Intermittent many-body dynamics at equilibrium

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.

Periodic orbits in Fermi-Pasta-Ulam-Tsingou systems

The Fermi-Pasta-Ulam-Tsingou (FPUT) paradox is the phenomenon whereby a one-dimensional chain of oscillators with nonlinear couplings shows long-lived nonergodic behavior prior to thermalization. The trajectory of the system in phase space, with a long-wavelength initial condition, closely follows that of the Toda model over short times, as both systems seem to relax quickly to a non-thermal, metastable state. Over longer times, resonances in the FPUT spectrum drive the system toward equilibrium, away from the Toda trajectory. Similar resonances are observed in q-breather spectra, suggesting that q-breathers are involved in the route toward thermalization. In this article, we first review previous important results related to the metastable state, solitons, and q-breathers. We then investigate orbit bifurcations of q-breathers and show that they occur due to resonances, where the q-breather frequencies become commensurate as mΩ1=Ωk. The resonances appear as peaks in the breather energy spectrum. Furthermore, they give rise to new "composite periodic orbits," which are nonlinear combinations of multiple q-breathers that exist following orbit bifurcations. We find that such resonances are absent in integrable systems, as a consequence of the (extensive number of) conservation laws associated with integrability.

Equilibria and dynamics of two coupled chains of interacting dipoles

We explore the energy transfer dynamics in an array of two chains of identical rigid interacting dipoles. Varying the distance b between the two chains of the array, a crossover between two different ground-state (GS) equilibrium configurations is observed. Linearizing around the GS configurations, we verify that interactions up to third nearest neighbors should be accounted to accurately describe the resulting dynamics. Starting with one of the GS, we excite the system by supplying it with an excess energy ΔK located initially on one of the dipoles. We study the time evolution of the array for different values of the system parameters b and ΔK. Our focus is hereby on two features of the energy propagation: the redistribution of the excess energy ΔK among the two chains and the energy localization along each chain. For typical parameter values, the array of dipoles reaches both the equipartition between the chains and the thermal equilibrium from the early stages of the time evolution. Nevertheless, there is a region in parameter space (b,ΔK) where even up to the long computation time of this study, the array does neither reach energy equipartition nor thermalization between chains. This fact is due to the existence of persistent chaotic breathers.

Thermalization of Two- and Three-Dimensional Classical Lattices

Understanding how systems achieve thermalization is a fundamental task in statistical physics. This Letter presents both analytical and numerical evidence showing that thermalization can be universally achieved in sufficiently large two- and three-dimensional lattices via weak nonlinear interactions. Thermalization time follows a universal scaling law unaffected by lattice structures, types of interaction potentials, or whether the lattice is ordered or not. Moreover, this study highlights the critical impact of dimensionality and degeneracy on thermalization dynamics.

Dynamical chaos in the integrable Toda chain induced by time discretization

We use the Toda chain model to demonstrate that numerical simulation of integrable Hamiltonian dynamics using time discretization destroys integrability and induces dynamical chaos. Specifically, we integrate this model with various symplectic integrators parametrized by the time step τ and measure the Lyapunov time TΛ (inverse of the largest Lyapunov exponent Λ). A key observation is that TΛ is finite whenever τ is finite but diverges when τ→0. We compare the Toda chain results with the nonintegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In addition, we observe a breakdown of the simulations at times TB≫TΛ due to certain positions and momenta becoming extremely large ("Not a Number"). This phenomenon originates from the periodic driving introduced by symplectic integrators and we also identify the concrete mechanism of the breakdown in the case of the Toda chain.

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.

The Metastable State of Fermi-Pasta-Ulam-Tsingou Models

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan-Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi-Pasta-Ulam-Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure-the spectral entropy (η)-and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model.

How close are integrable and nonintegrable models: A parametric case study based on the Salerno model

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for "generic" initial data, how close are the integrable to the nonintegrable models? Our more precise formulation of this question is, How well is the constancy of formerly conserved quantities preserved in the nonintegrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.

