We show that quantum chaotic many-body systems possess the thermodynamic arrow of time in the thermodynamic limit. Berry's conjecture in quantum chaotic systems and equivalence of ensembles imply the Kelvin statement of the second law of thermodynamics at leading order in perturbation theory. We verify this result using numerical calculations. We also show that this gives rise to new constraints on the off-diagonal terms in eigenstate thermalization hypothesis (ETH) statement. We call the new constraints collectively as ETH-monotonicity. These constraints arise because pure entropic consideration is not enough for the emergence of the thermodynamic arrow of time.
We show that quantum chaotic many-body systems possess the thermodynamic arrow of time in the thermodynamic limit. Berry's conjecture in quantum chaotic systems and equivalence of ensembles imply the Kelvin statement of the second law of thermodynamics at leading order in perturbation theory. We verify this result using numerical calculations. We also show that this gives rise to new constraints on the off-diagonal terms in eigenstate thermalization hypothesis (ETH) statement. We call the new constraints collectively as ETH-monotonicity. These constraints arise because pure entropic consideration is not enough for the emergence of the thermodynamic arrow of time.
A profound quest of statistical mechanics is the origin of irreversibility - the arrow of time. New stimulants have been provided, thanks to unprecedented degree of control reached in experiments with isolated quantum systems and rapid theoretical developments of manybody localization in disordered interacting systems. The proposal of (many-body) eigenstate thermalization (ET) for these systems reinforces the common belief that either interaction or extrinsic randomness is required for thermalization. Here, we unveil a quantum thermalization mechanism challenging this belief. We find that, provided one-body quantum chaos is present, as a pure many-body state evolves the arrow of time can emerge, even without interaction or randomness. In times much larger than the Ehrenfest time that signals the breakdown of quantum-classical correspondence, quantum chaotic motion leads to thermal [Fermi-Dirac (FD) or Bose-Einstein (BE)] distributions and thermodynamics in individual eigenstates. Our findings lay dynamical foundation of statistical mechanics and thermodynamics of isolated quantum systems.
The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds simultaneously for all m-body operators with m<α∗N in the thermodynamic limit for some nonzero constant α∗>0. We first show the existence of such nontrivial constants for idealized (pseudo) random-matrix descriptions of many-body eigenstates. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH simultaneously for all few-body operators, including their thermal and quantum fluctuations.
We study the off-diagonal matrix elements of real-space observables in time-reversal-invariant single-particle chaotic systems. By analyzing the semiclassical expression for the off-diagonal variance derived from Berry's conjecture, we show that the banded structure of the observable matrix emerges naturally. For local observables, we identify a characteristic bandwidth associated with a late-time timescale inversely proportional to the particle velocity. We further show that, for systems with steep-wall confinement, the predicted magnitude follows the entropy scaling of the eigenstate thermalization hypothesis (ETH), multiplied by an additional kinetic-energy-dependent factor that is independent of spatial dimension and is not captured by conventional many-body ETH. We illustrate these results through a case study of quantum billiards and verify the semiclassical scaling numerically in a generalized quarter-Sinai billiard. Our results elucidate the dynamical implications of Berry's conjecture and provide a comparison between single-particle eigenstate thermalization and many-body ETH.
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.
We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were Gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon eigenstate thermalization. We show that a generic initial state will approach thermal equilibrium at least as fast as O(\ensuremath{\Elzxh}/\ensuremath{\Delta})${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\Delta} is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a sound foundation for quantum statistical mechanics.
Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies which, we conjecture, applies to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system-a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio V_{A}/V, while the Renyi entropy densities depend nonlinearly on V_{A}/V. Surprisingly, Renyi entropies S_{n} for n>1 are convex functions of the subsystem size, with a volume law coefficient that depends on V_{A}/V, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum-mechanical systems, and is closely related to the eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform an exact diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions.