Nonlocal Dynamics Research Articles

Strongly Nonlocal Dislocation Dynamics in Crystals

We consider the equation where L s is an integro-differential operator of order 2s, with s ∈ (0, 1), W is a periodic potential, and σϵ is a small external stress. The solution v represents the atomic dislocation in the Peierls–Nabarro model for crystals, and we specifically consider the case s ∈ (0, 1/2), which takes into account a strongly nonlocal elastic term. We study the evolution of such dislocation function for macroscopic space and time scales, namely we introduce the function We show that, for small ϵ, the function v ϵ approaches the sum of step functions. From the physical point of view, this shows that the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. We also show that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.

How to test the gauge-invariant non-local quantum dynamics of the Aharonov–Bohm effect

The gauge invariant non local quantum dynamics of the Aharonov-Bohm effect can be tested experimentally by measuring the instantaneous shift of the velocity distribution occurring when the particle passes by the flux line. It is shown that in relativistic quantum theory it is possible to measure the instantaneous velocity with accuracy sufficient to detect the change of the velocity distribution. In non relativistic quantum theory the instantaneous velocity can be measured to any desired accuracy.

Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well

This paper is a direct offspring of the work of Garbaczewski and Stephanovich [“Lévy flights and nonlocal quantum dynamics,” J. Math. Phys. 54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed.

Dynamics of the Logistic Equation with Delay and Delay Control

Asymptotic methods investigate the dynamic properties of the logistic equation with delay and delay control. The possibility to effectively control the characteristics of the relaxation cycle has been illustrated. We introduce a new method of studying the dynamics, provided that the ratio of the delayed control is large enough. The original problem of the dynamics equation with delays has been reduced to the nonlocal special dynamics of nonlinear boundary-value problems of a parabolic type.

Normalization of a System with Two Large Delays

In this paper, the local dynamics of differential equations is studied with two asymptotically large proportional delays. Depending on the parameters, critical cases in the problem of the stability of the equilibrium state are identified. In all critical cases, special evolutionary equations (quasinormal forms) are built. Their nonlocal dynamics determine the local behavior of solutions of the original equations.

Spatially distributed control of the dynamics of the logistic delay equation

The influence exerted by a small spatially inhomogeneous control on the dynamics of the logistic delay equation is studied. This paper consists of two parts. The first deals with the case where the logistic delay equation has a stable relaxation cycle. It is shown that a small control function can give rise to complex relaxation objects, namely, to a large number of different attractors. In the second part, the local dynamics of the stability problem is analyzed in a neighborhood of equilibrium in a close-to-critical case of “infinite” dimension. Special quasi-normal forms are constructed whose nonlocal dynamics determine the local behavior of solutions to the original equation. Some results of a numerical analysis are presented.

Dynamics of the logistic delay equation with a large spatially distributed control coefficient

The local dynamics of the logistic delay equation with a large spatially distributed control coefficient is asymptotically studied. The basic bifurcation scenarios are analyzed depending on the relations between the parameters of the equation. It is shown that the equilibrium states can lose stability even for asymptotically small values of the delay parameter. The corresponding critical cases can have an infinite dimension. Special nonlinear parabolic equations are constructed whose nonlocal dynamics determine the local behavior of solutions to the original boundary value problem.

New insights from comparing statistical theories for inertial particles in turbulence: II. Relative velocities

Part I of this two-part series compared two theories for the radial distribution function (RDF), a statistical measure of the clustering of inertial particles in isotropic turbulence. In Part II, we will contrast three theoretical models for the relative velocities of inertial particles in isotropic turbulence, one by Zaichik et al (2009 New. J. Phys. 11 103018), the second by Pan et al (2010 J. Fluid Mech. 661 73) and the third by Gustavsson et al (2011 Phys. Rev. E. 84 045304). We find that in general they describe the relative velocities in qualitatively similar ways, capturing the influence of the non-local dynamics on the formation of caustics and non-smooth scaling behavior in the dissipation range. We then compare the theoretical predictions with direct numerical simulation data and find that although they capture the qualitative behavior of the data consistently, they differ quantitatively, and we discuss the possible sources of error in each of the theories. Finally, we consider how the Zaichik et al theory predicts that the formation of caustics modifies the form of the RDF, and show that the theory describes the behavior of the RDF for particles whose response time scales with the inertial range time scales of the turbulence.

