Equilibration of quasi-integrable systems

A dynamical system approach to relaxation in glass-forming liquids

The "classical" thermodynamic and statistical mechanical theories of Gibbs and Boltzmann are both predicated on axiomatic assumptions whose applicability is hard to ascertain. Theoretical objections and an increasing number of observed deviations from these theories have led to sustained efforts to develop an improved mathematical and physical foundation for them, and the search for appropriate extensions that are generally applicable to condensed materials at low temperatures (T) and high material densities where the assumptions of these theories start to become particularly questionable. These theoretical efforts have largely focused on minimal models of condensed material systems, such as the Fermi-Ulam-Pasta-Tsingou model, and other simplified models of condensed materials that are amenable to numerical and analytic treatments and that can serve to illuminate essential features of relaxation processes in condensed materials under conditions approaching integrable dynamics where clear departures from classical thermodynamics and dynamics can be generally expected. These studies indicate an apparently general multi-step relaxation process, corresponding to an initial "fast" relaxation process (termed the fast β-relaxation in the context of cooled liquids), followed by a longer "equipartition time", namely, the α-relaxation time τα in the context of cooled liquids. This relaxation timescale can be enormously longer than the fast β-relaxation time τβ so that τα is the primary parameter governing the rate at which the material comes into equilibrium, and thus is a natural focus of theoretical attention. Since the dynamics of these simplified dynamical systems, originally intended as simplified models of real crystalline materials exhibiting anharmonic interactions, greatly resemble the observed relaxation dynamics of both heated crystals and cooled liquids, we adapt this dynamical system approach to the practical matter of estimating relaxation times in both cooled liquids and crystals at elevated temperatures, which we identify as weakly non-integrable dynamical systems.

On the timescales in the chaotic dynamics of a 4D symplectic map

In this work, we investigate different timescales of chaotic dynamics in a multi-parametric 4D symplectic map. We compute the Lyapunov time and a macroscopic timescale, the instability time, for a wide range of values of the system's parameters and many different ensembles of initial conditions in resonant domains. The instability time is obtained by plain numerical simulations and by its estimates from the diffusion time, which we derive in three different ways: through a normal and an anomalous diffusion law and by the Shannon entropy, whose formulation is briefly revisited. A discussion about which of the four approaches provide reliable values of the timescale for a macroscopic instability is addressed. The relationship between the Lyapunov time and the instability time is revisited and studied for this particular system where in some cases, an exponential or polynomial law has been observed. The main conclusion of the present research is that only when the dynamical system behaves as a nearly ergodic one such relationship arises and the Lyapunov and instability times are global timescales, independent of the position in phase space. When stability regions prevent the free diffusion, no correlations between both timescales are observed, they are local and depend on both the position in phase space and the perturbation strength. In any case, the instability time largely exceeds the Lyapunov time. Thus, when the system is far from nearly ergodic, the timescale for predictable dynamics is given by the instability time, being the Lyapunov time its lower bound.

Unusual ergodic and chaotic properties of trapped hard rods

We investigate ergodicity, chaos, and thermalization for a one-dimensional classical gas of hard rods confined to an external quadratic or quartic trap, which breaks microscopic integrability. To quantify the strength of chaos in this system, we compute its maximal Lyapunov exponent numerically. The approach to thermal equilibrium is studied by considering the time evolution of particle position and velocity distributions and comparing the late-time profiles with the Gibbs state. Remarkably, we find that quadratically trapped hard rods are highly nonergodic and do not resemble a Gibbs state even at extremely long times, despite compelling evidence of chaos for four or more rods. On the other hand, our numerical results reveal that hard rods in a quartic trap exhibit both chaos and thermalization, and equilibrate to a Gibbs state as expected for a nonintegrable many-body system.

Quasiperiodicity in the α-Fermi-Pasta-Ulam-Tsingou problem revisited: An approach using ideas from wave turbulence

The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α-FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α-FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α-FPUT problem. We argue that certain aspects of the α-FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.

Exploring Integrability-Chaos Transition with a Sequence of Independent Perturbations

A gas of interacting particles is a paradigmatic example of chaotic systems. It is shown here that, even if all but one particle are fixed in generic positions, the excited states of the moving particle are chaotic. They are characterized by the number of principal components (NPC)-the number of integrable system eigenstates involved into the nonintegrable one, which increases linearly with the number of strong scatterers. This rule is a particular case of the general effect of an additional perturbation on the system chaotic properties. The perturbation independence criteria supposing the system chaoticity increase are derived here as well. The effect can be observed in experiments with photons or cold atoms as the decay of observable fluctuation variance, which is inversely proportional to NPC and, therefore, to the number of scatterers. This decay indicates that the eigenstate thermalization is approached. The results are confirmed by numerical calculations for a harmonic waveguide with zero-range scatterers along its axis.

