Temporal Nonlocality Research Articles

Random fractional kinematic wave equations of overland flow: The HPM solutions and applications

This paper presents new findings from analyses of a random fractional kinematic wave equation (rfKWE) for overland flow. The rfKWE is featured with orders of temporal and spatial fractional derivatives and with the roughness parameter, the effective rainfall intensity and infiltration rate as random variables. The new solutions are derived with the aid of a numerical method named the homotopy perturbation method (HPM) and approximate solutions are presented for different situations. The solutions are evaluated with data from overland flow flumes with simulated rainfall in the laboratory. The results suggest that on an infiltrating surface the temporal nonlocality of overland flow represented by the temporal order of fractional derivatives diminishes over time while the spatial nonlocality manifested by the spatial order of fractional derivatives continue if there is overland flow. It shows that the widely used unit discharge-height relationship is a special case of the solution of the rfKWE. Procedures are demonstrated for determining the fractional roughness coefficient, nf, the order of spatial fractional derivative, ρ, and the steady-state infiltration rate during the overland flow, As. The analyses of the data show that the mean spatial order of fractional derivative is ρ=1.25, the mean flow pattern parameter m=1.50, and the mean fractional roughness coefficient is nf=0.002 which is smaller than the conventional roughness coefficient, n=0.108. With these average values of the parameters and their standard deviations, simulations were performed to demonstrate the use of the methods, which is also a comparison of the classic KWE and rfKWE models.

On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

Effect of temporal nonlocality on wave propagation behaviors of viscoelastic FGM nanoshells

Effect of temporal nonlocality on wave propagation behaviors of viscoelastic FGM nanoshells

Electron kinetics in a positive column of AC discharges in a dynamic regime

We have performed hybrid kinetic-fluid simulations of a positive column in alternating current (AC) argon discharges over a range of driving frequencies f and gas pressure p for the conditions when the spatial nonlocality of the electron energy distribution function (EEDF) is substantial. Our simulations confirmed that the most efficient conditions of plasma maintenance are observed in the dynamic regime when time modulations of mean electron energy (temperature) are substantial. The minimal values of the root mean square electric field and the electron temperature have been observed at f/p values of about 3 kHz Torr−1 in a tube of radius R = 1 cm. The ionization rate and plasma density reached maximal values under these conditions. The numerical solution of a kinetic equation allowed accounting for the kinetic effects associated with spatial and temporal nonlocality of the EEDF. Using the kinetic energy of electrons as an independent variable, we solved an anisotropic tensor diffusion equation in phase space. We clarified the role of different flux components during electron diffusion in phase space over surfaces of constant total energy. We have shown that the kinetic theory uncovers a more exciting and rich physics than the classical ambipolar diffusion (Schottky) model. Non-monotonic radial distributions of excitation rates, metastable densities, and plasma density have been observed in our simulations at pR > 6 Torr cm. The predicted off-axis plasma density peak in the dynamic regime has never been observed in experiments so far. We hope our results stimulate further experimental studies of the AC positive column. The kinetic analysis could help uncover new physics even for such a well-known plasma object as a positive column in noble gases.

Magnetization dynamics and Peierls instability in topological Josephson structures

We study a long topological Josephson junction with a ferromagnetic strip between two superconductors. The low-energy theory exhibits a non-local in time and space interaction between chiral Majorana fermions, mediated by the magnonic excitations in the ferromagnet. While short ranged interactions turn out to be irrelevant by power counting, we show that sufficiently strong and long-ranged interactions may induce a $\mathbb{Z}_2$-symmetry breaking. This spontaneous breaking leads to a tilting of the magnetization perpendicular to the Majorana propagation direction and the opening of a fermionic gap (Majorana mass). It is analogous to the Peierls instability in the commensurate Fr\"ohlich model and reflects the nontrivial interplay between Majorana modes and magnetization dynamics. Within a Gaussian fluctuation analysis, we estimate critical values for the temporal and spatial non-locality of the interaction, beyond which the symmetry breaking is stable at zero temperature -- despite the effective one-dimensionality of the model. We conclude that non-locality, i.e., the stiffness of the magnetization in space and time, stabilizes the symmetry breaking. In the stabilized regime, we expect the current-phase relation to exhibit an experimentally accessible discontinuous jump. At nonzero temperatures, as usual in the 1D Ising model, the long-range order is destroyed by solitonic excitations, which in our case carry each a Majorana zero mode. In order to estimate the correlation length, we investigate the solitons within a self-consistent mean-field approach.

