Chain Of Anharmonic Oscillators Research Articles

Wave turbulence and collective behavior models for wave equations with short- and long-range interactions

In this work, we discuss a situation which could lead to both wave turbulence and collective behavior kinetic equations. The wave turbulence kinetic models appear in the kinetic limit when the wave equations have local differential operators. Viewing wave equations on the lattice as chains of anharmonic oscillators and replacing the local differential operators (short-range interactions) by non-local ones (long-range interactions), we arrive at a new Vlasov-type kinetic model in the mean field limit under the molecular chaos assumption reminiscent of models for collective behavior in which anharmonic oscillators replace individual particles.

Topological interface states due to spontaneous symmetry breaking in a chain of anharmonic oscillators

In this work, we propose a one-dimensional system of coupled anharmonic oscillators with nonlinear coupling and study a spontaneous formation of interface excitations in the linearized spectrum of such periodic array. We show that during its evolution such system can undergo a topological transition from the disordered and topologically trivial phase into the nontrivial one. The topological transition is accompanied by the formation of the interface state in the spectrum of linearized excitations of the equilibrium state. Employing the technique based on the calculation of mean chiral displacement we demonstrate that the in-gap interface state exhibits a topological nature and is analogous to that in the well-known Su-Schrieffer-Heeger model. Our results demonstrate the formation of topological interface state induced by spontaneous symmetry breaking mechanism and open novel possibilities for designing various nonlinear electronic, photonic and polaritonic systems with disorder-robust control of information flows.

Traveling solitons in long-range oscillator chains

We investigate the existence and propagation of solitons in a long-range extension of the quartic Fermi–Pasta–Ulam (FPU) chain of anharmonic oscillators. The coupling in the linear term decays as a power-law with an exponent . We obtain an analytic perturbative expression of traveling envelope solitons by introducing a non linear Schrödinger equation for the slowly varying amplitude of short wavelength modes. Due to the non analytic properties of the dispersion relation, it is crucial to develop the theory using discrete difference operators. Those properties are also the ultimate reason why kink-solitons may exist but are unstable, at variance with the short-range FPU model. We successfully compare these approximate analytic results with numerical simulations for the value which was chosen as a case study.

Energy rectification in quantum graded spin chains: Analysis of the XXZ model.

In this work, with focus on the energy-transport properties in quantum, low-dimensional, graded materials, we address the investigation of the energy (and spin) current in XXZ open chains with graded inner structures and driven out of equilibrium by magnetization pumping applied at the ends. We study several types of graded structures in different situations in order to show a ubiquitous occurrence of energy rectification, even for the system under a homogeneous magnetic field. Due to technical difficulties, we carry out the computation for small chains, but we present arguments that indicate the extension of some results to larger systems. Recalling the generic existence of energy rectification in classical, graded materials, which are described by anharmonic chains of oscillators, and recalling also the anharmonicity of these XXZ models, which involve quartic terms in more transparent representation in terms of fermionic creation and annihilation operators, we may say that our results extend the ubiquity of energy rectification occurrence in classical graded materials to the case of quantum systems.

Quasi-Static Hydrodynamic Limits

We consider hydrodynamic limits of interacting particles systems with open boundaries, where the exterior parameters change in a time scale slower than the typical relaxation time scale. The limit deterministic profiles evolve quasi-statically. These limits define rigorously the thermodynamic quasi static transformations also for transition between non-equilibrium stationary states. We study first the case of the symmetric simple exclusion, where duality can be used, and then we use relative entropy methods to extend to other models like zero range systems. Finally we consider a chain of anharmonic oscillators in contact with a thermal Langevin bath with a temperature gradient and a slowly varying tension applied to one end.

Cross-correlations between phonon modes in anharmonic oscillator chains: role in heat transport.

