Nonlocal Motion Research Articles

Quantifying nonlocal bedload transport: A regional-based nonlocal model for bedload transport from local to global scales

Recent studies have emphasized the importance of nonlocal models in characterizing bedload transport in natural rivers, particularly in mixed-size gravel beds or steep hillslopes. Nonlocality denotes that a quantity (flux) at a specific location x is dependent on the conditions in the surrounding area, as opposed to solely at the location itself. This concept applies in bedload transport even in planar flumes, where particles are entrained at an upstream position and travel a finite distance, ultimately contributing nonlocally to the sediment flux. However, existing bedload transport models, such as the advection–diffusion equation (ADE) or the fractional derivative equation (FDE) models, are inadequate in characterizing the nonlocal transport behavior of bedload at a regional scale. Large errors may arise from the lack of an accurate description of the nonlocal bedload transport processes at regional scales. This study proposes a regional-based nonlocal bedload transport model, which is conceptualized from the probabilistic Exner-based equations and the peridynamic (PD) differential operator. The PD model encapsulates the nonlocal motion of bedload sediments on the basis of the PD differential operator, by utilizing a pre-defined weight function and influence domain. Comparisons demonstrate that the PD model serves as a generalized tool connecting the local and the global models with different PD functions and influence domains. Its variability on kernel function and influence domain, enable it conveniently describe sub-, super-, and normal diffusion behaviors of bedload transport.

Bending Vibration Analysis of Nanobeams using the Nonlocal Motion Equations Solved by an Integral Approach

This paper deals with the dynamic characteristics for bending vibrations of Euler-Bernoulli type nanobeams taking into account the scale effects via the nonlocal motion equations. An integral method, based on the use of Green’s functions, has been used in order to obtain the corresponding eigenvalue problem. The proposed integral approach is an approximate matrix method. Effects of different boundary conditions and of an elastic foundation have been also included. The presented numerical examples show good agreement when compared to results from literature. The proposed method can be used in the case of nanodevices analysis modeled as beams (MEMS, NEMS).

On the dynamics of small-sized structures

In this study, free vibration analyses of small-scaled trusses and frames are firstly carried out based on nonlocal elasticity of Eringen. Nonlocal matrix motion formulation is derived by using linear algebraic equations. Finite element method based weighted residual is utilized to solve the resulting equations. Various numerical studies are presented for nondimensional natural frequencies of different truss and frame models. A detailed parametric study is performed to investigate the influences of nonlocal parameter, geometric properties, direction angle, mode numbers, and length-to-diameter ratio on the natural frequencies of micro/nano trusses and frames. It is revealed that there is a significant relationship between the size-dependent dynamic response of these structures and the geometrical and structural properties of them.

Long-range vortex transfer in superconducting nanowires

Under high-enough values of perpendicularly-applied magnetic field and current, a type-II superconductor presents a finite resistance caused by the vortex motion driven by the Lorentz force. To recover the dissipation-free conduction state, strategies for minimizing vortex motion have been intensely studied in the last decades. However, the non-local vortex motion, arising in areas depleted of current, has been scarcely investigated despite its potential application for logic devices. Here, we propose a route to transfer vortices carried by non-local motion through long distances (up to 10 micrometers) in 50 nm-wide superconducting WC nanowires grown by Ga+ Focused Ion Beam Induced Deposition. A giant non-local electrical resistance of 36 Ω has been measured at 2 K in 3 μm-long nanowires, which is 40 times higher than signals reported for wider wires of other superconductors. This giant effect is accounted for by the existence of a strong edge confinement potential that hampers transversal vortex displacements, allowing the long-range coherent displacement of a single vortex row along the superconducting channel. Experimental results are in good agreement with numerical simulations of vortex dynamics based on the time-dependent Ginzburg-Landau equations. Our results pave the way for future developments on information technologies built upon single vortex manipulation in nano-superconductors.

