Abstract This work explores fundamental statistical and thermodynamic properties of short-and long-range-interacting systems. The purpose of this study is twofold. Firstly, we rigorously prove that the probability distribution of arbitrary few-body observables is restricted by a Gaussian concentration bound (or Chernoff–Hoeffding inequality) above some threshold temperature. This bound is then derived for arbitrary Gibbs states of systems that include long-range interactions Secondly, we establish a quantitative relationship between the concentration bound of the Gibbs state and the equivalence of canonical and micro-canonical ensembles. We then evaluate the difference in the averages of thermodynamic properties between the canonical and the micro-canonical ensembles. Under the assumption of the Gaussian concentration bound on the canonical ensemble, the difference between the ensemble descriptions is upper-bounded by n − 1 log ( n 3 ∕ 2 Δ − 1 ) 1 ∕ 2 with n being the system size and Δ being the width of the energy shell of the micro-canonical ensemble This limit gives a non-trivial upper bound exponentially small energy width with respect to the system size. By combining these two results, we prove the ensemble equivalence as well as the weak eigenstate thermalization in arbitrary long-range-interacting systems above a threshold temperature.

中文翻译：

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包括长程相互作用在内的通用量子多体系统中的高斯浓度界和集合等价

摘要 这项工作探索了短程和长程相互作用系统的基本统计和热力学特性。这项研究的目的是双重的。首先，我们严格证明任意少体可观测量的概率分布受高于某个阈值温度的高斯浓度界（或 Chernoff-Hoeffding 不等式）的限制。然后为包含长程相互作用的系统的任意吉布斯状态推导出该界限。其次，我们在吉布斯状态的浓度界限与正则和微正则系综的等价性之间建立了定量关系。然后，我们评估规范和微规范集合之间热力学特性平均值的差异。在正则系综上的高斯集中界的假设下，系综描述之间的差异上限为 n − 1 log ( n 3 ∕ 2 Δ − 1 ) 1 ∕ 2 其中 n 是系统大小，Δ 是微正则系综的能量壳的宽度这个限制给出了一个非平凡的上限，相对于系统大小呈指数级小能量宽度。通过结合这两个结果，我们证明了在阈值温度以上的任意远程相互作用系统中的系综等价性以及弱本征态热化。系综描述之间的差异上限为 n − 1 log ( n 3 ∕ 2 Δ − 1 ) 1 ∕ 2 其中 n 是系统大小，Δ 是微正则系综的能量壳的宽度 这个限制给出相对于系统大小呈指数级小能量宽度的非平凡上限。通过结合这两个结果，我们证明了在阈值温度以上的任意远程相互作用系统中的系综等价性以及弱本征态热化。系综描述之间的差异上限为 n − 1 log ( n 3 ∕ 2 Δ − 1 ) 1 ∕ 2 其中 n 是系统大小，Δ 是微正则系综的能量壳的宽度 这个限制给出相对于系统大小呈指数级小能量宽度的非平凡上限。通过结合这两个结果，我们证明了在阈值温度以上的任意远程相互作用系统中的系综等价性以及弱本征态热化。