Product durability management using surface engineering technologies

The problem of managing the durability of products due to the creation of regions in materials with a given temporal and spatial non-locality is considered. A mathematical model of softening of materials under the action of external loads is proposed, allowing to take into account the special role in the formation of damaged places of exit of grain boundaries and their triple junctions on the outer surfaces of parts, as well as their interaction with stress concentrators formed after mechanical processing.
 The developed model representations are used to ensure the operational uniformity of structural elements with concentrators under stress by methods of surface engineering, in particular, when determining the role of changes in the quantitative parameters of the microstructure on the wear resistance of the surface layers of rolling stock parts of railway transport containing stressed concentrators. The optimal depth of the plasma hardening of the subsurface layers of the rims of wheel pairs of locomotives has been established, depending on the size, placement and geometric characteristics of the tension concentrators. It is shown that taking into account the dynamics of structural changes during the degradation of materials in the conditions of operation according to the proposed model ratios allows to achieve the specified life cycle of parts at minimal costs.
 Increasing the durability of products is achieved by optimizing technological modes of surface engineering, which ensure the formation of structures with a smaller gradient of properties.

Исключение интегрального члена в уравнениях одной эредитарной системы, связанной с задачей гидромагнитного динамо

В работе изучается двумерная система интегро-дифференциальных уравнений, которая является простейшей эредитарной моделью двумодового гидромагнитного динамо. Учет пространственной и временной нелокальности взаимодействий в динамо-системах сейчас активно исследуется. В маломодовых приближениях уравнений динамо можно рассматривать только временную нелокальность, т.е. эредитарность (память). Память в исследуемой системе реализована в виде обратной связи, распределенной по всем прошлым состояниям системы. Обратная связь представлена с помощью интегрального члена типа свертки от квадратичной комбинации фазовых переменных с ядром достаточно общего вида. Этот член моделирует подавление турбулентного генератора поля (α-эффекта) квадратичной формой от фазовых переменных. В реальных динамо-системах такое подавление обеспечивается силой Лоренца. Основной результат работы – доказательство возможности исключения интегрального члена для одного класса ядер. Такие ядра являются решениями однородного линейного дифференциального уравнения с постоянными коэффициентами. Доказано, что исследуемую интегро-дифференциальную систему можно заменить дифференциальной системой большей размерности с подходящими начальными условиями на дополнительные фазовые переменные. Если ядро является решением уравнения n-го порядка, то размерность системы может достигать 3n−2 и зависит от начальных условий, которым удовлетворяет ядро. В работе используются классические методы теории дифференциальных уравнений. Приводятся примеры динамических систем, возникающих при некоторых ядрах в результате исключения интегрального члена. Результаты работы можно использовать для верификации вычислительных алгоритмов и программных кодов, разработанных для решения интегро-дифференциальных уравнений.We study a two-dimensional system of integro-differential equations, which is the simplest hereditary model of a two-mode hydromagnetic dynamo. Accounting for the spatial and temporal nonlocality of interactions in dynamo systems is currently being actively studied. In the low-mode approximations of the dynamo equations, one can consider only temporal nonlocality, i.e. heredity (memory). Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by a convolution-type integral term of a quadratic combination of phase variables with a fairly general kernel. This term models the quenching of the turbulent field generator (α-effect) by a quadratic form in phase variables. In real dynamo systems, such quenchingn is provided by the Lorentz force. The main result of the work is a proof of the possibility of eliminating the integral term for one class of kernels. Such kernels are solutions of a homogeneous linear differential equation with constant coefficients. It is proved that the studed integro-differential system can be replaced by a higher-dimensional differential system with suitable initial conditions for additional phase variables. If the kernel is a solution to an n-order equation, then the dimension of the system can reach 3n−2 and depends on the initial conditions that the kernel satisfies. The work uses classical methods of the theory of differential equations. Examples are given of dynamical systems that arise for some kernels as a result of the elimination of the integral term. The results of the work can be used to verify computational algorithms and program codes developed for solving integro-differential equations.

