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Physical review. E

Chaotic route to classical thermalization: A real-space analysis

Yue Liu, Dahai He

Published Jun 6, 2024• 3 citations
Publisher

Abstract

Most of the previous studies on classical thermalization focus on the wave-vector space, encountering limitations when extended beyond quasi-integrable regions. In this study, we propose a scheme to study the thermalization of the classical Hamiltonian chain of interacting oscillators in real space by developing a thermalization indicator proposed by Parisi [Europhys. Lett. 40, 357 (1997)0295-507510.1209/epl/i1997-00471-9], which approaches zero in the thermal state. Upon reaching the steady state characterized by the generalized Gibbs ensemble for a harmonic chain, a quench protocol is implemented to change the Hamiltonian to a nonintegrable form instantaneously, thereby preparing nonequilibrium initial states. This approach enables investigations of thermalization in real space, particularly valuable for exploring regions beyond quasi-integrability. For the FPUT-β lattice, we observe that the thermalization time as a function of the nonintegrable strength follows a -2 scaling law in the quasi-integrable region and -1/4 in the strongly integrable region. Moreover, numerical results reveal the thermalization time is proportional to the Lyapunov time, which bridges microscopic chaotic dynamics and the macroscopic thermalization process.

DOI
10.1103/physreve.109.064115

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Most of the previous studies on classical thermalization focus on the wave-vector space, encountering limitations when extended beyond quasi-integrable regions. In this study, we propose a scheme to study the thermalization of the classical Hamiltonian chain of interacting oscillators in real space by developing a thermalization indicator proposed by Parisi [Europhys. Lett. 40, 357 (1997)0295-507510.1209/epl/i1997-00471-9], which approaches zero in the thermal state. Upon reaching the steady state characterized by the generalized Gibbs ensemble for a harmonic chain, a quench protocol is implemented to change the Hamiltonian to a nonintegrable form instantaneously, thereby preparing nonequilibrium initial states. This approach enables investigations of thermalization in real space, particularly valuable for exploring regions beyond quasi-integrability. For the FPUT-β lattice, we observe that the thermalization time as a function of the nonintegrable strength follows a -2 scaling law in the quasi-integrable region and -1/4 in the strongly integrable region. Moreover, numerical results reveal the thermalization time is proportional to the Lyapunov time, which bridges microscopic chaotic dynamics and the macroscopic thermalization process.
We devote our studies to the subject of weakly nonintegrable dynamics of systems with a macroscopic number of degrees of freedom. Our main points of interest are the relations between the timescales of thermalization and the timescales of chaotization; the choice of appropriate observables and the structure of equations coupling them; identifying the classes of weakly nonintegrable dynamics and developing tools to diagnose the properties specific to such classes. We discuss the traditional in the field methods for thermalization timescale computation and employ them to study the scaling the timescale with the proximity to the integrable limit. We then elaborate on a novel framework based on the full Lyapunov spectra computation for large systems as a powerful tool for the characterization of weak nonintegrability. In particular, the Lyapunov spectrum scaling offers a quantitative description allowing us to infer the structure of the underlying network of observables. Proximity to integrable limit is associated with the rapid growth of thermalization timescales and, thus, potential numerical challenges. We solve these challenges by performing numerical tests using computationally efficient model - unitary maps. The great advantage of unitary maps for numerical applications is time-discrete error-free evolution. We use these advantages to perform large timescale and system size computations in extreme proximity to the integrable limit. To demonstrate the scope of obtained results we report on the application of the developed framework to Hamiltonian 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.
Nonintegrability is a necessary condition for the thermalization of a generic Hamiltonian system. In practice, the integrability can be broken in various ways. As illustrating examples, we numerically studied the thermalization behaviors of two types of one-dimensional (1D) diatomic chains in the thermodynamic limit. One chain was the diatomic Toda chain whose nonintegrability was introduced by unequal masses. The other chain was the diatomic Fermi-Pasta-Ulam-Tsingou-β chain whose nonintegrability was introduced by quartic nonlinear interaction. We found that these two different methods of destroying the integrability led to qualitatively different routes to thermalization in the near-integrable region, but the thermalization time, T_{eq}, followed the same scaling law; T_{eq} was inversely proportional to the square of the perturbation strength. This law also agreed with the existing results of 1D monatomic lattices. All these results imply that there is a universal scaling law of thermalization that is independent of the method of breaking integrability.
After a quench, observables in an integrable system may not relax to the standard thermal values, but can relax to the ones predicted by the generalized Gibbs ensemble (GGE) [M. Rigol et al., Phys. Rev. Lett. 98, 050405 (2007)]. The GGE has been shown to accurately describe observables in various one-dimensional integrable systems, but the origin of its success is not fully understood. Here we introduce a microcanonical version of the GGE and provide a justification of the GGE based on a generalized interpretation of the eigenstate thermalization hypothesis, which was previously introduced to explain thermalization of nonintegrable systems. We study relaxation after a quench of one-dimensional hard-core bosons in an optical lattice. Exact numerical calculations for up to 10 particles on 50 lattice sites (≈10(10) eigenstates) validate our approach.
Thermalization (generalized thermalization) in nonintegrable (integrable) quantum systems requires two ingredients: equilibration and agreement with the predictions of the Gibbs (generalized Gibbs) ensemble. We prove that observables that exhibit eigenstate thermalization in single-particle sector equilibrate in many-body sectors of quantum-chaotic quadratic models. Remarkably, the same observables do not exhibit eigenstate thermalization in many-body sectors (we establish that there are exponentially many outliers). Hence, the generalized Gibbs ensemble is generally needed to describe their expectation values after equilibration, and it is characterized by Lagrange multipliers that are smooth functions of single-particle energies.
We study sudden quantum quenches in which the initial states are selected to be either eigenstates of an integrable Hamiltonian that is nonmappable to a noninteracting one or a nonintegrable Hamiltonian, while the Hamiltonian after the quench is always integrable and mappable to a noninteracting one. By studying weighted energy densities and entropies, we show that quenches starting from nonintegrable (chaotic) eigenstates lead to an ``ergodic'' sampling of the eigenstates of the final Hamiltonian, while those starting from the integrable eigenstates do not (or at least it is not apparent for the system sizes accessible to us). This goes in parallel with the fact that the distribution of conserved quantities in the initial states is thermal in the nonintegrable cases and nonthermal in the integrable ones, and means that, in general, thermalization occurs in integrable systems when the quench starts form an eigenstate of a nonintegrable Hamiltonian (away from the edges of the spectrum), while it fails (or requires larger system sizes than those studied here to become apparent) for quenches starting at integrable points. We test those conclusions by studying the momentum distribution function of hard-core bosons after a quench.