Time-dependent External Force Research Articles

The Riemann problem for the one-dimensional compressible flow of a van der Waals gas

We address the Riemann problem for the one-dimensional compressive flow of a van der Waals gas, in which the external force is a given continuous function of time. The explicit forms for shock wave, contact discontinuity and rarefaction wave are presented. The influence of van der Waals parameter on these elementary waves is discussed. Furthermore, we obtain six kinds of solutions to this Riemann problem and establish a condition for the occurrence of vacuum in the solutions. Particularly, all of the solutions are not self-similar in (t, x)-plane due to the presence of the time-dependent external force.

Uniform stability of a non-autonomous semilinear Bresse system with memory

The Bresse system is a recognized mathematical model for vibrations of a circular arched beam that contains the class of Timoshenko beams when the arch’s curvature is zero. It turns out that the majority of mathematical analysis to Bresse systems are concerned with the asymptotic stability of linear homogeneous problems. Under this scenario, we consider a nonlinear Bresse system modeling arched beams with memory effects, in a nonlinear elastic foundation. Then we establish uniform decay rates of the energy under time-dependent external forces.

Subdiffusion in an external force field.

The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time-dependent force, we study the subdiffusion described by the subordinated Langevin equation with white Gaussian noise or, equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as the ensemble- and time-averaged mean squared displacements, position autocorrelation function, correlation coefficient, and generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are nonstationary, nonergodic, and aging.

Curing the self-force runaway problem in finite-difference integration

The electromagnetic self-force equation of motion is known to be afflicted by the so-called runaway problem. A similar problem arises in the semiclassical Einstein's field equation and plagues the self-consistent semiclassical evolution of spacetime. Motivated to overcome the latter challenge, we first address the former (which is conceptually simpler), and present a pragmatic finite-difference method designed to numerically integrate the self-force equation of motion while curing the runaway problem. We restrict our attention here to a charged point-like mass in a one-dimensional motion, under a prescribed time-dependent external force $F_{ext}(t)$. We demonstrate the implementation of our method using two different examples of external force: a Gaussian and a Sin^4 function. In each of these examples we compare our numerical results with those obtained by two other methods (a Dirac-type solution and a reduction-of-order solution). Both external-force examples demonstrate a complete suppression of the undesired runaway mode, along with an accurate account of the radiation-reaction effect at the physically relevant time scale, thereby illustrating the effectiveness of our method in curing the self-force runaway problem.

Pullback dynamics of 3D Navier–Stokes equations with nonlinear viscosity

This paper is concerned with pullback dynamics of 3D Navier–Stokes equations with variable viscosity and subject to time-dependent external forces. Our main result establishes the existence of finite-dimensional pullback attractors in a general setting involving tempered universes. We also present a sufficient condition on the viscosity coefficients that guarantees the attractors are nontrivial. We end the paper by showing the upper semi-continuity of pullback attractors as the non-autonomous perturbation vanishes.

Time periodic solutions to the beam equation with weak damping

In this paper, we study the beam equation with weak damping in n-dimensional space. In the case of time-dependent periodic external force imposed, we prove the existence and uniqueness of time periodic solutions that have the same period as the time-dependent periodic external force in some suitable function space for all space dimensions n ≥ 1. The proof is based on the spectral analysis for the solution operators and the contraction mapping theorem. Moreover, we show the time asymptotic stability of time periodic solutions by continuous argument.

Study on solving the ill-posed problem of force load reconstruction

Force load reconstruction methodology is used to estimate the unknown time-dependent external load forces in a damped system based on the structural response. However, the measurement errors of responses and estimation error of the system can affect the reliability of the results in several such inverse problems. An implicit Landweber method was proposed for single-source and multi-source force load reconstructions, and the concept of response sensitivity was used to reconstruct the dynamic force load. Numerical simulations and transonic wind tunnel tests of a spacecraft model demonstrate the effectiveness and robustness of the proposed method.

