Time-dependent External Force Research Articles

Optimal synchronization to a limit cycle.

In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic trajectory-the limit cycle-after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimizing the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed-limit inequality, which expresses a trade-off between the connection time and cost-in terms of the non-conservative work. We show how the above results can be generalized to the broader family of non-linear oscillators given by the Liénard equation. Finally, we also look into the problem of minimizing the total work done by the external force.

Derivation of the nonequilibrium generalized Langevin equation from a time-dependent many-body Hamiltonian.

It has become standard practice to describe systems that remain far from equilibrium even in their steady state by Langevin equationswith colored noise which is chosen independently from the friction contribution. Since these Langevin equationsare typically not derived from first-principle Hamiltonian dynamics, it is not clear whether they correspond to physically realizable scenarios. By exact Mori projection in phase space we derive the nonequilibrium generalized Langevin equation(GLE) for an arbitrary phase-space dependent observable A from a generic many-body Hamiltonian with a time-dependent external force h(t) acting on the same observable A. This is the same Hamiltonian from which the standard fluctuation-dissipation theorem is derived, which reflects the generality of our approach. The observable A could, for example, be the position of an atom, of a molecule or of a macroscopic object, the distance between two such entities or a more complex phase-space function such as the reaction coordinate of a chemical reaction or of the folding of a protein. The Hamiltonian could, for example, describe a fluid, a solid, a viscoelastic medium, or even a turbulent inhomogeneous environment. The GLE, which is a closed-form equationof motion for the observable A, is obtained in explicit form to all orders in h(t) and without restrictions on the type of many-body Hamiltonian or the observable A. If the dynamics of the observable A corresponds to a Gaussian process, the resultant GLE has a similar form as the equilibrium Mori GLE, and in particular the friction memory kernel is given by the two-point autocorrelation function of the sum of the complementary and the external force h(t). This is a nontrivial and useful result, as many observables that characterize nonequilibrium systems display Gaussian statistics. For non-Gaussian nonequilibrium observables correction terms appear in the GLE and in the relation between the force autocorrelation and the friction memory kernel, which are explicitly given in terms of cubic correlation functions of A. Interpreting the external force h(t) as a stochastic process, we derive nonequilibrium corrections to the fluctuation-dissipation theorem and present methods to extract all GLE parameters from experimental or simulation time-series data, thus making our nonequilibrium GLE a practical tool to study and model general nonequilibrium systems.

Application of functionally graded graphene nanoplatelets to improve transient vibrations of the car’s roof under mechanical loading: Introducing machine learning algorithm for transient problems

Doubly curved structures under external force due to their applicable structure can be used in the roofs of various structures. Good knowledge about the transient dynamics of this kind of structure under external shock loading is very important in the engineering field. So, in the current work, steady and transient vibrations of composite doubly curved structures under time-dependent external force for the first time are presented. The current structure is reinforced by graphene nanoplatelets with high efficiency to improve the mechanical properties of the system in the thickness direction due to external shock loading in this direction. The peak pulse pressure, the Laplace transform, and some mathematical work to get time-dependent vibration responses of the composite doubly curved structures under external force. With the aid of different pre-processing of machine learning algorithms (Z-Score, Rrobust, Min-Max, and Quantile transformation), the results are predicted. The optimization algorithm, neural network structure, learning rate, mean of square error, and other training data are obtained for the current machine learning algorithm. The outputs of the machine learning method show that instead of simulating the complex structures in various situations with the aid of mathematical simulation, the researchers can use machine learning algorithms to correctly simulate this kind of system in different situations and material properties to obtain optimum conditions for doubly curved structure structures. The findings in the results section suggest that the time-dependent displacement and stress fields of the system are significantly influenced by the geometrical and physical characteristics of the current composite curved structure.