Counterdiabatic driving in the classical β-Fermi-Pasta-Ulam-Tsingou chain

Shortcuts to adiabaticity (STAs) have been used to make rapid changes to a system while eliminating or minimizing excitations in the system's state. In quantum systems, these shortcuts allow us to minimize inefficiencies and heating in experiments and quantum computing protocols, but the theory of STAs can also be generalized to classical systems. We focus on one such STA, approximate counterdiabatic (ACD) driving, and numerically compare its performance in two classical systems: a quartic anharmonic oscillator and the β Fermi-Pasta-Ulam-Tsingou lattice. In particular, we modify an existing variational technique to optimize the approximate driving and then develop classical figures of merit to quantify the performance of the driving. We find that relatively simple forms for the ACD driving can dramatically suppress excitations regardless of system size. ACD driving in classical nonlinear oscillators could have many applications, from minimizing heating in bosonic gases to finding optimal local dressing protocols in interacting field theories.

Chaos and thermalization in a classical chain of dipoles

We explore the connection between chaos, thermalization, and ergodicity in a linear chain of N interacting dipoles. Starting from the ground state, and considering chains of different numbers of dipoles, we introduce single site excitations with excess energy ΔK. The time evolution of the chaoticity of the system and the energy localization along the chain is analyzed by computing, up to a very long time, the statistical average of the finite-time Lyapunov exponent λ(t) and the participation ratio Π(t). For small ΔK, the evolution of λ(t) and Π(t) indicates that the system becomes chaotic at approximately the same time as Π(t) reaches a steady state. For the largest considered values of ΔK the system becomes chaotic at an extremely early stage in comparison with the energy relaxation times. We find that this fact is due to the presence of chaotic breathers that keep the system far from equipartition and ergodicity. Finally, we show numerically and analytically that the asymptotic values attained by the participation ratio Π(t) fairly correspond to thermal equilibrium.

Fragile many-body ergodicity from action diffusion

Weakly nonintegrable many-body systems can restore ergodicity in distinctive ways depending on the range of the interaction network in action space. Action resonances seed chaotic dynamics into the networks. Long-range networks provide well connected resonances with ergodization controlled by the individual resonance chaos time scales. Short-range networks instead yield a dramatic slowing down of ergodization in action space, and lead to rare resonance diffusion. We use Josephson junction chains as a paradigmatic study case. We exploit finite time average distributions to characterize the thermalizing dynamics of actions. We identify an action resonance diffusion regime responsible for the slowing down. We extract the diffusion coefficient of that slow process and measure its dependence on the proximity to the integrable limit. Independent measures of correlation functions confirm our findings. The observed fragile diffusion is relying on weakly chaotic dynamics in spatially isolated action resonances. It can be suppressed, and ergodization delayed, by adding weak action noise, as a proof of concept.

Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi-Pasta-Ulam-Tsingou recurrence

Modulation instability, breather formation, and the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A "cascading mechanism" is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with "eye-shaped" breathers in one waveguide and "four-petal" modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.

Thermalization in the one-dimensional Salerno model lattice

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

Wave-Turbulence Origin of the Instability of Anderson Localization against Many-Body Interactions

Whether Anderson localization is robust against many-body interactions and, closely related, whether a disordered many-body system can be thermalized are long outstanding issues. In this Letter, we address these issues with the wave-turbulence theory. We show that, in general, the thermalization time in one-dimensional disordered lattice systems is inversely proportional to the squared interaction strength in the thermodynamic limit. It leads to the conclusion that such systems can always be thermalized by arbitrarily weak many-body interactions and thus the localized states are unstable.

Structural localization in the classical and quantum Fermi-Pasta-Ulam model

We study the statistics and short-time dynamics of the classical and the quantum Fermi-Pasta-Ulam chain in the thermal equilibrium. We analyze the distributions of single-particle configurations by integrating out the rest of the system. At low temperatures, we observe a systematic increase in the mobility of the chain when transitioning from classical to quantum mechanics due to zero-point energy effects. We analyze the consequences of quantum dispersion on the dynamics at short times of configurational correlation functions.

