Dynamical Glass and Ergodization Times in Classical Josephson Junction Chains

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.

Generalized N-rotor problems, synchronized subsystems, and associated solitons

We consider systems of N particles interacting on the unit circle through 2π-periodic potentials. An example is the N-rotor problem that arises as the classical limit of coupled Josephson junctions and for various energies is known to have a wide range of behaviors such as global chaos and ergodicity, together with families of periodic solutions and transitions from order to chaos. We focus here on selected initial values for generalized systems in which the second order Euler-Lagrange equations reduce to first order equations, which we show by example can describe an ensemble of oscillators with associated emergent phenomena such as synchronization. A specific case is that of the Kuramoto model with well-known synchronization properties. We further demonstrate the relation of these models to field theories in 1+1 dimensions that allow static kink solitons satisfying first order Bogomolny equations, well-known in soliton physics, which correspond to the first order equations of the generalized N-rotor models. For the nonlinear pendulum, for example, the first order equations define the separatrix in the phase portrait of the system and correspond to kink solitons in the sine-Gordon equation.

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.

Chaotic route to classical thermalization: A real-space analysis

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.

Thermalization slowing down in multidimensional Josephson junction networks

We characterize thermalization slowing down of Josephson junction networks in one, two, and three spatial dimensions for systems with hundreds of sites by computing their entire Lyapunov spectra. The ratio of Josephson coupling E_{J} to energy density h controls two different universality classes of thermalization slowing down, namely, the weak-coupling regime, E_{J}/h→0, and the strong-coupling regime, E_{J}/h→∞. We analyze the Lyapunov spectrum by measuring the largest Lyapunov exponent and by fitting the rescaled spectrum with a general ansatz. We then extract two scales: the Lyapunov time (inverse of the largest exponent) and the exponent for the decay of the rescaled spectrum. The two universality classes, which exist irrespective of network dimension, are characterized by different ways the extracted scales diverge. The universality class corresponding to the weak-coupling regime allows for the coexistence of chaos with a large number of near-conserved quantities and is shown to be characterized by universal critical exponents, in contrast with the strong-coupling regime. We expect our findings, which we explain using perturbation theory arguments, to be a general feature of diverse Hamiltonian systems.

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.

Long-lived solitons and their signatures in the classical Heisenberg chain

Motivated by the Kardar-Parisi-Zhang (KPZ) scaling recently observed in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations in this model. We find that the Heisenberg chain, although well known to be nonintegrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with ln(1+S_{i}·S_{j}) interactions. We explicitly construct infinitely long-lived stationary solitons, and provide an adiabatic construction procedure for moving soliton solutions, which shows that Ishimori solitons have a long-lived Heisenberg counterpart when they are not too narrow and not too fast moving. Finally, we demonstrate their presence in thermal states of the Heisenberg chain, even when the typical soliton width is larger than the spin correlation length, and argue that these excitations likely underlie the KPZ scaling.

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.

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.

Measurement and memory in the periodically driven complex Ginzburg-Landau equation

In the present work we illustrate that classical but nonlinear systems may possess features reminiscent of quantum ones, such as memory, upon suitable external perturbation. As our prototypical example, we use the two-dimensional complex Ginzburg-Landau equation in its vortex glass regime. We impose an external drive as a perturbation mimicking a quantum measurement protocol, with a given "measurement rate" (the rate of repetition of the drive) and "mixing rate" (characterized by the intensity of the drive). Using a variety of measures, we find that the system may or may not retain its coherence, statistically retrieving its original glass state, depending on the strength and periodicity of the perturbing field. The corresponding parametric regimes and the associated energy cascade mechanisms involving the dynamics of vortex waveforms and domain boundaries are discussed.

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.

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.

Analytical approach to Lyapunov time: Universal scaling and thermalization

Based on the geometrization of dynamics and self-consistent phonon theory, we develop an analytical approach to derive the Lyapunov time, the reciprocal of the largest Lyapunov exponent, for general nonlinear lattices of coupled oscillators. The Fermi-Pasta-Ulam-Tsingou-like lattices are exemplified by using the method, which agree well with molecular dynamical simulations for the cases of quartic and sextic interactions. A universal scaling behavior of the Lyapunov time with the nonintegrability strength is observed for the quasi-integrable regime. Interestingly, the scaling exponent of the Lyapunov time is the same as the thermalization time, which indicates a proportional relationship between the two timescales. This relation illustrates how the thermalization process is related to the intrinsic chaotic property.

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}.

Thermal rectification induced by geometrical asymmetry: A two-dimensional multiparticle Lorentz gas model

We propose a two-dimensional (2D) multiparticle Lorentz gas model by combining the direct simulation Monte Carlo method with the Lorentz gas model, where the normal thermal transport under Fourier's law is confirmed by length-independent thermal conductivity. For this 2D multiparticle Lorentz gas model, the thermal rectification effect is obtained with the asymmetrical setup of a trapezoidal shape, which is a purely geometric effect. Furthermore, we find a scaling behavior between the rectification ratio and the geometrical parameters of the trapezoidal shape, which is verified by numerical simulations.