Chain Of Oscillators Research Articles

Equilibrium perturbations for stochastic interacting systems

We consider the equilibrium perturbations for two stochastic systems: the d-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order N−α to the equilibrium profile and speed up the process by N1+κ for parameters 0<κ≤α. Under some additional constraints on κ and α, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.

Classical n-body system in volume variables II: Four-body case

It is evident that the positions of four bodies in [Formula: see text]-dimensional space can be identified with vertices of a tetrahedron. Square of volume of the tetrahedron, weighted sum of squared areas of four facets and weighted sum of squared edges are called the volume variables. A family of translation-invariant potentials which depend on volume variables alone had been considered as well as solutions of the Newton equations which solely depend on volume variables. For the case of zero angular momentum [Formula: see text], the corresponding Hamiltonian, which describes these solutions, is derived. Three examples are studied in detail: (I) the (super)integrable four-body closed chain of harmonic oscillators for [Formula: see text] (the harmonic molecule); (II) a generic, two volume variable-dependent potential whose trajectories possess a constant moment of inertia ([Formula: see text]) and (III) the four-body anharmonic oscillator for [Formula: see text]. This work is the second of the sequel: the first one [A. M. Escobar-Ruiz, R. Linares, A. V. Turbiner and W. Miller Jr., Int. J. Mod. Phys. A 36, 2150140 (2021)] was dedicated to study the three-body classical problem in volume variables.

Numerical evidence for the approximation of dissipative systems by gyroscopically coupled oscillator chains

Numerical evidence for the approximation of dissipative systems by gyroscopically coupled oscillator chains

Silicon Micromachined Waveguide Circuit for a 2 THz Schottky Receiver: Progress and Challenges

Space instruments that operate in the THz range can enable unique measurements for a better understanding of the chemical and physical processes taking place in our Universe. One such application of this technology is a compact 2.06 THz receiver that can measure the ionized oxygen line to determine wind velocity in Earth's upper atmosphere. Metal machined waveguide circuits have long been utilized to build and demonstrate functional receivers in the submillimeter-wave range. However, as the operational frequency is increased, extremely challenging requirements are placed on metal machining in terms of the required precision, surface roughness, and alignment tolerance. This work describes the design and implementation of the first-of-its-kind, vertically-integrated, 2-THz Schottky diode mixer using precise silicon micromachine technology. A traditional e-plane split waveguide package is re-designed to utilize the three-dimensional (3-D) capability of stacking silicon micromachined parts. The silicon microfabrication process has been optimized to produce smooth and precise features for packaging a 2 THz subharmonic GaAs Schottky diode mixer, providing surface roughness better than 1-micron rms with less than 5% variation on critical dimensions. The subharmonic mixer and a fully solid-state local oscillator (LO) chain is currently being implemented to validate silicon micromachining for THz packaging. The impact of micromachining variation on mixer performance is explored through simulations over a range of dimensions on the most sensitive regions of the Si module.

A combined FD-HB approximation method for steady-state vibrations in large dynamical systems with localised nonlinearities

The approximation of steady-state vibrations within non-linear dynamical systems is well-established in academics and is becoming increasingly important in industry. However, the complexity and the number of degrees of freedom of application-oriented industrial models demand efficient approximation methods for steady-state solutions. One possible approach to that problem are hybrid approximation schemes, which combine advantages of standard methods from the literature. The common ground of these methods is their description of the steady-state dynamics of a system solely based on the degrees of freedom affected directly by non-linearity—the so-called non-linear degrees of freedom. This contribution proposes a new hybrid method for approximating periodic solutions of systems with localised non-linearities. The motion of the non-linear degrees of freedom is approximated using the Finite Difference method, whilst the motion of the linear degrees of freedom is treated with the Harmonic Balance method. An application to a chain of oscillators showing stick-slip oscillations is used to demonstrate the performance of the proposed hybrid framework. A comparison with both pure Finite Difference and Harmonic Balance method reveals a noticeable increase in efficiency for larger systems, whilst keeping an excellent approximation quality for the strongly non-linear solution parts.

Activity driven transport in harmonic chains

The transport properties of an extended system driven by active reservoirs is an issue of paramount importance, which remains virtually unexplored. Here we address this issue, for the first time, in the context of energy transport between two active reservoirs connected by a chain of harmonic oscillators. The couplings to the active reservoirs, which exert correlated stochastic forces on the boundary oscillators, lead to fascinating behavior of the energy current and kinetic temperature profile even for this linear system. We analytically show that the stationary active current (i) changes non-monotonically as the activity of the reservoirs are changed, leading to a negative differential conductivity (NDC), and (ii) exhibits an unexpected direction reversal at some finite value of the activity drive. The origin of this NDC is traced back to the Lorentzian frequency spectrum of the active reservoirs. We provide another physical insight to the NDC using nonequilibrium linear response formalism for the example of a dichotomous active force. We also show that despite an apparent similarity of the kinetic temperature profile to the thermally driven scenario, no effective thermal picture can be consistently built in general. However, such a picture emerges in the small activity limit, where many of the well-known results are recovered.