Regular regimes of the harmonic three-mass system

The symmetric harmonic three-mass system with finite rest lengths, despite its apparent simplicity, displays a wide array of interesting dynamics for different energy values. At low energy the system shows regular behavior that produces a deformation-induced rotation with a constant averaged angular velocity. As the energy is increased this behavior makes way to a chaotic regime with rotational behavior statistically resembling Lévy walks and random walks. At high enough energies, where the rest lengths become negligible, the chaotic signature vanishes and the system returns to regularity, with a single dominant frequency. The transition to and from chaos, as well as the anomalous power-law statistics measured for the angular displacement of the harmonic three-mass system are largely governed by the structure of regular solutions of this mixed Hamiltonian system. Thus, a deeper understating of the system's irregular behavior requires mapping out its regular solutions. In this work we provide a comprehensive analysis of the system's regular regimes of motion, using perturbative methods to derive analytical expressions of the system as almost-integrable in its low- and high-energy extremes. The compatibility of this description with the full system is shown numerically. In the low-energy regime, the Birkhoff normal form method is utilized to circumvent the low-order 1:1 resonance of the system, and the conditions for Kolmogorov-Arnold-Moser theory are shown to hold. The integrable approximations provide the back-bone structure around which the behavior of the full nonlinear system is organized and provide a pathway to understanding the origin of the power-law statistics measured in the system.

The β Fermi-Pasta-Ulam-Tsingou recurrence problem

We perform a thorough investigation of the first Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in the β-FPUT chain for both positive and negative β. We show numerically that the rescaled FPUT recurrence time T_r=t_r/(N+1)³ depends, for large N, only on the parameter S≡Eβ(N+1). Our numerics also reveal that for small |S|, T_r is linear in S with positive slope for both positive and negative β. For large |S|, T_r is proportional to |S|^-1/2 for both positive and negative β but with different multiplicative constants. We numerically study the continuum limit and find that the recurrence time closely follows the |S|^-1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions that exist only in the negative β case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for T_r that depends only on S. In the latter regime, we show that T_r∝|S|^-1/2 is predicted by the soliton theory in the continuum limit. We then investigate the existence of the FPUT recurrences and show that their disappearance surprisingly depends only on Eβ for large N, not S. Finally, we end by discussing the striking differences in the amount of energy mixing between positive and negative β and offer some remarks on the thermodynamic limit.

Universal scaling of the thermalization time in one-dimensional lattices

We show that, in the thermodynamic limit, a one-dimensional (1D) nonlinear lattice can always be thermalized for arbitrarily small nonlinearity, thus proving the equipartition theorem for a class of systems. Particularly, we find that in the lattices with nearest-neighbor interaction potential V(x)=x^{2}/2+λx^{n}/n with n≥4, the thermalization time, T_{eq}, follows a universal scaling law; i.e., T_{eq}∝λ^{-2}ε^{-(n-2)}, where ε is the energy per particle. Numerical simulations confirm that it is accurate for an even n, while a certain degree of deviation occurs for an odd n, which is attributed to the extra vibration modes excited by the asymmetric interaction potential. This finding suggests that although the symmetry of interactions will not affect the system reaching equipartition eventually, it affects the process toward equipartition. Based on the scaling law found here, a unified formula for the thermalization time of a 1D general nonlinear lattice is obtained.

Dynamical glass in weakly nonintegrable Klein-Gordon chains

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means μ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/μ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲μ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.

N-break states in a chain of nonlinear oscillators

In the present work we explore a prestretched oscillator chain where the nodes interact via a pairwise Lennard-Jones potential. In addition to a homogeneous solution, we identify solutions with one or more (so-called) "breaks," i.e., jumps. As a function of the canonical parameter of the system, namely, the precompression strain d, we find that the most fundamental one-break solution changes stability when the monotonicity of the Hamiltonian changes with d. We provide a proof for this (motivated by numerical computations) observation. This critical point separates stable and unstable segments of the one-break branch of solutions. We find similar branches for two- through five-break branches of solutions. Each of these higher "excited state" solutions possesses an additional unstable pair of eigenvalues. We thus conjecture that k-break solutions will possess at least k-1 (and at most k) pairs of unstable eigenvalues. Our stability analysis is corroborated by direct numerical computations of the evolutionary dynamics.

