics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2308.12822 |