This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.

中文翻译：

---

从量子混沌和本征态热化到统计力学和热力学

本综述对本征态热化假说 (ETH)、其基础及其对统计力学和热力学的影响进行了教学性介绍。在第一部分中，ETH 作为量子混沌和随机矩阵理论 (RMT) 思想的自然延伸而被介绍。为此，我们简要概述了经典和量子混沌，以及 RMT 及其一些最重要的预测。后者包括能级统计、本征态分量和可观测的矩阵元素。在这些基础上，我们介绍了 ETH，并表明它允许人们在不调用外部浴的概念的情况下描述孤立的混沌系统中的热化。我们从多体晶格系统的研究中检查了本征态热化的数值证据。我们还介绍了猝灭的概念，作为使孤立系统脱离平衡的一种手段，并讨论了量子猝灭的数值实验结果。评论的第二部分探讨了量子混沌和 ETH 对热力学的影响。推导出基本的热力学关系，包括热力学第二定律、基本热力学关系、涨落定理、涨落-耗散关系以及爱因斯坦和昂萨格关系。特别是，它表明量子混沌可以证明单个哈密顿本征态的这些关系，从而将它们扩展到任意的平稳统计系综。在某些情况下，可以将其适用范围扩展到标准热平衡域之外。然后我们展示了如何使用这些关系在连续驱动的系统中获得非平凡的通用能量分布。在评论的最后，我们简要讨论了可积量子系统弛豫后的弛豫动力学和描述，为此违反了 ETH。我们展示了可积性量子猝灭的数值实验和分析研究的结果。我们介绍了广义 Gibbs 系综的概念，并讨论了它与弱相互作用系统中预热化思想的联系。我们展示了可积性量子猝灭的数值实验和分析研究的结果。我们介绍了广义 Gibbs 系综的概念，并讨论了它与弱相互作用系统中预热化思想的联系。我们展示了可积性量子猝灭的数值实验和分析研究的结果。我们介绍了广义 Gibbs 系综的概念，并讨论了它与弱相互作用系统中预热化思想的联系。