Chaos and Anderson localization in disordered classical chains: Hertzian versus Fermi-Pasta-Ulam-Tsingou models

We numerically investigate the dynamics of strongly disordered 1D lattices under single-particle displacements, using both the Hertzian model, describing a granular chain, and the α+β Fermi-Pasta-Ulam-Tsingou model (FPUT). The most profound difference between the two systems is the discontinuous nonlinearity of the granular chain appearing whenever neighboring particles are detached. We therefore sought to unravel the role of these discontinuities in the destruction of Anderson localization and their influence on the system's chaotic dynamics. Our results show that the dynamics of both models can be characterized by: (i) localization with no chaos; (ii) localization and chaos; (iii) spreading of energy, chaos, and equipartition. The discontinuous nonlinearity of the Hertzian model is found to trigger energy spreading at lower energies. More importantly, a transition from Anderson localization to energy equipartition is found for the Hertzian chain and is associated with the "propagation" of the discontinuous nonlinearity in the chain. On the contrary, the FPUT chain exhibits an alternate behavior between localized and delocalized chaotic behavior which is strongly dependent on the initial energy excitation.

Dynamical Glass and Ergodization Times in Classical Josephson Junction Chains

Models of classical Josephson junction chains turn integrable in the limit of large energy densities or small Josephson energies. Close to these limits the Josephson coupling between the superconducting grains induces a short-range nonintegrable network. We compute distributions of finite-time averages of grain charges and extract the ergodization time T_{E} which controls their convergence to ergodic δ distributions. We relate T_{E} to the statistics of fluctuation times of the charges, which are dominated by fat tails. T_{E} is growing anomalously fast upon approaching the integrable limit, as compared to the Lyapunov time T_{Λ}-the inverse of the largest Lyapunov exponent-reaching astonishing ratios T_{E}/T_{Λ}≥10^{8}. The microscopic reason for the observed dynamical glass is rooted in a growing number of grains evolving over long times in a regular almost integrable fashion due to the low probability of resonant interactions with the nearest neighbors. We conjecture that the observed dynamical glass is a generic property of Josephson junction networks irrespective of their space dimensionality.

Behavior and breakdown of higher-order Fermi-Pasta-Ulam-Tsingou recurrences

We numerically investigate the existence and stability of higher-order recurrences (HoRs), including super-recurrences, super-super-recurrences, etc., in the α and β Fermi-Pasta-Ulam-Tsingou (FPUT) lattices for initial conditions in the fundamental normal mode. Our results represent a considerable extension of the pioneering work of Tuck and Menzel on super-recurrences. For fixed lattice sizes, we observe and study apparent singularities in the periods of these HoRs, speculated to be caused by nonlinear resonances. Interestingly, these singularities depend very sensitively on the initial energy and the respective nonlinear parameters. Furthermore, we compare the mechanisms by which the super-recurrences in the two models breakdown as the initial energy and respective nonlinear parameters are increased. The breakdown of super-recurrences in the β-FPUT lattice is associated with the destruction of the so-called metastable state and thus with relaxation towards equilibrium. For the α-FPUT lattice, we find this is not the case and show that the super-recurrences break down while the lattice is still metastable and far from equilibrium. We close with comments on the generality of our results for different lattice sizes.

