In this short review we propose a critical assessment of the role of chaos for the thermalization of Hamiltonian systems with high dimensionality. We discuss this problem for both classical and quantum systems. A comparison is made between the two situations: some examples from recent and past literature are presented which support the point of view that chaos is not necessary for thermalization. Finally, we suggest that a close analogy holds between the role played by Kinchin's theorem for high-dimensional classical systems and the role played by Von Neumann's theorem for many-body quantum systems.
In this short review we propose a critical assessment of the role of chaos for the thermalization of Hamiltonian systems with high dimensionality. We discuss this problem for both classical and quantum systems. A comparison is made between the two situations: some examples from recent and past literature are presented which support the point of view that chaos is not necessary for thermalization. Finally, we suggest that a close analogy holds between the role played by Kinchin's theorem for high-dimensional classical systems and the role played by Von Neumann's theorem for many-body quantum systems.
In their seminal work, Fermi, Pasta, Ulam and Tsingou explored the connection between statistical mechanics and dynamical properties, such as chaos and ergodicity. Even today, seventy years later, the topic is not fully understood: while most results of statistical mechanics require the ergodic hypothesis to be rigorously proved, there are many indications that these predictions, both in and out of equilibrium, hold even in the absence of a rigorous form of ergodicity. Motivated by the above considerations, in this work we reconsider the point of view that the relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom and the choice of extensive observables, while the details of the dynamics do not play an essential role. This is the idea behind Khinchin's famous proof of the typicality of macroscopic observables at equilibrium. We extend this perspective to the context of non equilibrium, by investigating the thermalization properties of both harmonic (integrable) and nonharmonic (chaotic) oscillator chains initially prepared in out-of-equilibrium conditions. In integrable systems, thermalization occurs, or not, depending on the observable. In the chaotic regime, instead, thermalization is reached by any observable, although the relaxation timescale might be larger than the observation time.
Abstract Classical arguments for thermalization of isolated systems do not apply in a straightforward way to the quantum case. Recently, there has been interest in diagnostics of quantum chaos in many-body systems. In the classical case, chaos is a popular explanation for the legitimacy of the methods of statistical physics. In this work, we relate a previously proposed criteria of quantum chaos in the unitary time evolution operator to the entanglement entropy growth for a far-from-equilibrium initial pure state. By mapping the unitary time evolution operator to a doubled state, chaos can be characterized by suppression of mutual information between subsystems of the past and that of the future. We show that when this mutual information is small, a typical unentangled initial state will evolve to a highly entangled final state. Our result provides a more concrete connection between quantum chaos and thermalization in many-body systems.
We will discuss various aspects of thermalization, chaos and hydrodynamics in one dimensional classical Hamiltonian systems. We study two problems. First, we will revisit the Fermi-Pasta-Ulam-Tsingou (FPUT) problem in order to understand what thermalization is and discuss what leads the system starting from a typical initial condition to a thermal state. Here we discuss the various possibilities like the role of averaging, the choice of observables, the choice of initial conditions and the role of chaos in thermalization. Then, we study the evolution of a blast wave in an alternate mass hard particle (AHP) gas and study the evolution of conserved fields in the system at short times. We find a good agreement with the Taylor-von Neumann-Sedov (TvNS) solution, which was studied during the Second World War in the context of atomic explosions, everywhere except near the core of the blast. We then model this behaviour by using the Navier-Stokes-Fourier (NSF) equations.
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.
Many-body quantum chaos results from interparticle interactions in mesoscopic systems at sufficiently high level density. A short review shows that, going beyond generic properties of local spectral statistics, the ideas of many-body quantum chaos serve as a rich source of new theoretical approaches, computational and experimental tools. Chosen examples include the enhancement of weak perturbations, thermalization in finite systems, exponential convergence of large Hamiltonian matrices and correlations between the states of different symmetry.
We provide a pedagogical introduction to eigenstate thermalization. This phenomenon, which occurs in generic quantum many-body systems, allows one to understand why thermalization takes place in isolated systems under unitary dynamics. We motivate eigenstate thermalization using random matrix theory and discuss recent complementary results for the volume-law entanglement entropy of Haar-random states. We discuss numerical results that highlight the corresponding behaviors in quantum many-body systems.