First passage time on pattern formation in a non-local Fisher population dynamics

Abstract We use stochastic dynamics to develop the patterned attractor of a non-local extended system. This is done analytically using the stochastic path perturbation approach scheme, where a theory of perturbation in the small noise parameter is introduced to analyze the random escape of the stochastic field from the unstable state. Emphasis is placed on the specific mode selection that these types of systems exhibit. Concerning the stochastic propagation of the front we have carried out Monte Carlo simulations which coincide with our theoretical predictions.

Violation of the “information–disturbance relationship” in finite-time quantum measurements

The effect of measurement attributes (quantum level of precision, finite duration) on the classical and quantum correlations is analyzed for a pair of qubits immersed in a common reservoir. We show that the quantum discord is enhanced as the precision of the measuring instrument is increased, and both the classical correlation and the quantum discord experience noticeable changes during finite-time measurements performed on a neighboring partition of the entangled system. The implication of these results on the “information–disturbance relationship” is examined, with critical analysis of the delicate roles played by quantum non-locality and non-Markovian dynamics in the violation of this relationship, which appears surprisingly for a range of measurement attributes. This work highlights that the fundamental limits of quantum mechanical measurements can be altered by exchanges of non-classical correlations such as the quantum discord with external sources, which has relevance to cryptographic technology.

Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects.

Plasmons in metals can oscillate on a sub-wavelength length scale and this large-k response constitutes an inherent prerequisite for fascinating effects such as perfect imaging and intriguing wave phenomena associated with the epsilon-near-zero (ENZ) regime. While there is no upper cut-off within the local-response approximation (LRA) of the plasma polarization, nonlocal dynamics suppress response beyond ω/vF, where vF is the Fermi velocity of the electron gas. Nonlocal response has previously been found to pose limitations to field-enhancement phenomena. Accounting for nonlocal hydrodynamic response, we show that perfect imaging is surprisingly only marginally affected by nonlocal properties of a metal slab, even for a deep subwavelength case and an extremely thin film. Similarly, for the ENZ response we find no indications of nonlocal response jeopardizing the basic behaviors anticipated from the LRA. Finally, our study of waveguiding of gap plasmons even shows a positive nonlocal influence on the propagation length.

Examining non-locality and quantum coherent dynamics induced by a common reservoir

If two identical emitters are coupled to a common reservoir, entanglement can be generated during the decay process. When using Bell's inequality to examine the non-locality, however, it is possible that the bound cannot be violated in some cases. Here, we propose to use the steering inequality to examine the non-locality induced by a common reservoir. Compared with the Bell inequality, we find that the steering inequality has a better tolerance for examining non-locality. In view of the dynamic nature of the entangling process, we also propose to observe the quantum coherent dynamics by using the Leggett-Garg inequalities. We also suggest a feasible scheme, which consists of two quantum dots coupled to nanowire surface plasmons, for possible experimental realization.

Nonlocal formalism for nanoplasmonics: Phenomenological and semi-classical considerations

The plasmon response of metallic nanostructures is anticipated to exhibit nonlocal dynamics of the electron gas when exploring the true nanoscale. We extend the local-response approximation (based on Ohm's law) to account for a general short-range nonlocal response of the homogeneous electron gas. Without specifying further details of the underlying physical mechanism we show how this leads to a Laplacian correction term in the electromagnetic wave equation. Within the hydrodynamic model we demonstrate this explicitly and we identify the characteristic nonlocal range to be ξNL∼vF/ω where vF is the Fermi velocity and ω is the optical angular frequency. For noble metals this gives significant corrections when characteristic device dimensions approach ∼1–10nm, whereas at more macroscopic length scales plasmonic phenomena are well accounted for by the local Drude response.