Intrinsic ratchets: A Hamiltonian approach

An asymmetric Brownian particle subjected to an external time-dependent force may acquire a net drift velocity, and thus operate as a motor or ratchet, even if the external force is represented by an unbiased time-periodic function or by a zero-centered noise. For an adequate description of such ratchets, a conventional Langevin equation linear in the particle's velocity is insufficient, and one needs to take into account the first nonlinear correction to the dissipation force which emerges beyond the weak coupling limit. We derived microscopically the relevant nonlinear Langevin equation by extending the standard projection operation technique beyond the weak coupling limit. The particle is modeled as a rigid cluster of atoms and its asymmetry may be geometrical, compositional (when a cluster is composed of atoms of different types), or due to a combination of both factors. The drift velocity is quadratic in the external force's amplitude and increases with decreasing the force's frequency (for a periodic force) and inverse correlation time (for a fluctuating force). The maximum value of the drift velocity is independent on the particle's mass and achieved in the adiabatic limit, i.e. for an infinitely slow change of the external field.

Reciprocal relations associated with linear responses to mechanical and thermal perturbations in nonequilibrium Langevin systems

We study universal relationships in linear responses to instances when mechanical and thermal perturbations are applied to a nonequilibrium steady state. For perturbations in nonequilibrium Langevin systems, we consider a time-dependent external force exerted on a particle along with a time-dependent temperature for the heat bath. In this system, by calculating heat currents and particle velocity, we obtain the kinetic coefficients (Onsager's coefficients) and the time correlation functions using the Fokker--Planck equations. With these quantities, we derive the reciprocal relation associated with the violation of the fluctuation response relation. We obtain the relation by integrating the extent of the violation with respect to the frequency. We also find the reciprocity in the first moment of the frequency integration. These frequency sum rules hold for both overdamped and underdamped regimes for any type of potential and any strength of the external force.

Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential

In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier–Stokes equations, nonlinearly coupled with a convective nonlocal Cahn–Hilliard equation. The system rules the evolution of the (volume-averaged) velocity $$\varvec{u}$$ of the mixture and the (relative) concentration difference $$\varphi $$ of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force $$\varvec{v}$$ acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map $$\varvec{v}\mapsto [\varvec{u},\varphi ]$$, and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with Rocca (SIAM J Control Optim 54:221–250, 2016). There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and Gal (WIAS preprint series No. 2309, Berlin, 2016).

Superposition principle for the simultaneous optimization of collective responses.

We study the dynamics of sets of independent systems, all of which are coupled to the same time-dependent external force. Using optimal control theory, we compute the most efficient temporal pulse shape for this force that can maximize simultaneously the collective response of these systems. This response can be a weighted sum of all amplitudes at the final interaction time. Remarkably, it turns out that for certain systems this optimal force for the collective response can be related to the individual forces that would optimize each system separately. We illustrate this superposition principle for the simultaneous optimization of collective responses with numerical and also analytical solutions for sets of damped linear and nonlinear oscillators. We also apply this principle to predict the optimal temporal profile of a laser pulse that can maximize the final macroscopic polarization (total dipole moment) of a set of quantum mechanical two-level atoms.

Quantum mechanical response to a driven Caldeira-Leggett bath

We determine the frequency-dependent response characteristics of a quantum system to a driven Caldeira-Leggett bath. The bath degrees of freedom are explicitly driven by an external time-dependent force, in addition to the direct time-dependent forcing of the system itself. After general considerations of driven Caldeira-Leggett baths, we consider the Rubin model of a chain of quantum particles coupled by linear springs as an important model of a quantum dissipative system. We show that in the presence of time-dependent driving of the chain, this model can be mapped to a quantum system which couples to a driven Caldeira-Leggett bath. The effect of the bath driving is captured by a time-dependent force on the central system, which is, in principle, non-Markovian in nature. We study two specific examples, the exactly solvable case of a harmonic potential and a quantum two-state system for which we assume a weak system-bath coupling. We evaluate the dynamical response to a periodic driving of the system and the bath. The dynamic susceptibility is shown to be altered qualitatively by the bath drive: The dispersive part is enhanced at low frequencies and acquires a maximum at zero frequency. The absorptive part develops a shoulder-like behavior in this frequency regime. These features seem to be generic for quantum systems in a driven Caldeira-Leggett bath.

Scattering of accelerated wave packets

Wave-packet scattering from a stationary potential is significantly modified when the wave-packet is subject to an external time-dependent force during the interaction. In the semiclassical limit, wave--packet motion is simply described by Newtonian equations and the external force can, for example, cancel the potential force making a potential barrier transparent. Here we consider wave-packet scattering from reflectionless potentials, where in general the potential becomes reflective when probed by an accelerated wave-packet. In the particular case of the recently-introduced class of complex Kramers-Kronig potentials we show that a broad class of time dependent forces can be applied without inducing any scattering, while there is a breakdown of the reflectionless property when there is a broadband distribution of initial particle momentum, involving both positive and negative components.