Cross-scale energy transfer of chaotic oscillator chain in stiffness-dominated range

The investigations on the dynamics of hierarchical oscillator chains provide basic theories for the fields of acoustic metamaterial and passive guidance of energy flow. In the present study, the characteristics of stiffness-dominated energy transfer of a reduced cross-scale oscillator chain that generates chaotic vibrations are studied. The semi-analytical solutions are obtained by the complexification-averaging method and the least-square method for thoroughly searching the branches of responses. Then, the energy transfer patterns of the oscillator chain are analyzed based on Runge–Kutta method. Moreover, chaotic responses are determined using the displacements and Lyapunov exponents of the system. The average energy and normalized energy flux are defined to evaluate the energy distribution in each oscillator and the energy transfer between oscillators, respectively. The semi-analytical results show that the oscillator chain produces complex unstable branches of response around the resonance frequency of the linear oscillator. It is found that the detached higher branches of response of the chain are similar to that of a system consisting of one linear oscillator and one cubic oscillator. When chaotic responses occur, the energy transfer patterns that reflected by average energy and normalized energy flux are similar in the same branch of response for different excitation frequencies. However, energy transfer patterns in different branches of response are quite different even at the same excitation frequency, and the higher branch generates the lower energy transfer. Furthermore, analyses of the input energy and the scale between cubic oscillators demonstrate that variations of the two parameters slightly affect the normalized energy flux.

Coaxial Micro Ring Arrays Fabricated on Self‐Assembled Microtube Cavities for Resonant Light Modulation

AbstractComposite optical microcavities, as a versatile platform for light manipulation, have been theoretically revealed that the configuration of concentrically coupled resonant trajectories can significantly enhance the coupling strength, which is however challenging in experimental preparation. Here coaxial whispering gallery resonator chains were designed and fabricated with optically coupled modes split apart by more than the free spectral range (FSR) of the cavity itself. Because such three dimensional (3D) coupled optical systems are extremely difficult to obtain with traditional processing techniques an approach is developed to release and roll‐up pre‐strained, nanostructured silicon nitride membranes into microtubular optical cavities. Parallel nano‐ridges modified on the patterned membranes form into coaxial micro ring arrays after the membrane rolls up into a microtubular structure. Energetic degeneracy, as a consequence of mode splitting exceeding the FSR, leads to interference between bonding and antibonding modes with different mode identities (mode number), resulting in novel light modulation in the form of hybrid super wave modes, which display a distinct hopping behavior resembling linear stretch modes of an oscillator chain.

Scaling and universality in Brownian motion on a stochastic harmonic oscillator chain.

Diffusion of a Brownian particle along a stochastic harmonic oscillator chain is investigated. In contrast to the usually discussed Brownian motion driven by Gaussian white noise, the particle at high temperatures performs long Lévy flights. At high temperatures T the diffusion coefficient scales as D∼T^{2+α}, where the parameter α determine the average damping force ∝1/(T^{α}P) on the particle at large momentum P and at high temperature. The exponent α depends on the particle-chain interaction and chain properties. It is shown that the mean time t[over ¯]_{f} necessary to perform a flight of l lattice constant scales with l as t[over ¯]_{f}∝l^{2/3} at high temperatures and flight lengths. Last, the flight length probability distribution is found to decay as 1/l^{β} with the exponent β=4/3 being universal, i.e., independent of the model parameters.

Collective escape and homoclinic bifurcation phenomena in a nonlinear oscillators chain

The phenomenon of deterministic collective escape of particles from the cubic on-site potential well in the presence of both uniform damping and a periodic force is studied. Using analytical techniques such as the separation of time and space as well as the Melnikov theorem, the condition on the periodic force for which a single particle exhibits an irregular motion induced by the homoclinic bifurcation (HB) is derived. Numerical simulation showed that this irregular motion can lead to a strong localization of energy on all the coupled particles allowing them to collectively cross the energy barrier. Moreover, the critical value of the driving force inducing collective escape increases as the potential energy barrier increases and decreases as its frequency increases. Depending on the frequency range of the driving frequency, the collective escape and HB can occur simultaneously; otherwise, the HB prevails.

Vibrational analysis of harmonic oscillator chains with random topological constraints

The effect of topological disorder on the dynamic behaviour of topologically constrained oscillator chains is investigated through a normal mode analysis. A measure for mode localization based on the skewness of the density of energy states of normal modes is proposed, which correlates well with main configuration characteristics. Appropriate configuration sampling is achieved by employing the Ising model together with the Metropolis algorithm.

Neimark Sacker bifurcations and non-linear energy exchange in chains of non-linear oscillators

We treat a chain of oscillators with linear stiffness and internal and external cubic non-linearities. The method of harmonic balance is used to determine the non-linear modes of the Hamiltonian system. For each equilibrium point, a stability analysis is performed by means of the associated monodromy matrix. The numerical results shows that branch points, limit points and Neimark–Sacker bifurcations exist in the system. The aim of the paper is to study how they drive the energy in the system: branch points make energy transfer between non-linear modes possible, while Neimark–Sacker bifurcations can lead to chaotic behaviour. As the number of oscillators increases, the energy required to reach the first Neimark–Sacker bifurcation follows a remarkable regularity.

Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions

We consider one-dimensional infinite chains of harmonic oscillators with random exchanges of momenta and long-range interaction potentials which have polynomial decay rate |x|−θ , x → ∞, θ > 1 where is the interaction range. The dynamics conserve total momentum, total length and total energy. We prove that the systems evolve macroscopically on superballistic space-time scale when 1 < θ < 3, when θ = 3, and hyperbolic space-time scale (yɛ −1, tɛ −1) when θ > 3. Combining our results and the results in (Suda 2021 Ann. Inst. Henri Poincare B 57 2268–2314), we show the existence of two different space-time scales on which the systems evolve. In addition, we prove the fluctuations of the normal modes of the superballistic wave equation, which are analogues of the Riemann invariants and capture fluctuations along the characteristics. For the normal modes, the space-time scale is superdiffusive when 2 < θ ⩽ 4 and diffusive when θ > 4.

Asymptotic Scattering by Poissonian Thermostats

In the present paper, we consider an infinite chain of harmonic oscillators coupled with a Poisson thermostat attached at a point. The kinetic limit for the energy density of the chain, given by the Wigner distribution, satisfies a transport equation outside the thermostat location. A boundary condition emerges at this site, which describes the reflection-transmission-scattering of the wave energy scattered off by the thermostat. Formulas for the respective coefficients are obtained. Unlike the case of the Langevin thermostat studied in Komorowski et al. (Arch. Ration. Mech. Anal. 237, 497–543, 2020), the Poissonian thermostat scattering generates in the limit a continuous cloud of waves of frequencies different from that of the incident wave.

Non-homogeneous Chain of Harmonic Oscillators

Non-homogeneous Chain of Harmonic Oscillators

Hydrodynamic Limit for a Disordered Quantum Harmonic Chain

We study the hydrodynamic limit, in the hyperbolic space-time scaling, for a one-dimensional unpinned chain of quantum harmonic oscillators with random masses. To the best of our knowledge, this is among the first examples where one can prove the hydrodynamic limit for a quantum system rigorously. In fact, we prove that after hyperbolic rescaling of time and space, the distribution of the elongation, momentum, and energy averaged under the proper locally Gibbs state converges to the solution of the Euler equation. Moreover, our result indicates that the temperature profile is frozen in any space-time scale; in particular, the thermal diffusion coefficient vanishes. There are two main phenomena in this chain that enable us to deduce this result. First is the Anderson localization, which decouples the mechanical and thermal energy, providing the closure of the equation for energy. The second phenomenon is similar to some sort of decay of correlation phenomena, which let us circumvent the difficulties arising from the fact that our Gibbs state is not a product state due to the quantum nature of the system.

Energy transport in one-dimensional oscillator arrays with hysteretic damping

Energy transport in 1-dimensional oscillator arrays has been extensively studied to date in the conservative case, as well as under weak viscous damping. When driven at one end by a sinusoidal force, such arrays are known to exhibit the phenomenon of supratransmission, i.e. a sudden energy surge above a critical driving amplitude. In this paper, we study 1-dimensional oscillator chains in the presence of hysteretic damping, and include nonlinear stiffness forces that are important for many materials at high energies. We first employ Reid's model of local hysteretic damping, and then study a new model of nearest neighbor dependent hysteretic damping to compare their supratransmission and wave packet spreading properties in a deterministic as well as stochastic setting. The results have important quantitative differences, which should be helpful when comparing the merits of the two models in specific engineering applications.

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.

Wide Tuning-range LO Chain with a Hybrid Inductor for LTE and SAW-less GSM Transceiver

A low phase noise, multi-standard local oscillator (LO) chain for LTE and GSM is presented. The LO chain is composed of a dynamic divider and a CMOS VCO with a hybrid inductor using both bondwire and a planar spiral inductor in the same area. This approach improves both tuning range and phase noise characteristics without additional area loss. Based on this concept, the LO chain with a hybrid inductor VCO is implemented in a 65nm CMOS process. The measured performance satisfies wide tunability and most stringent GSM transmitter phase noise specification, which proves that the LO chain can be used for LTE and SAW-less GSM transceiver.

On a hybrid approximation concept for self‐excited periodic oscillations of large‐scale dynamical systems

AbstractWhen approximating periodic solutions in the context of large‐scale dynamical systems involving strong local nonlinearities, efficiency is of special interest. Hence, the literature suggests a combination of two approximation methods for increasing the ratio of computational cost to accuracy. Within this contribution, a combination of Finite Difference and Harmonic Balance method is proposed. Due to the usage of Harmonic Balance it is shown, that the resulting equations only depend on the degrees of freedom that are affected by nonlinear forces. As an application, a self‐excited limit cycle of a chain of oscillators is approximated and results are compared against numerical time integration to highlight qualitative accuracy.