Quantum Teleportation of Dynamics and Effective Interactions between Remote Systems

Most protocols for quantum information processing consist of a series of quantum gates, which are applied sequentially. In contrast, interactions between matter and fields, for example, as well as measurements such as homodyne detection of light are typically continuous in time. We show how the ability to perform quantum operations continuously and deterministically can be leveraged for inducing nonlocal dynamics between two separate parties. We introduce a scheme for the engineering of an interaction between two remote systems and present a protocol that induces a dynamics in one of the parties that is controlled by the other one. Both schemes apply to continuous variable systems, run continuously in time, and are based on real-time feedback.

Nonlocal response in plasmonic waveguiding with extreme light confinement

Abstract We present a novel wave equation for linearized plasmonic response, obtained by combining the coupled real-space differential equations for the electric field and current density. Nonlocal dynamics are fully accounted for, and the formulation is very well suited for numerical implementation, allowing us to study waveguides with subnanometer cross-sections exhibiting extreme light confinement. We show that groove and wedge waveguides have a fundamental lower limit in their mode confinement, only captured by the nonlocal theory. The limitation translates into an upper limit for the corresponding Purcell factors, and thus has important implications for quantum plasmonics.

Lévy flights and nonlocal quantum dynamics

We develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacian-based framework of the Schrödinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, m ⩾ 0) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy's synthesis of “covariant particle equations” is extended to encompass free Maxwell theory, which however is devoid of any “particle” content. Links with the photon wave mechanics are explored.

Structure of nonlocality of plasma turbulence

Various indications on the weakly nonlocal character of turbulent plasma transport both from experimental fluctuation measurements from Tore Supra and observations from the full-f, flux-driven gyrokinetic code GYSELA are reported. A simple Fisher equation model of this weakly nonlocal dynamics can be formulated in terms of an evolution equation for the turbulent entropy density, which contains the basic phenomenon of radial turbulence spreading in addition to avalanche-like dynamics via coupling to profile modulations. A derivation of this model, which contains the so-called beach effect, a diffusive and convective flux components for the flux of turbulence intensity, in addition to linear group propagation is given, starting from the drift-kinetic equation. The proposed model has the form of a transport equation for turbulence intensity, and may be considered as an addition to transport modelling. The kinetic fluxes given, can be computed using model closures, or local gyrokinetics. The model is also used in a particular setup that represents the near edge region as a relatively stable zone between the core and edge region where the energy injection is locally more substantial. It is observed that with constant, physical coefficients, the model gives a convincing qualitative profile of fluctuation intensity when the turbulence is coming from the core region with either a group velocity or a convective flux.

On the Honesty in Nonlocal and Discrete Fragmentation Dynamics in Size and Random Position

A discrete initial-value problem describing multiple fragmentation processes, where the fragmentation rate is size and position dependent and where new particles are spatially randomly distributed according to some probabilistic law, is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. The existence of semigroups is established for each parameter and “glued” together so as to obtain a semigroup to the full space. Under certain conditions on each fragmentation rate, we used Kato’s Theorem in to show the existence of the generator and we provide sufficient conditions for honesty.

A general formulation of the reversible stress tensor for a nonlocal fluid

The nonlocal stress tensor is an indispensable constitutive equation required to close the thermodynamic system of nonlocal fluid dynamics. A nonlocal functional variational principle is employed to derive a general expression for the thermodynamically reversible stress tensor for a two-phase, single component, nonlocal fluid. The Euler–Lagrange equation and Noether’s current are used to obtain the general form of the stress tensor, which is then used to derive a wide range of functional forms found in the literature. We also clarify some existing ambiguities. The general form of the nonlocal stress tensor is able to represent micro scale intermolecular interactions, and provides an efficient mesoscale numerical tool for multi-scale analysis using Lattice Boltzmann simulation.

Decoherence dynamics of measurement-induced nonlocality and comparison with geometric discord for two qubit systems

We check the decoherence dynamics of Measurement-induced Nonlocality(in short, MIN) and compare it with geometric discord for two qubit systems. There are quantum states, on which the action of dephasing channel cannot destroy MIN in finite or infinite time. We check the additive dynamics of MIN on a qubit state under two independent noise. Geometric discord also follows such additive dynamics like quantum discord. We have further compared non-Markovian evolution of MIN and geometric discord under dephasing and amplitude damping noise for pure state and it shows distinct differences between their dynamics.