Equilibration of quasi-integrable systems

We study the slow relaxation of isolated quasi-integrable systems, focusing on the classical problem of Fermi-Pasta-Ulam-Tsingou (FPU) chain. It is well known that the initial energy sharing between different linear modes can be inferred by the integrable Toda chain. Using numerical simulations, we show explicitly how the relaxation of the FPU chain toward equilibration is determined by a slow drift within the space of Toda's integrals of motion. We analyze the whole spectrum of Toda modes and show how they dictate, via a generalized Gibbs ensemble, the quasistatic states along the FPU evolution. This picture is employed to devise a fast numerical integration, which can be generalized to other quasi-integrable models. In addition, we discuss how a fluctuation theorem, recently derived in Goldfriend and Kurchan [Europhys. Lett. 124, 10002 (2018)10.1209/0295-5075/124/10002], describes the large deviations as the system flows in the entropy landscape.

Asymptotic Prethermalization in Periodically Driven Classical Spin Chains

We reveal a continuous dynamical heating transition between a prethermal and an infinite-temperature stage in a clean, chaotic periodically driven classical spin chain. The transition time is a steep exponential function of the drive frequency, showing that the exponentially long-lived prethermal plateau, originally observed in quantum Floquet systems, survives the classical limit. Even though there is no straightforward generalization of Floquet's theorem to nonlinear systems, we present strong evidence that the prethermal physics is well described by the inverse-frequency expansion. We relate the stability and robustness of the prethermal plateau to drive-induced synchronization not captured by the expansion. Our results set the pathway to transfer the ideas of Floquet engineering to classical many-body systems, and are directly relevant for photonic crystals and cold atom experiments in the superfluid regime.

Chaotic Dynamics in a Quantum Fermi-Pasta-Ulam Problem

We investigate the emergence of chaotic dynamics in a quantum Fermi—Pasta—Ulam problem for anharmonic vibrations in atomic chains applying semi-quantitative analysis of resonant interactions complemented by exact diagonalization numerical studies. The crossover energy separating chaotic high energy phase and localized (integrable) low energy phase is estimated. It decreases inversely proportionally to the number of atoms until approaching the quantum regime where this dependence saturates. The chaotic behavior appears at lower energies in systems with free or fixed ends boundary conditions compared to periodic systems. The applications of the theory to realistic molecules are discussed.

Glassy dynamics in disordered oscillator chains

The escape of energy injected into one site in a disordered chain of nonlinear oscillators is examined numerically. When the disorder has a "fractal" pattern, the decay of the residual energy at the injection site can be fit to a stretched exponential with an exponent that varies continuously with the control parameter. At low temperature, we see evidence that energy can be trapped for an infinite time at the original site, i.e., classical many body localization.

Weakly Nonergodic Dynamics in the Gross-Pitaevskii Lattice

The microcanonical Gross-Pitaevskii (also known as the semiclassical Bose-Hubbard) lattice model dynamics is characterized by a pair of energy and norm densities. The grand canonical Gibbs distribution fails to describe a part of the density space, due to the boundedness of its kinetic energy spectrum. We define Poincaré equilibrium manifolds and compute the statistics of microcanonical excursion times off them. The tails of the distribution functions quantify the proximity of the many-body dynamics to a weakly nonergodic phase, which occurs when the average excursion time is infinite. We find that a crossover to weakly nonergodic dynamics takes place inside the non-Gibbs phase, being unnoticed by the largest Lyapunov exponent. In the ergodic part of the non-Gibbs phase, the Gibbs distribution should be replaced by an unknown modified one. We relate our findings to the corresponding integrable limit, close to which the actions are interacting through a short range coupling network.

Double Scaling in the Relaxation Time in the β-Fermi-Pasta-Ulam-Tsingou Model

We consider the original β-Fermi-Pasta-Ulam-Tsingou system; numerical simulations and theoretical arguments suggest that, for a finite number of masses, a statistical equilibrium state is reached independently of the initial energy of the system. Using ensemble averages over initial conditions characterized by different Fourier random phases, we numerically estimate the time scale of equipartition and we find that for very small nonlinearity it matches the prediction based on exact wave-wave resonant interaction theory. We derive a simple formula for the nonlinear frequency broadening and show that when the phenomenon of overlap of frequencies takes place, a different scaling for the thermalization time scale is observed. Our result supports the idea that the Chirikov overlap criterion identifies a transition region between two different relaxation time scalings.