Characterizing the time-dependent external force on the cars’ hood door in accident using deep neural networks

The car hood door is a very important part of a car, especially in the time of an accident. So, in this report for the first time, the innovative applicable model is presented to simulate the transient dynamics of a car’s hood door under axial mechanical shock loading at the time of the accident. Due to axial mechanical shock excitation, it is very important to improve the stability of the structure in the axial direction. So, the composite structure is reinforced by graphene nanoplatelets in the axial direction. For modeling the current structure, the differential conditions are translated into Laplace space in order to determine how the system will react as a function of time. At this stage, Laplace space is employed to infer a temporal perception of the system's reaction using Abate and Dubner's modified message of strategy. To verify the results, the current results are compared to open-source results from the literature and deep neural networks (DNN). To anticipate the system's vibrational behavior, DNN incorporates a supervised neural network based on physical data. In this situation, data-driven research and solutions are used to determine natural frequencies. The results highlight how important GPL characteristics are to transient and forced vibrations of the composite system. The findings of the present research may serve as a reference point and as helpful suggestions for the next structural design methods with improved qualities.

Targeting Staphylococcal Cell-Wall Biosynthesis Protein FemX Through Steered Molecular Dynamics and Drug-Repurposing Approach.

Staphylococcus aureus-mediated infection is a serious threat in this antimicrobial-resistant world. S. aureus has become a "superbug" by challenging conventional as well as modern treatment strategies. Nowadays, drug repurposing has become a new trend for the discovery of new drug molecules. This study focuses on evaluating FDA-approved drugs that can be repurposed against S. aureus infection. Steered molecular dynamics (SMD) has been performed for Lumacaftor and Olaparib against staphylococcal FemX to understand their binding to the active site. A time-dependent external force or rupture force has been applied to the ligands to calculate the force required to dislocate the ligand from the binding pocket. SMD analysis indicates that Lumacaftor has a high affinity for the substrate binding pocket in comparison to Olaparib. Umbrella sampling exhibits that Lumacaftor possesses a higher free energy barrier to displace it from the ligand-binding site. The bactericidal activity of Lumacaftor and Olaparib has been tested, and it shows that Lumacaftor has moderate activity along with biofilm inhibition potential (MIC value with conc. 128 μg/mL). Pharmacokinetic and toxicology evaluations indicate that Lumacaftor has higher pharmacokinetic potential with lower toxicity. This is the first experimental report where staphylococcal FemX has been targeted for the discovery of new drugs. It is suggested that Lumacaftor may be a potential lead molecule against S. aureus.

Statistical mechanics of the GENERIC framework under external forcing.

The General Equation for Non-Equilibrium Reversible Irreversible Coupling (generic) framework provides a thermodynamically consistent approach to describe the evolution of coarse-grained variables. This framework states that Markovian dynamic equations governing the evolution of coarse-grained variables have a universal structure that ensures energy conservation (first law) and entropy increase (second law). However, the presence of external time-dependent forces can break the energy conservation law, requiring modifications to the framework's structure. To address this issue, we start from a rigorous and exact transport equation for the average of a set of coarse-grained variables derived from a projection operator technique in the presence of external forces. Under the Markovian approximation, this approach provides the statistical mechanics underpinning of the generic framework under external forcing conditions. By doing so, we can account for the effects of external forcing on the system's evolution while ensuring thermodynamic consistency.

Relations between timescales of stochastic thermodynamic observables

Abstract Any real physical process that produces entropy, dissipates energy as heat, or generates mechanical work must do so on a finite timescale. Recently derived thermodynamic speed limits place bounds on these observables using intrinsic timescales of the process. Here, we derive relationships for the thermodynamic speeds for any composite stochastic observable in terms of the timescales of its individual components. From these speed limits, we find bounds on thermal efficiency of stochastic processes exchanging energy as heat and work and bound the rate of entropy change in a system with entropy production and flow. Using the time set by an external clock, we find bounds on the first time to reach any value for the entropy production. As an illustration, we compute these bounds for Brownian particles diffusing in space subject to a constant-temperature heat bath and a time-dependent external force.

Qualitative analysis of a mechanical system of coupled nonlinear oscillators

In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to + ∞ , provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations.

Statistical dynamics of driven systems of confined interacting particles in the overdamped-motion regime

Systems consisting of confined, interacting particles doing overdamped motion admit an effective description in terms of nonlinear Fokker-Planck equations. The behavior of these systems is closely related to the S power-law entropies and can be interpreted in terms of the S-based thermostatistics. The connection between overdamped systems and the S measures provides valuable insights on diverse physical problems, such as the dynamics of interacting vortices in type-II superconductors. The S-thermostatistical approach to the study of many-body systems described by nonlinear Fokker-Planck equations has been intensively explored in recent years, but most of these efforts were restricted to systems affected by time-independent external potentials. Here, we extend this treatment to systems evolving under time-dependent external forces. We establish a lower bound on the work done by these forces when they drive the system during a transformation. The bound is expressed in terms of a free energy based on the S entropy and is satisfied even if the driving forces are not derivable from a potential function. It constitutes a generalization, for systems governed by nonlinear Fokker-Planck equations involving general time-dependent external forces, of the H-theorem satisfied by these systems when the external forces arise from a time-independent potential.

