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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. 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. PACS: 03.65.-w; 05.30.-d; 05.45.Mt; 05.70.-a Keywords: quantum statistical mechanics; eigenstate thermalization; quantum chaos; random matrix theory; quantum quench; quantum thermodynamics; generalized Gibbs ensemble Downloaded by [Northern Illinois University] at 04:12 01 August 2016 Contents PAGE 1Introduction 241 2Chaosandrandommatrixtheory(RMT) 244 2.1 Classical chaos 244 2.2 Random matrix theory 246 2.2.1 Chaotic eigenfunctions 249 2.2.2 The structure of the matrix elements of operators 250 2.3 Berry–Tabor conjecture 251 2.4 The semi-classical limit and Berry’s conjecture 253 *Corresponding author. Email: [email protected] c 2016 Informa UK Limited, trading as Taylor & Francis Group ⃝ 240 L. D’Alessio et al. 3Quantumchaosinphysicalsystems 255 3.1 Examples of Wigner–Dyson and Poisson statistics 255 3.2 The structure of many-body eigenstates 259 3.3 Quantum chaos and entanglement 261 3.4 Quantum chaos and delocalization in energy space 265 4Eigenstatethermalization 269 4.1 Thermalization in quantum systems 269 4.2 The eigenstate thermalization hypothesis (ETH) 271 4.2.1 ETH and thermalization 272 4.2.2 ETH and the quantum ergodic theorem 274 4.3 Numerical experiments in lattice systems 275 4.3.1 Eigenstate thermalization 275 4.3.1.1 Diagonal matrix elements 276 4.3.1.2 Off-diagonal matrix elements 277 4.3.2 Quantum quenches and thermalization in lattice systems 281 4.3.2.1 Dynamics 282 4.3.2.2 Post relaxation 283 5Quantumchaosandthelawsofthermodynamics 284 5.1 General setup and doubly stochastic evolution 284 5.1.1 Properties of master equations and doubly stochastic evolution 287 5.2 General implications of doubly stochastic evolution 291 5.2.1 The infinite temperature state as an attractor of doubly stochastic evolution 291 5.2.2 Increase of the diagonal entropy under doubly stochastic evolution 292 5.2.3 The second law in the Kelvin formulation for passive density matrices 294 5.3 Implications of doubly stochastic evolution for chaotic systems 295 5.3.1 The diagonal entropy and the fundamental thermodynamic relation 295 5.3.2 The fundamental relation vs. the first law of thermodynamics 298 6Quantumchaos,fluctuationtheorems,andlinearresponserelations 300 6.1 Fluctuation theorems 300 6.1.1 Fluctuation theorems for systems starting from a Gibbs state 302 6.1.2 Fluctuation theorems for quantum chaotic systems 303 6.2 Detailed balance for open systems 305 6.3 Einstein’s energy drift–diffusion relations for isolated and open systems 307 6.4 Fokker–Planck equation for the heating of a driven isolated system 309 6.5 Fluctuation theorems for two (or more) conserved quantities 310 6.6 Linear response and Onsager relations 312 Downloaded by [Northern Illinois University] at 04:12 01 August 2016 6.7 Nonlinear response coefficients 313 6.8 ETH and the fluctuation–dissipation relation for a single eigenstate 314 6.9 ETH and two-observable correlation functions 318 7ApplicationofEinstein’srelationtocontinuouslydrivensystems 319 7.1 Heating a particle in a fluctuating chaotic cavity 320 7.2 Driven harmonic system and a phase transition in the distribution function 325 7.3 Two equilibrating systems 328 8IntegrablemodelsandthegeneralizedGibbsensemble(GGE) 329 8.1 Constrained equilibrium: the GGE 330 8.1.1 Noninteracting spinless fermions 331 8.1.2 Hard-core bosons 332 8.2 Generalized eigenstate thermalization 334 8.2.1 Truncated GGE for the transverse field Ising model 336 Advances in Physics 241 8.3 Quenches in the XXZ model 337 8.4 Relaxation of weakly nonintegrable systems: prethermalization and quantum kinetic equations 340 Acknowledgments 344 Disclosure statement 344 Funding 344 Notes 344 References 346 Appendix A. The kicked rotor 353 Appendix B. Zeros of the Riemann zeta function 355 Appendix C. The infinite temperature state as an attractor 357 Appendix D. Birkhoff’s theorem and doubly stochastic evolution 358 Appendix E. Proof of W 0forpassivedensitymatricesanddoublystochasticevolution 359 Appendix F. Derivation⟨ of⟩≥ the drift–diffusion relation for continuous processes 361 Appendix G. Derivation of Onsager relations 362 Verily at the first Chaos came to be, but next wide-bosomed Earth, the ever-sure foundations of all. Hesiod, Theogony 1. Introduction Despite the huge success of statistical mechanics in describing the macroscopic behavior of phys- ical systems [1,2], its relation to the underlying microscopic dynamics has remained a subject of debate since the foundations were laid [3,4]. One of the most controversial topics has been the reconciliation of the time reversibility of most microscopic laws of nature and the apparent irreversibility of the laws of thermodynamics. Let us first consider an isolated classical system subject to some macroscopic constraints (such as conservation of the total energy and confinement to a container). To derive its equilib- rium properties, within statistical mechanics, one takes a fictitious ensemble of systems evolving under the same Hamiltonian and subject to the same macroscopic constraints. Then, a probabil- ity is assigned to each member of the ensemble, and the macroscopic behavior of the system is computed by averaging over the fictitious ensemble [5]. For an isolated system, the ensemble is typically chosen to be the microcanonical one. To ensure that the probability of each configu- ration in phase space does not change in time under the Hamiltonian dynamics, as required by Downloaded by [Northern Illinois University] at 04:12 01 August 2016 equilibrium, the ensemble includes, with equal probability, all configurations compatible with the macroscopic constraints. The correctness of the procedure used in statistical mechanics to describe real systems is, however, far from obvious. In actual experiments, there is generally no ensemble of systems – there is one system – and the relation between the calculation just outlined and the measurable outcome of the underlying microscopic dynamics is often unclear. To address this issue, two major lines of thought have been offered. In the first line of thought, which is found in most textbooks, one invokes the ergodic hypoth- esis [6](refinementssuchasmixingarealsoinvoked[7]). This hypothesis states that during its time evolution an ergodic system visits every region in phase space (subjected to the macroscopic constraints) and that, in the long-time limit, the time spent in each region is proportional to its volume. Time averages can then be said to be equal to ensemble averages, and the latter are the ones that are ultimately computed [6]. The ergodic hypothesis essentially implies that the “equal probability” assumption used to build the microcanonical ensemble is the necessary ingredient to 242 L. D’Alessio et al. capture the long-time average of observables. This hypothesis has been proved for a few systems, such as the Sinai billiard [8,9], the Bunimovich stadium [10], and systems with more
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