Universal Kardar-Parisi-Zhang transient diffusion in nonequilibrium anharmonic chains

The well known nonlinear fluctuating hydrodynamics theory has grouped diffusions in anharmonic chains into two universality classes: one is the Kardar-Parisi-Zhang (KPZ) class for chains with either asymmetric potential or nonzero static pressure and the other is the Gaussian class for chains with symmetric potential at zero static pressure, such as Fermi-Pasta-Ulam-Tsingou (FPUT)-β chains. However, little is known of the nonequilibrium transient diffusion in anharmonic chains. Here, we reveal that the KPZ class is the only universality class for nonequilibrium transient diffusion, manifested as the KPZ scaling of the side peaks of momentum correlation (corresponding to the sound modes correlation), which was completely unexpected in equilibrium FPUT-β chains. The underlying mechanism is that the nonequilibrium soliton dynamics cause nonzero transient pressure so that the sound modes satisfy approximately the noisy Burgers equation, in which the collisions of solitons was proved to yield the KPZ dynamic exponent of the soliton dispersion. Therefore, the unexpected KPZ universality class is obtained in the nonequilibrium transient diffusion in FPUT-β chains and the corresponding carriers of nonequilibrium transient diffusion are attributed to solitons.

Burgers Turbulence in the Fermi-Pasta-Ulam-Tsingou Chain

We prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterized by a transient Burgers turbulence regime on a wide range of time and energy scales. This regime is present at long wavelengths and energy per particle small enough that equipartition is not reached on a fast timescale. In this range, we prove that the driving mechanism to thermalization is the formation of a shock that can be predicted using a pair of generalized Burgers equations. We perform a perturbative calculation at small energy per particle, proving that the energy spectrum of the chain E_{k} decays as a power law, E_{k}∼k^{-ζ(t)}, on an extensive range of wave numbers k. We predict that ζ(t) takes first the value 8/3 at the Burgers shock time, and then reaches a value close to 2 within two shock times. The value of the exponent ζ=2 persists for several shock times before the system eventually relaxes to equipartition. During this wide time window, an exponential cutoff in the spectrum is observed at large k, in agreement with previous results. Such a scenario turns out to be universal, i.e., independent of the parameters characterizing the system and of the initial condition, once time is measured in units of the shock time.

Thermalization dynamics of macroscopic weakly nonintegrable maps

We study thermalization of weakly nonintegrable nonlinear unitary lattice dynamics. We identify two distinct thermalization regimes close to the integrable limits of either linear dynamics or disconnected lattice dynamics. For weak nonlinearity, the almost conserved actions correspond to extended observables which are coupled into a long-range network. For weakly connected lattices, the corresponding local observables are coupled into a short-range network. We compute the evolution of the variance ^σ ( T ) of finite time average distributions for extended and local observables. We extract the ergodization time scale _T which marks the onset of thermalization, and determine the type of network through the subsequent decay of ^σ ( T ). We use the complementary analysis of Lyapunov spectra [M. Malishava and S. Flach, Phys. Rev. Lett. 128, 134102 (2022)] and compare the Lyapunov time _T with _T. We characterize the spatial properties of the tangent vector and arrive at a complete classification picture of weakly nonintegrable macroscopic thermalization dynamics.

Lyapunov Spectrum Scaling for Classical Many-Body Dynamics Close to Integrability

We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.

Finite-time fluctuation theorem for oscillatory lattices driven by a temperature gradient

The finite-time fluctuation theorem (FT) for the master functional, total entropy production, and medium entropy is studied in the one-dimensional Fermi-Pasta-Ulam-Tsingou-β (FPUT-β) chain coupled with two heat reservoirs at different temperatures. Through numerical simulations and theoretical analysis, we find that the nonequilibrium steady-state distribution of the one-dimensional FPUT-β chain violates the time-reversal symmetry. Thus, unlike the master functional, the total entropy production fails to satisfy the fluctuation relation for finite time. Meanwhile, we discuss the range of medium entropy production which obeys the conventional steady-state fluctuation theorem (SSFT) in the infinite time limit. Furthermore, we find that the generalized SSFT for medium entropy monotonically approaches the conventional SSFT as the time interval increases, irrespective of temperature difference, anharmonicity, and system size. Interestingly, the medium entropy production rate shows a nonmonotonic variation with anharomonicity, which comes from a competition mechanism of the phonon transport. Correspondingly, the difference between the generalized SSFT and the conventional SSFT shows similar nonmonotonic behaviors.

Quasi-integrable systems are slow to thermalize but may be good scramblers

Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time-the clearest example being the Solar System-but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent, defined by the evolution of the four-point out-of-time-order correlator (OTOC), of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. (1) We find that in the semiclassical limit the quantum Lyapunov exponent is given by the classical one: it scales as ε^{1/3}, with ε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε^{-1}). (2) We also find that in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and (3) for sufficiently small perturbations the ε^{1/3} dependence is also suppressed-another purely quantum effect which we explain. These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and kT/ℏ may stay finite at a low temperature T.