Pullback Dynamics of Non-autonomous Timoshenko Systems

This paper is concerned with the Timoshenko system, a recognized model for vibrations of thin prismatic beams. The corresponding autonomous system has been widely studied. However, there are only a few works dedicated to its non-autonomous counterpart. Here, we investigate the long-time dynamics of Timoshenko systems involving a nonlinear foundation and subjected to perturbations of time-dependent external forces. The main result establishes the existence of a pullback exponential attractor, which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. The upper-semicontinuity of attractors, as the non-autonomous forces tend to zero, is also studied.

Stochastic efficiency of an isothermal work-to-work converter engine.

We investigate the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.

Recurrent solutions of the derivative Ginzburg–Landau equation with boundary forces

ABSTRACTIn this paper, we will establish the existence of the bounded solutions, periodic solutions, quasi-periodic solutions and almost periodic solutions for the derivative Ginzburg–Landau equation with time-dependent boundary external forces. A smoothing effect for the semigroup associated with linear Ginzburg–Landau operator is the crucial tool in establishing the main results.

Nonstationary Navier–Stokes equations with singular time-dependent external forces

We establish a sufficient condition for the existence of solutions to the incompressible Navier–Stokes equations, with singular time-dependent external forces defined in terms of capacity CapH1,2(E).

Nonlinear oscillations of viscoelastic microplates

The nonlinear oscillations of viscoelastic microplates is addressed in this paper based on the modified couple stress theory (MCST). Employing the Kirchhoff plate theory, both out-of-plane and in-plane motions as well as the corresponding inertia are taken into account; an internal damping mechanism, based on Kelvin–Voigt model, is employed to model the behaviour of the material. The strain energy, the kinetic energy, the work done by the viscous parts of the classical and non-classical stresses, and the work of the external time-dependent force are obtained and implemented into Hamilton's framework so as to derive a set of fully coupled nonlinear partial differential equations (PDEs) for motions in the out-of-plane and in-plane directions. The Galerkin scheme is utilized to reduce the set of viscoelastically coupled nonlinear PDEs into a set of nonlinear ordinary differential equations (ODEs). Thereupon, this set of equations is transformed into a new set of size-dependent viscoelastically coupled nonlinear first-order ODEs and then is solved with the aid of a continuation scheme. The nonlinear oscillations is thoroughly investigated through conducting extensive numerical simulations and plotting force-response and frequency-response diagrams of the viscoelastic microsystem. The results reveal that the contributions of the nonlinear damping terms, arising due to employing a viscoelastic model, in the response of the viscoelastic microsystem increase substantially when the forcing amplitude is increased. Moreover, the concurrent presence of the nonlinear amplitude-dependent damping mechanism and the length-scale parameter affects the resonant response of the microplate significantly in both linear and nonlinear senses.

Dynamics of wave equations with moving boundary

This paper is concerned with long-time dynamics of weakly damped semilinear wave equations defined on domains with moving boundary. Since the boundary is a function of the time variable the problem is intrinsically non-autonomous. Under the hypothesis that the lateral boundary is time-like, the solution operator of the problem generates an evolution process U(t,τ):Xτ→Xt, where Xt are time-dependent Sobolev spaces. Then, by assuming the domains are expanding, we establish the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the forcing terms. Our assumptions allow nonlinear perturbations with critical growth and unbounded time-dependent external forces.

Dynamics of a quantum two-state system in a linearly driven quantum bath

When an open quantum system is driven by an external time-dependent force, the coupling of the driving to the central system is usually included whereas the impact of the driving field on the bath is neglected. We investigate the effect of a quantum bath of linearly driven harmonic oscillators on the relaxation dynamics of a quantum two-level system which itself is not directly driven. In particular, we calculate the frequency-dependent response of the system when the bath is subject to a Dirac and a Gaussian driving-pulse. We show that a time-retarded effective force on the system is induced by the driven bath which depends on the full history of the perturbation and the spectral characteristics of the underlying bath. In particular, when a structured Ohmic bath with a pronounced Lorentzian peak is considered, the dynamical response of the system to a driven bath is qualitatively different as compared to the undriven bath. Specifically, additional resonances appear which can be directly associated to a Jaynes-Cummings-like effective energy spectrum.