Modelling of viscoelastic materials using non-ordinary state-based peridynamics

This paper proposes a framework for implementing viscoelastic constitutive model from the classical continuum mechanics (CCM) theory within non-ordinary state-based peridynamics (NOSBPD). The motivation stems from the inadequacy of CCM to model very complex material behaviours such as initiation and propagation of cracks and nonlocal behaviour due to size effects. The proposed formulation leverages on the constitutive correspondence between NOSBPD and CCM to incorporate a CCM viscoelastic constitutive model based on hereditary integral into NOSBPD. The combination of hereditary constitutive model and NOSBPD effectively makes this formulation a nonlocal time–space viscoelastic framework where temporal nonlocality is incorporated by a hereditary viscoelastic model which stipulates that the behaviour of a material at any point in time depends on both the present action and the complete history of previous actions on the material, and spatial nonlocality on the other hand is incorporated via the nonlocal mechanism provided by the NOSBPD. For model validation, three benchmark problems were solved using the proposed framework. Results obtained were compared to results from analytical solution and solutions from referenced literature. In addition, parametric study was conducted to determine the influence of nonlocality on numerical prediction. Conclusions drawn from the validation studies presented are that the proposed framework is able to predict viscoelastic responses that agree well with local macro models as well as nonlocal micromodels/nanomodels as reported in the literature.

Field Form of the Dynamics of Classical Many- and Few-Body Systems: From Microscopic Dynamics to Kinetics, Thermodynamics and Synergetics

A method is proposed for describing the dynamics of systems of interacting particles in terms of an auxiliary field, which in the static mode is equivalent to given interatomic potentials, and in the dynamic mode is a classical relativistic composite field. It is established that for interatomic potentials, the Fourier transform of which is a rational algebraic function of the wave vector, the auxiliary field is a composition of elementary fields that satisfy the Klein-Gordon equation with complex masses. The interaction between particles carried by the auxiliary field is nonlocal both in space variables and in time. The temporal non-locality is due to the dynamic nature of the auxiliary field and can be described in terms of functional-differential equations of retarded type. Due to the finiteness mass of the auxiliary field, the delay in interactions between particles can be arbitrarily large. A qualitative analysis of the dynamics of few-body and many-body systems with retarded interactions has been carried out, and a non-statistical mechanisms for both the thermodynamic behavior of systems and synergistic effects has been established.

Multiband homogenization of metamaterials in real-space: Higher-order nonlocal models and scattering at external surfaces

This work develops a dynamic homogenization approach for metamaterials. It finds an approximate macroscopic homogenized equation with constant coefficients posed in space and time; however, the resulting homogenized equation is higher order in space and time. The homogenized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the higher-order space derivatives lead to additional continuity conditions at the boundary between a homogeneous material and a metamaterial. These provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute. Next, the setting of two bands with a bandgap is considered. The homogenized equation has also higher-order time derivatives. Notably, the homogenized model provides a single equation that is valid over both bands and the bandgap. The continuity conditions are applied to wave scattering at a boundary, and show good agreement with the exact fine-scale solution. Using that the order of the highest time derivative is proportional to the number of bands considered, a nonlocal-in-time structure is conjectured for the homogenized equation in the limit of infinite bands. This suggests that homogenizing over finer length and time scales -- with the temporal homogenization being carried out through the consideration of higher bands in the dispersion relation -- is a mechanism for the emergence of macroscopic spatial and temporal nonlocality, with the extent of temporal nonlocality being related to the number of bands considered.

СONTROL OF A NONLOCAL IN TIME FINITE ELEMENT MODEL OF THE DYNAMIC BEHAVIOR OF A COMPOSITE BEAM BASED ON THE RESULTS OF A NUMERICAL EXPERIMENT

The article presents numerical methods for controlling the parameters of temporal nonlocality of computer models of rod structures made of composite materials. The finite element method is the most widely used numerical method for solving practical problems of the analysis of mechanical systems. A nonlocal in time internal damping model is integrated into the algorithm of this method. The one-dimensional model of the Euler-Bernoulli beam is presented in the article. The equilibrium equation of a moving mechanical system is solved numerically using an implicit scheme. In the article the damping matrix obtained from the condition of stationarity of the total deformation energy was used. The article presents the study of non-local in time damping model properties. The model is integrated into the finite element method. The non-local model is algorithmized and programmed in the MATLAB software package.

