Abstract A study of quantum-classical correspondence for the onset of thermalization in an isolated system of four interacting particles is presented. Provided that classical dynamics of the system is chaotic, thermalization is present and fits the expected statistical mechanics results. Quantum classical correspondence is established for the classical analogue of the single particle distributions of occupation numbers. The latter quantity is useful to test thermalization and it allows to obtain the microcanonical Boltzmann factor in the quantum and semi-classical cases. The correspondence between them and the statistical mechanics results are surprisingly good considering that we have only four interacting particles. The resulting Boltzmann factor for the occupation number distribution depends on the eigenfunction width, which is a semi-classical parameter explicitly calculated and consistent with the quantum one.
Abstract A study of quantum-classical correspondence for the onset of thermalization in an isolated system of four interacting particles is presented. Provided that classical dynamics of the system is chaotic, thermalization is present and fits the expected statistical mechanics results. Quantum classical correspondence is established for the classical analogue of the single particle distributions of occupation numbers. The latter quantity is useful to test thermalization and it allows to obtain the microcanonical Boltzmann factor in the quantum and semi-classical cases. The correspondence between them and the statistical mechanics results are surprisingly good considering that we have only four interacting particles. The resulting Boltzmann factor for the occupation number distribution depends on the eigenfunction width, which is a semi-classical parameter explicitly calculated and consistent with the quantum one.
In this review the problem of statistical description of isolated quantum systems of interacting particles is discussed. Main attention is paid to a recently developed approach which is based on chaotic properties of compound states in the basis of non-interacting particles. In order to demonstrate the most important aspects of this approach, the matrix model of two-body random interaction between Fermi-particles has been used. Different problems have been considered such as the onset of chaos and statistical equilibrium, the relation between the structure of eigenstates and distribution of occupation numbers, the emergence of the Fermi-Dirac distribution in isolated systems of finite number of particles and many others. The application of the approach to dynamical systems with the classical limit is discussed as well.
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.
The occupation number is a key observable for diagnosing thermalization, as it connects directly to standard statistical laws such as Fermi--Dirac, Bose--Einstein, and Boltzmann distributions. In the context of spin systems, it represents the population of the sublevels of the magnetization in the z-direction. We use this quantity to probe the onset of thermalization in the isolated quantum and classical one-dimensional spin-1 Ising model with transverse and longitudinal fields. Thermalization is achieved when the long-time average of the occupation number converges to the microcanonical prediction as the chain length L increases, consistent with the emergence of ergodicity. However, the finite-size scaling analysis in the quantum model is challenged by the exponential growth of the Hilbert space with L. To overcome this limitation, we turn to the corresponding classical model, which enables access to much larger system sizes. By tracking the dynamics of individual spins on their three-dimensional Bloch spheres and employing tools from random matrix theory, we establish a quantitative criterion for classical ergodicity in interacting spin systems. We find that deviations from classical ergodicity decay algebraically with system size. This power-law scaling then provides a quantitative bound on the approach to thermal equilibrium in the quantum model.
The occupation number is a key observable for diagnosing thermalization, as it connects directly to standard statistical laws such as Fermi-Dirac, Bose-Einstein, and Boltzmann distributions. In the context of spin systems, it represents the population of the sublevels of the magnetization in the z direction. We use this quantity to probe the onset of thermalization in an isolated one-dimensional quantum spin-1 Ising model with transverse and longitudinal fields and in its classical counterpart. Thermalization is achieved when the long-time average of the occupation number converges to the microcanonical prediction as the chain length L increases, consistent with the emergence of ergodicity. However, the finite-size scaling analysis in the quantum model is challenged by the exponential growth of the Hilbert space with L. To overcome this limitation, we turn to the classical model, which enables access to much larger system sizes. By tracking the dynamics of individual spins on their three-dimensional Bloch spheres and employing tools from random matrix theory, we establish a quantitative criterion for classical ergodicity in interacting spin systems. We find that deviations from classical ergodicity decay algebraically with system size. This power-law scaling then provides a quantitative bound on the approach to thermal equilibrium in the quantum model.
The onset of thermalization in a closed system of randomly interacting bosons at the level of a single eigenstate is discussed. We focus on the emergence of Bose-Einstein distribution of single-particle occupation numbers, and we give a local criterion for thermalization dependent on the eigenstate energy. We show how to define the temperature of an eigenstate, provided that it has a chaotic structure in the basis defined by the single-particle states. The analytical expression for the eigenstate temperature as a function of both interparticle interaction and energy is complemented by numerical data.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.