From Quantum Chaos and Eigenstate Thermalization to Statistical
Advances in Physics,2016 Vol. 65, No. 3, 239–362, http://dx.doi.org/10.1080/00018732.2016.1198134 REVIEW ARTICLE From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics a,b c b a Luca D’Alessio ,YarivKafri,AnatoliPolkovnikov and Marcos Rigol ∗ aDepartment of Physics, The Pennsylvania State University, University Park, PA 16802, USA; bDepartment of Physics, Boston University, Boston, MA 02215, USA; cDepartment of Physics, Technion, Haifa 32000, Israel (Received 22 September 2015; accepted 2 June 2016) 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 ran- dom 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. Build- ing 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 equi- librium, 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 fun- damental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations.
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