Finite time blow-up in the higher dimensional parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system

In this work, we mainly study nonnegative classical solutions to a Neumann initial-boundary value problem for the following parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded and smooth domain Ω⊂Rn(n≥1) with χ,ξ≥0, nonnegative initial data (u0,w0) and homogeneous Neumann boundary conditions.In this setup, we first show pure haptotaxis (χ=0) cannot induce any blow-up and pattern for any n≥1, showing haptotaxis is overbalanced by diffusion on boundedness and longtime behavior. Then, in the radial setting with Ω=BR⊂Rn(n≥3), it is well-known that small mass of ‖u0‖L1(Ω) can lead to blow-up in the corresponding Keller-Segel chemotaxis-only model obtained by setting w≡0, known as generic mass blow-up phenomenon [23]. Herein, in the presence of the temporal nonlocality brought by haptotaxis, we show, in an explicit realm of the form χ‖u0‖L1(Ω)>A(n,R)ξ‖w0‖L∞(Ω), that the aforementioned generic mass finite time blow-up phenomenon preserves. This seems to be the first rigorous blow-up result in relevant chemotaxis-haptotaxis models. The blow-up result also suggests that haptotatic cross-diffusion may not be negligible compared to chemotactic aggregation in ≥3D, in contrast to the recently detected 2D negligibility of haptotaxis in a minimal chemotaxis-haptotaxis model [16].

Temporal Quantum Memory and Non-Locality of Two Trapped Ions under the Effect of the Intrinsic Decoherence: Entropic Uncertainty, Trace Norm Nonlocality and Entanglement

The engineering properties of trapped ions and their capacity to engender numerous quantum information resources determine many aspects of quantum information processing. We devise a setup of coherent and even coherent fields acting on two trapped ions to generate quantum memory, non-locality, and entanglement. Various effects, such as intrinsic decoherence, Lamb–Dicke regime, and dipole–dipole interaction are investigated. The inter-coupling of trapped ions, as well as the generation and dynamics of correlations between them, are analyzed. Using quantum memory assisted entropic uncertainty, trace-norm measurement induced non-locality, and concurrence, we find that the coherent and even coherent fields successfully generate non-local correlations in trapped-ions, with the latter being more resourceful for the dynamics and preservation of the non-local correlations. Furthermore, we observe that the entropic uncertainty and the trace norm induced non-locality present symmetrical dynamics. The dipole–dipole interaction improves correlation’s generation, robustness, and entropic uncertainty suppression.

NeuroLISP: High-level symbolic programming with attractor neural networks

Despite significant improvements in contemporary machine learning, symbolic methods currently outperform artificial neural networks on tasks that involve compositional reasoning, such as goal-directed planning and logical inference. This illustrates a computational explanatory gap between cognitive and neurocomputational algorithms that obscures the neurobiological mechanisms underlying cognition and impedes progress toward human-level artificial intelligence. Because of the strong relationship between cognition and working memory control, we suggest that the cognitive abilities of contemporary neural networks are limited by biologically-implausible working memory systems that rely on persistent activity maintenance and/or temporal nonlocality. Here we present NeuroLISP, an attractor neural network that can represent and execute programs written in the LISP programming language. Unlike previous approaches to high-level programming with neural networks, NeuroLISP features a temporally-local working memory based on itinerant attractor dynamics, top-down gating, and fast associative learning, and implements several high-level programming constructs such as compositional data structures, scoped variable binding, and the ability to manipulate and execute programmatic expressions in working memory (i.e., programs can be treated as data). Our computational experiments demonstrate the correctness of the NeuroLISP interpreter, and show that it can learn non-trivial programs that manipulate complex derived data structures (multiway trees), perform compositional string manipulation operations (PCFG SET task), and implement high-level symbolic AI algorithms (first-order unification). We conclude that NeuroLISP is an effective neurocognitive controller that can replace the symbolic components of hybrid models, and serves as a proof of concept for further development of high-level symbolic programming in neural networks.