Machine learning-based statistical closure models for turbulent dynamical systems.

We propose a machine learning (ML) non-Markovian closure modelling framework for accurate predictions of statistical responses of turbulent dynamical systems subjected to external forcings. One of the difficulties in this statistical closure problem is the lack of training data, which is a configuration that is not desirable in supervised learning with neural network models. In this study with the 40-dimensional Lorenz-96 model, the shortage of data is due to the stationarity of the statistics beyond the decorrelation time. Thus, the only informative content in the training data is from the short-time transient statistics. We adopt a unified closure framework on various truncation regimes, including and excluding the detailed dynamical equations for the variances. The closure framework employs a Long-Short-Term-Memory architecture to represent the higher-order unresolved statistical feedbacks with a choice of ansatz that accounts for the intrinsic instability yet produces stable long-time predictions. We found that this unified agnostic ML approach performs well under various truncation scenarios. Numerically, it is shown that the ML closure model can accurately predict the long-time statistical responses subjected to various time-dependent external forces that have larger maximum forcing amplitudes and are not in the training dataset. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Nonlinear continuum mechanics of thick hyperelastic sandwich beams using various shear deformable beam theories

In this study, the time-dependent mechanics of multilayered thick hyperelastic beams are investigated for the first time using five different types of shear deformation models for modelling the beam (i.e. the Euler–Bernoulli, Timoshenko, third-order, trigonometric and exponential shear deformable models), together with the von Kármán geometrical nonlinearity and Mooney–Rivlin hyperelastic strain energy density. The laminated hyperelastic beam is assumed to be resting on a nonlinear foundation and undergoing a time-dependent external force. The coupled highly nonlinear hyperelastic equations of motion are obtained by considering the longitudinal, transverse and rotation motions and are solved using a dynamic equilibrium technique. Both the linear and nonlinear time-dependent mechanics of the structure are analysed for clamped–clamped and pinned–pinned boundaries, and the impact of considering the shear effect using different shear deformation theories is discussed in detail. The influence of layering, each layer’s thickness, hyperelastic material positioning and many other parameters on the nonlinear frequency response is analysed, and it is shown that the resonance position, maximum amplitude, coupled motion and natural frequencies vary significantly for various hyperelastic and layer properties. The results of this study should be useful when studying layered soft structures, such as multilayer plastic packaging and laminated tubes, as well as modelling layered soft tissues.

The Navier\u2013Stokes equations on the plane with time-dependent external forces

We are concerned with the non-stationary Navier–Stokes equations on the whole plane with external forces which are non-decaying in time, and give a sufficient condition on the external forces for the existence of a solution which exists for whole time. Typical examples are time-periodic solutions and solutions almost periodic in time. The stability under perturbation is also verified.

An extremum problem for a linear integro-differential system describing creeping flows of a viscoelastic fluid

We consider an optimal control problem for an integro-differential system (with a quadratic cost functional) modeling a three-dimensional creeping flow of an incompressible viscoelastic fluid in a bounded domain with impermeable solid walls. The fluid flow is controlled by the time-dependent external force. The concept of the control operator is proposed. We prove a theorem on the existence of a unique optimal control under the assumption that the set of admissible controls is convex and that it is closed in a suitable function space. Moreover, we obtain a variational ine-quality for the optimal control. The proof of this theorem is based on the application of the Faedo–Galerkin approximation scheme taking into account energy estimates of approximate solutions and using the lemma on the existence and uniqueness of the metric projection of a point onto a closed convex set in a real Hilbert space.

Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains

Electron pumping through energy-gapped systems is restricted for vanishing local density of states at the Fermi level. In this paper, we propose a topological Su–Schrieffer–Heeger (SSH) chain between unbiased leads as an effective electron pump. We analyze the electron transport properties of topologically trivial and nontrivial systems in the presence of external time-dependent forces in the form of one-Gaussian or two-Gaussian perturbations (train impulses). We have found that the topologically trivial chain stands for much better charge pump than other normal or nontrivial chains. It is important that, during the perturbation, electrons are pumped through the mid-gap temporary states or through the induced sidebands states outside the energy gap. We also analyze the local density of states dynamics during the quench transition between different topological phases of the SSH chain. It turns out that after the quench, the edge topological states migrate through other sites and can temporarily exist in a topologically trivial part of the system. The tight-binding Hamiltonian and the evolution operator technique are used in our calculations.

Numerically “exact” simulations of entropy production in the fully quantum regime: Boltzmann entropy vs von Neumann entropy

We present a scheme to evaluate thermodynamic variables for a system coupled to a heat bath under a time-dependent external force using the quasi-static Helmholtz energy from the numerically "exact" hierarchical equations of motion (HEOM). We computed the entropy produced by a spin system strongly coupled to a non-Markovian heat bath for various temperatures. We showed that when changes to the external perturbation occurred sufficiently slowly, the system always reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the von Neumann entropy for an isothermal process, as well as various thermodynamic variables, such as changes in internal energies, heat, and work, for a system in quasi-static equilibrium based on the HEOM. We found that although the characteristic features of the system entropies in the Boltzmann and von Neumann cases as a function of the system-bath coupling strength are similar, those for the total entropy production are completely different. The total entropy production in the Boltzmann case is always positive, whereas that in the von Neumann case becomes negative if we chose a thermal equilibrium state of the total system (an unfactorized thermal equilibrium state) as the initial state. This is because the total entropy production in the von Neumann case does not properly take into account the contribution of the entropy from the system-bath interaction. Thus, the Boltzmann entropy must be used to investigate entropy production in the fully quantum regime. Finally, we examined the applicability of the Jarzynski equality.

Recurrent solutions of the Schrödinger-KdV system with boundary forces

In this paper, we consider the Schrodinger-KdV system with time-dependent boundary external forces. We give conditions on the external forces sufficient for the unique existence of small solutions bounded for all time. Then, we investigate the existence of bounded solution, periodic solution, quasi-periodic solution and almost periodic solution for the Schrodinger-KdV system. The main difficulty is the nonlinear terms in the equations, in order to overcome this difficulty, we establish some properties for the semigroup associated with linear operator which is a crucial tool.

Creating deterministic collisions between two orbiting bodies

We aim to create deterministic collisions between orbiting bodies by applying a time-dependent external force to one or both bodies, whether the bodies are mutually repulsive, as in the two- or multi-electron atomic case or mutually attractive, as in the planetary-orbit case. Specifically, we have devised a mathematical framework for causing deterministic collisions by launching an inner orbiting body to a higher energy such that this inner body is guaranteed to collide with the outer body. Our method first expresses the problem mathematically as coupled nonlinear differential equations with a time-dependent driving force and solves to find a feasible solution for the force function. Although our calculation is based strictly on classical physics, our approach is suitable for the case of helium with two highly excited electrons and is also valid for creating collisions in the gravitational case such as for our solar system.

Quantum theory of radiation by nonstationary systems with application to high-order harmonic generation

In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.

Dynamics of 2D Incompressible Non-autonomous Navier–Stokes Equations on Lipschitz-like Domains

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier–Stokes equations with a non-homogeneous boundary condition on Lipschitz-like domains. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimates on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is perturbed from a stationary force by a time-dependent perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we investigate the regularity of these attractors when smoother initial data are given. Our results are new even for smooth domains.

Optimal distributed control of a 2D simplified Ericksen–Leslie system for the nematic liquid crystal flows

We study the initial boundary value problem of a simplified Ericksen–Leslie system modeling the incompressible nematic liquid crystal flows in two dimensions of space, where the equations of the velocity field are characterized by a time-dependent external force g(t) and a no-slip boundary condition, and the equations for the molecular orientation are subjected to a time-dependent Dirichlet boundary condition h(t). Based on the recently addressed well-posedness and regularity results of the system, we present a rigorous proof to show the existence of optimal distributed controls, the control-to-state operator is Fréchet differentiable and first-order necessary optimality conditions for an associated optimal control problem.