Jerk in Planetary Systems and Rotational Dynamics, Nonlocal Motion Relative to Earth and Nonlocal Fluid Dynamics in Rotating Earth Frame

Some results following from the implications of nonlocal-in-time kinetic energy approach introduced recently by Suykens in the framework of rotational dynamics and motion in a non-inertial frame are discussed. Their roles in treating aspects concerning the nonlocal motion relative to Earth, the free-fall problem, the Foucault pendulum and the motion of a massive body in a rotating tube are analyzed. Governing nonlocal equations of fluid dynamics in particular the nonlocal-in-time Navier–Stokes equations are constructed under the influence of Earth rotation. Their properties are analyzed and a number of features were revealed and discussed accordingly.

Suppression of the 0−π transition in a Josephson junction with parallel double-quantum-dot barriers

With the help of the numerical renormalization group method, we theoretically investigate the Josephson phase transition in a parallel junction with one quantum dot embedded in each arm. It is found that in the cases of uniform dot levels and dot-superconductor couplings, the Josephson phase transition will be suppressed. This is manifested as the fact that with the enhancement of the electron correlation, the supercurrent only arrives at its $\pi'$ phase but cannot enter its $\pi$ phase. Moreover, when the dot levels are detuned, one $\pi'$-phase island appears in the phase diagram. Such a result is attributed to the nonlocal motion of the Cooper pair in this structure. We believe that this work can be helpful in understanding the Josephson phase transition modified by the electron correlation and quantum interference.

On nonlocal complex Maxwell equations and wave motion in electrodynamics and dielectric media

A formulation of Maxwell’s equations on the complex domain is presented in this paper which is based on the extension of the nonlocal-in-time kinetic energy approach recently introduced by Suykens. New wave equations with time-dependent source terms are obtained in a uniform optical medium. A number of physical effects were raised and their implications in classical electrodynamics besides to the propagation of waves in dielectric media are analyzed and discussed accordingly.

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

ABSTRACTIn this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.

GENERAL INTEGRAL EQUATIONS OF MICROMECHANICS OF HETEROGENEOUS MATERIALS

One considers a linear composite medium, which consists of a homogeneous matrix containing either the periodic or random set of heterogeneities. An operator form of the general integral equation (GIE) is obtained for the general cases of local and nonlocal problems, static and wave motion phenomena for composite materials with periodic and random (statistically homogeneous and inhomogeneous, so-called graded) structures containing coated or uncoated inclusions of any shape and orientation with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled, homogeneous or inhomogeneous external fields of different physical nature. The GIE, connecting the driving fields and fluxes in a point being considered and the fields in the surrounding points, are obtained for both the random and periodic fields of heterogeneities in the infinite media. The new GIE is presented in a general form of perturbations introduced by the heterogeneities and taking into account a possible imperfection of interface conditions. The mentioned perturbations can be found by any available numerical method which has advantages and disadvantages and it is crucial for the analyst to be aware of their range of applications. The method of obtaining of the GIE is based on a centering procedure of subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods. One proves the absolute convergence of the proposed GIEs which are presented in two equivalent forms for both the driving fields and fluxes. Some particular cases, asymptotic representations, and simplifications of proposed GIE are presented for the particular constitutive equations such as linear thermoelastic cases with the perfect and imperfect interfaces, conductivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocal elastic properties of constituents, and the wave propagation in composites with electromagnetic, optic, and mechanical responses.

Solution of general integral equations of micromechanics of heterogeneous materials

One considers a linear composite materials (CM), which consists of a homogeneous matrix containing a random set of heterogeneities. An operator form of solution of the general integral equation (GIE) for the general cases of local and nonlocal problems, static and wave motion phenomena for composite materials with random (statistically homogeneous and inhomogeneous, so-called graded) structures containing coated or uncoated inclusions of any shape and orientation with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled, homogeneous or inhomogeneous external fields of different physical nature. The GIE, connecting the driving fields and fluxes in a point being considered and the fields in the surrounding points, are obtained for the random fields of heterogeneities in the infinite media. Estimations of the effective properties and both the first and second statistical moments of fields in the constituents of CMs are presented in a general form of perturbations introduced by the heterogeneities and taking into account a possible imperfection of interface conditions. The solution methods of GIEs are obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known methods of micromechanics. Some particular cases, asymptotic representations, and simplifications of proposed methods are presented for the particular constitutive equations such as linear thermoelastic cases with the perfect and imperfect interfaces, conductivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocal elastic properties of constituents, and the wave propagation in composites with electromagnetic, optic and mechanical responses.