Trinity of relational quantum dynamics

The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum dynamics: 1) relational observables in the clock-neutral picture of Dirac quantization, 2) Page and Wootters' (PW) Schr\"odinger picture formalism, and 3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. We develop a quantization procedure for relational Dirac observables using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of gauge-invariantly extending gauge-fixed quantities. We establish algebraic properties of these relational observables. We extend a recent clock-neutral approach to changing temporal reference frames, transforming relational observables and states, and demonstrate a clock dependent temporal nonlocality effect. We show that Kucha\v{r}'s criticism, alleging that the conditional probabilities of the PW formalism violate the constraint, is incorrect. They are a quantum analog of a gauge-fixed description of a gauge-invariant quantity and equivalent to the manifestly gauge-invariant evaluation of relational observables in the physical inner product. The trinity furthermore resolves a previously reported normalization ambiguity and clarifies the role of entanglement in the PW formalism. The trinity finally permits us to resolve Kucha\v{r}'s criticism that the PW formalism yields wrong propagators by showing how conditional probabilities of relational observables give the correct transition probabilities. Unlike previous proposals, our resolution does not invoke approximations, ideal clocks or ancilla systems, is manifestly gauge-invariant, and easily extends to an arbitrary number of conditionings.

How to Erase Quantum Monogamy?

The phenomenon of quantum erasure exposed a remarkable ambiguity in the interpretation of quantum entanglement. On the one hand, the data is compatible with the possibility of arrow-of-time violations. On the other hand, it is also possible that temporal non-locality is an artifact of post-selection. Twenty years later, this problem can be solved with a quantum monogamy experiment, in which four entangled quanta are measured in a delayed-choice arrangement. If Bell violations can be recovered from a “monogamous” quantum system, then the arrow of time is obeyed at the quantum level.

Heat Conduction Beyond the Fourier Law

The Fourier law correctly describes heat transport in most practical macroscopic problems. However, for heat transfer in rapid processes, heat transport on micro- and nanoscales, and heat transfer in materials with an internal structure (porous media and biological tissues), other models are required that take into account nonlinear effects, as well as temporal (memory) and spatial nonlocality. Such models are considered in this review, including models with time lag, phonon and thermodynamic models, as well as models based on differential equations in fractional derivatives.

The Meta-Dynamic Nature of Consciousness.

How, if at all, consciousness can be part of the physical universe remains a baffling problem. This article outlines a new, developing philosophical theory of how it could do so, and offers a preliminary mathematical formulation of a physical grounding for key aspects of the theory. Because the philosophical side has radical elements, so does the physical-theory side. The philosophical side is radical, first, in proposing that the productivity or dynamism in the universe that many believe to be responsible for its systematic regularities is actually itself a physical constituent of the universe, along with more familiar entities. Indeed, it proposes that instances of dynamism can themselves take part in physical interactions with other entities, this interaction then being “meta-dynamism” (a type of meta-causation). Secondly, the theory is radical, and unique, in arguing that consciousness is necessarily partly constituted of meta-dynamic auto-sensitivity, in other words it must react via meta-dynamism to its own dynamism, and also in conjecturing that some specific form of this sensitivity is sufficient for and indeed constitutive of consciousness. The article proposes a way for physical laws to be modified to accommodate meta-dynamism, via the radical step of including elements that explicitly refer to dynamism itself. Additionally, laws become, explicitly, temporally non-local in referring directly to quantity values holding at times prior to a given instant of application of the law. The approach therefore implicitly brings in considerations about what information determines states. Because of the temporal non-locality, and also because of the deep connections between dynamism and time-flow, the approach also implicitly connects to the topic of entropy insofar as this is related to time.

Fractional-Fractal Modeling of Filtration-Consolidation Processes in Saline Saturated Soils

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries.

Molecular excitations from meta-generalized gradient approximations in the Kohn-Sham scheme.

Meta-Generalized Gradient Approximations (meta-GGAs) can, in principle, include spatial and temporal nonlocality in time-dependent density functional theory at a much lower computational cost than functionals that use exact exchange. We here test whether a meta-GGA that has recently been developed with a focus on capturing nonlocal response properties and the particle number discontinuity can realize such features in practice. To this end, we extended the frequency-dependent Sternheimer formalism to the meta-GGA case. Using the Krieger-Li-Iafrate (KLI) approximation, we calculate the optical response for the selected paradigm molecular systems and compare the meta-GGA Kohn-Sham response to the one found with exact exchange and conventional (semi-)local functionals. We find that the new meta-GGA captures important properties of the nonlocal exchange response. The KLI approximation, however, emerges as a limiting factor in the evaluation of charge-transfer excitations.