We have computed current-current correlation functions in chains of anharmonic oscillators described by various models (FPU-β, FPU-αβ, ϕ4), considering both the total current and the currents associated with individual phonon modes, which are important in view of the Green-Kubo relation for heat conductivity. Our simulations show that, contrary to the common hypothesis, there are, under some circumstances, significant correlations between neighboring modes. These cross-mode correlations are the dominant contribution to the conductivity in the low anharmonicity regime. The inverse of the timescale over which they are significant, 1/τc, is related to the anharmonicity level in a way similar to the largest Lyapunov exponent, suggesting that the two quantities are related. Cross-mode correlations exist in both anomalous and regular heat-conducting systems although we are unable to observe a transition to the independent-mode regime in the latter case.

Thermal rectification and thermal resistive phase cross over in exponential mass graded materials

Concept of the functional graded materials (FGMs) has been explored by considering exponential mass variation along the chain of anharmonic oscillators in the study of heat transport at low dimensions. This exponential distribution of mass along the space invokes the diffusion of phonons transport which results to temperature gradient, asymmetric heat flow, thermal rectification and cross over between positive differential thermal resistance (PDTR) and negative differential thermal resistance (NDTR) in one-dimensional (1D) exponential mass graded chain. The temperature dependence thermal rectification achieved is 4−74% and also predicted that the thermal rectification can be controlled by tuning the higher and lower average temperature limits of two thermal reservoirs. It is also seen that in FGMs, the thermal conductivity does not change drastically against the average temperature of two heat baths. The cross over between PDTR and NDTR can be tuned either by mass ratio of one dimensional (1D) exponential mass graded anharmonic chain and/or by temperature difference between two heat baths. The figure of merit of the 1D structure can also be tuned by mass gradient, the higher mass gradient material will work as the potential candidate for better thermoelectric material.

Macroscopic energy diffusion for a chain of anharmonic oscillators

We study the energy diffusion in a chain of anharmonic oscillators where the Hamiltonian dynamics is perturbed by a local energy conserving noise. We prove that under diffusive rescaling of space-time, energy fluctuations diffuse and evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and some upper and lower bounds.

Heat-flow properties of systems with alternate masses or alternate on-site potentials

We address a central issue of phononics: the search of properties or mechanisms to manage the heat flow in reliable materials. We analytically study standard and simple systems modeling the heat flow in solids, namely, the harmonic, self-consistent harmonic and also anharmonic chains of oscillators, and we show an interesting insulating effect: While in the homogeneous models the heat flow decays as the inverse of the particle mass, in the chain with alternate masses it decays as the inverse of the square of the mass difference, that is, it decays essentially as the mass ratio (between the smaller and the larger one) for a large mass difference. A similar effect holds if we alternate on-site potentials instead of particle masses. The existence of such behavior in these different systems, including anharmonic models, indicates that it is a ubiquitous phenomenon with applications in the heat flow control.

Analytic approach to the (an)harmonic crystal chains with self-consistent stochastic reservoirs

We consider the harmonic and anharmonic chains of oscillators with self-consistent stochastic reservoirs and derive an integral representation (à la Feynman-Kac) for the correlations, in particular, for the heat flow. For the harmonic chain, we give a new proof that its thermal conductivity is finite in the steady state. Based on this integral representation for the correlations and a perturbative analysis, the approach is quite general and can be extended to more intricate systems.

Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators.

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures

We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of H\"ormander used in the study of hypoelliptic differential operators.

Equipartition thresholds in chains of anharmonic oscillators

We perform a detailed numerical study of the transition to equipartition in the Fermi-Pasta-Ulam quartic model and in a class of potentials of given symmetry using the normalized spectral entropy as a probe. We show that the typical time scale for the equipartition of energy among Fourier modes grows linearly with system size: this is the time scale associated with the smallest frequency present in the system. We obtain two different scaling behaviors, either with energy or with energy density, depending on the scaling of the initial condition with system size. These different scaling behaviors can be understood by a simple argument, based on the Chirikov overlap criterion. Some aspects of the universality of this result are investigated: symmetric potentials show a similar transition, regulated by the same time scale.