Asymptotic Behavior of a Nonlocal Diffusive Logistic Equation

The long time behavior of a logistic-type equation modeling the motion of cells is investigated. The equation we consider takes into account birth and death processes using a simple logistic effect as well as a nonlocal motion of cells using a nonlocal Darcy's law with regular kernel. Using the periodic framework we first investigate the well-posedness of the problem before deriving some information about its long time behavior. The lack of asymptotic compactness of the system is overcome by making use of the Young measure theory. This allows us to conclude that the semiflow converges for the Young measure topology.

On the nonlocal properties of relativistic ponderomotive force

A theory of relativistic ponderomotive force of transversely localized laser fields is presented by taking into account the nonlocal effects that correspond to higher order terms of the expansion parameter ϵ≡l/L, i.e., the ratio between particle excursion length and scale length of the field amplitude gradient, while the existing local theory is the first order of ϵ. As a method for preserving the Hamiltonian structure up to higher orders, we employ the variational principle in noncanonical phase space coordinates incorporated with the Lie transformation. By finding noncanonical coordinates and gauges, we obtain a new formula for the ponderomotive force that involves new terms in the third order represented by the second and third spatial derivatives. The force then depends not only on the local field gradient but also on the curvature and its variation which represent the effects of higher-order nonlocal particle motion. The higher-order terms are found to be subject to the symmetry of the field structure. The obtained formula is accessible to the regime in which the higher derivatives of the field amplitude regulate the interaction. We have applied the formula to laser fields exhibiting flat-top super Gaussian and concave hollow transverse structures which are effective in maintaining the long time scale interaction. The associated nonlocal effects are found to play a key role in determining the interactions. Comparison with the direct integration of the particle orbit demonstrates the validity of the derived formula.

Nonlinear vibration and instability of embedded double-walled carbon nanocones based on nonlocal Timoshenko beam theory

Nonlinear vibration and instability of embedded double-walled carbon nanocones subjected to axial load are investigated in this article based on Eringen's nonlocal theory and Timoshenko beam model. The elastic medium is simulated as Pasternak foundation and the van der Waals forces between the inner and the outer layers of double-walled carbon nanocones are taken into account. Using von Kármán geometric nonlinearity, energy method and Hamilton’s principle, the nonlocal nonlinear motion equations are obtained. The differential quadrature method is applied to discretize the motion equations, which are then solved to obtain the nonlinear frequency and critical fluid velocity of viscous-fluid-conveying double-walled carbon nanocones. A detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, thickness-to-length ratio, temperature change, apex angles, elastic medium and van der Waals forces on the dimensionless frequency and critical buckling load of double-walled carbon nanocones. The results show that the small-size effect on the nonlinear frequency is significant and cannot be neglected; also, the nonlinear frequency and critical buckling load decrease with increasing the cone apex-angle.

Nonlocal properties of ponderomotive force in strongly nonuniform high intensity laser fields

In order to study the particle dynamics in strongly nonuniform and/or localized laser fields, we extend the theory of the ponderomotive force by taking into account the laser field structure up to the higher order with respect to � , the ratio between the particle excursion length and the scale length of the laser field amplitude variation. This extension is realized by keeping the rigorous phase space Lagrangian structure utilizing the noncanonical Lie perturbation up to higher orders. As a result, the equations of motion in a oscillation center coordinate is derived up to the third order of � , which include the second and third spatial derivatives of the laser field amplitude. This denotes that the ponderomotive force depends not only on the local field gradient, but also on the curvature and its derivative. The additional higher-order force originates from the nonlocal particle motion around the oscillation center, which could have an influence especially in the case of tight focusing in the level of laser wavelength.

Asymptotic analysis of a diffuse interface relaxation to a nonlocal optimal partition problem

We present some asymptotic analysis of a diffuseinterface relaxation to a nonlocal optimal domain partition problemand the associated nonlocal interfacial motion when the interfacialwidth is approaching to zero.Motivated by careful numerical calculations,we first discuss several assumptions on thesteady state solutions of the coupled system ofdifferential equations which are supportedby numerical results. These assumptionsallow us to construct a suitable ansatz to the solutionswhich not only captures the leadingorder behavior but also provides sufficient estimates on the nextorder behavior so that more accurate estimates can be shown forinteresting physical quantities such as energies and eigenvalues.When adopted to the gradient flow system, the ansatzgives an estimate of the asymptotic convergence rate in timeto the equilibrium partitions.

Reversal of Nonlocal Vortex Motion in the Regime of Strong Nonequilibrium

We investigate nonlocal vortex motion in weakly pinning a-NbGe nanostructures, which is driven by a transport current I and remotely detected as a nonlocal voltage V{nl}. At a high I of a given polarity, V{nl} changes sign dramatically. This is followed by V{nl} becoming even in I, with the opposite sign at low and high temperatures T. These findings can be explained by a Nernst-like effect resulting from local electron overheating (low T), and a magnetization enhancement due to a nonequilibrium quasiparticle distribution that leads to a gap enhancement near the vortex core (high T).

Homogeneously precessing domain in3He-B: formation and properties

The long-range order realized in the superfluid phases of3He leads to a nonlocal motion of spin in these phases. In the B phase the nonlocality manifestsitself in the formation of a homogeneously precessing domain (HPD). This domain isformed under conditions of nuclear magnetic resonance. Within the domain spin precessescoherently—with the same frequency and phase even though the steady magnetic fieldcan be nonuniform. Coherence of precession is maintained by the spin currentcarried by the condensate of Cooper pairs. The key experiments, revealing themain properties of the HPD and the underlying theory are briefly reviewed inthis paper.

Wave-particle duality in complex space

The purpose of this paper was to justify the fact that deterministic corpuscular description of a free particle can be made reconciled with its dual probabilistic wave description in complex space. It is found that the known wave-particle duality can be best manifested in complex space by showing that the wave motion associated with a material particle is just the phenomenon of projection of its complex motion into real space. To verify this new interpretation of matter wave, the equation of motion for a particle moving in complex space is derived first, then it is solved to reveal how the interaction between the real and imaginary motion can produce the particle’s wave motion observed in real space. The derived complex equation of motion for a “free” particle indicates that a so-called free particle is only free from classical potential, but not free from the complex quantum potential. Due to the action of this complex quantum potential, a free particle may move either right or left in a classical way retaining its corpuscular property, or may oscillate between the two directions producing a non-local wave motion. A propagation criterion is derived in this paper to determine when a particle follows a classical corpuscular motion and when it follows a quantum wave motion. Based on this new interpretation, the internal mechanism producing polarization of matter wave and the formation of interference fringes can all be understood from the particle’s motion in complex space, and the reason why wave function can be served as a probability density function also becomes clear.

Steering for a class of dynamic nonholonomic systems

We derive control algorithms for a class of dynamic nonholonomic steering problems, characterized as mechanical systems with nonholonomic constraints and symmetries. We use a simplified geometric framework in developing two types of control algorithms for such systems. The first targets local configuration controllability, using a perturbation approach to establish results similar to steering using sinusoids. The second method builds on this approach by utilizing kinematic steering algorithms for mobile robots in generating more nonlocal motion plans for this class of systems.

Variational Evolution Problems and Nonlocal Geometric Motion

We consider two variational evolution problems related to Monge-Kantorovich mass transfer. These problems provide models for collapsing sandpiles and for compression molding. We prove the following connection between these problems and nonlocal geometric curvature motion: The distance functions to surfaces moving according to certain nonlocal geometric laws are solutions of the variational evolution problems. Thus we do the first step of the proof of heuristics developed in earlier works. The main techniques we use are differential‐equation methods in the Monge‐Kantorovich theory.