Nonlocal Interactions Research Articles

Segregation and domain formation in non-local multi-species aggregation equations

A system of aggregation equations describing nonlocal interaction of two species is studied. When interspecies repulsive forces dominate intra-species repulsion, phase segregation may occur. This leads to the formation of distinct phase domains, separated by moving interfaces.The one dimensional interface problem is formulated variationally, and conditions for existence and nonexistence are established. The singular limit of large and short-ranged repulsion in two dimensions is then considered, leading to a two-phase free boundary problem describing the evolution of phase interfaces. Long term dynamics are investigated computationally, demonstrating coarsening phenomenon quantitatively different from classical models of phase separation. Finally, the interplay between long-range interspecies attraction and interfacial energy is illustrated, leading to pattern formation.

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey-predator model.

Pattern formation is a central process that helps to understand the individuals' organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey-predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.

Spatio-Temporal Feature Pyramid Interactive Attention Network for Egocentric Gaze Prediction

Over the past few years, gaze prediction technology has rapidly developed in human visual attention mechanism research. However, there is still a large gap between the prediction models and the human visual system (HVS). This work makes four contributions to video egocentric gaze prediction research. First, we introduce a new benchmark named ECVG (egocentric video gaze), which consists of 3K high-quality, elaborately selected video sequences spanning a large range of scenes. Existing eye-tracking datasets limit human activities. In contrast, ECVG achieves the collection of eye movement data under real human activity. Second, we establish a visual attention-selection and gaze-tracking model, which summarizes the human visual attention process. Third, we propose a novel feature pyramid interactive attention 3D network (FPIANet) for egocentric gaze prediction, which directly captures the long-term relationship between spatiotemporal features at different time steps and achieves feature nonlocal interactions across temporal, spatial, and scales. In addition, we propose a multiscale interactive attention (MSIA) module, which first explicitly integrates the human top-down and bottom-up visual attention mechanisms to transform any spatiotemporal feature into another feature of the same size but with richer contexts. Furthermore, we propose a new gaze transfer path consistency evaluation and thoroughly examine the performance of our model on two large datasets (ECVG and LEDOV). The experimental results show that our model outperforms the existing state-of-the-art methods.

A length scale for non-local multi-scale gradient interactions in isotropic turbulence

Three-dimensional turbulent flows enhance velocity gradients via strong nonlinear interactions of the rate-of-strain tensor with the vorticity vector, and with itself. For statistically homogeneous flows, their total contributions to gradient production are related to each other by conservation of mass, and so are the total enstrophy and total dissipation. However, locally, they do not obey this relation and have different (often extreme) values, and for this reason both production mechanisms have been subject to numerous studies, often decomposed into multi-scale interactions. In general lines, their dynamics and contributions to the cascade processes and turbulent kinetic dissipation are different, which poses a difficulty for turbulence modelling. In this paper, we explore the consequence of the ‘Betchov’ relations locally, and show that they implicitly define a length scale. This length scale is found to be approximately three times the size of the turbulent structures and their interactions. It is also found that, while the non-locality of the dissipation and enstrophy at a given scale comes mostly from larger scales that do not cancel, the non-local production of strain and vorticity comes from multi-scale interactions. An important consequence of this work is that isotropic cascade models need not distinguish between vortex stretching and strain self-amplification, but can instead consider both entities as part of a more complex transfer mechanism, provided that their detailed point value is not required and a local average of reasonable size is sufficient.

Shaping magnetization dynamics in a planar square dot by adjusting its surface anisotropy

A planar square dot is one of the simplest structures confined to three dimensions. Despite its geometrical simplicity, the description of the spin wave modes in this structure is not trivial due to the competition of dipolar and exchange interactions. An additional factor that makes this description challenging are the boundary conditions depend both on non-local dipolar interactions and local surface parameters such as surface anisotropy. In the presented work, we showed how the surface anisotropy applied at the lateral faces of the dot can tune the frequency of fundamental mode in the planar CoFeB dot, magnetized in an out-of-plane direction. Moreover, we analyzed the spin wave profile of the fundamental mode and the corresponding dynamic stray field. We showed that the asymmetric application of surface anisotropy produces an asymmetric profile of dynamic stray field for square dot and can be used to tailor inter-dot coupling. The calculations were performed with the use of the finite-element method.

Multipolar Pseudochirality-Induced Optical Torque.

It has been observed that achiral nanoparticles, such as flat helices, may be subjected to an optical torque even when illuminated by normally incident linearly polarized light. However, the origin of this fascinating phenomenon has so far remained mostly unexplained. We therefore propose an exhaustive discussion that provides a clear and rigorous explanation for the existence of such a torque. Using multipolar theory and taking into account nonlocal interactions, we find that this torque stems from multipolar pseudochiral responses that generate both spin and orbital angular momenta. We also show that the nature of these peculiar responses makes them particularly dependent on the asymmetry of the particles. By elucidating the origin of this type of torque, this work may prove instrumental for the design of high-performance nano-rotors.

Closed-Form Wave Dispersion Relationships for Ordinary State-Based Peridynamics

Peridynamics (PD) is a new continuum mechanics formulation introduced to overcome limitations of classical continuum mechanics (CCM). This is mainly achieved by using integro-differential equations rather than partial differential equations. Another important difference of PD is its nonlocal nature with respect to local characteristic of CCM. Moreover, it has a length scale parameter, horizon, defining the range of nonlocal interactions between material points. This nonclassical behaviour also shows itself for dispersion relationships. As opposed to linear dispersion relationships for CCM, PD dispersion relationships are non-linear similar to the observed in experiments. In this study, closed-form dispersion relationships are provided for ordinary state-based peridynamics which is one of the most common PD formulations. Finally, derived closed-form solutions are used to demonstrate the dispersion relationships for various material systems including copper, gold, silver and platinum.

Non-Local Interactions Are Essential Elements for Dark Matter Halo Stability: A Cross-Model Study

This paper introduces a comprehensive methodology for examining the stability of dark matter (DM) halos, emphasizing the necessity for non-local inter-particle interactions, whether they are fundamental or effective in nature, to maintain halo stability. We highlight the inadequacy of vanilla cold, collisionless DM models in forecasting a stable halo without considering a “non-local” interaction in the halo’s effective free energy, which could potentially arise from factors like baryonic feedback, self-interactions, or the intrinsic quantum characteristics of dark particles. The stability prerequisite necessitates significant effective interactions between any two points within the halo, regardless of their distance from the center. The methodology proposed herein offers a systematic framework to scrutinize the stability of various DM models and refine their parameter spaces. We deduce that DM halos within a model, where the deviation from the standard cold, collisionless framework is confined to regions near the halo center, are unlikely to exhibit stability in their outer sectors. In our study, we demonstrate that the issue of instability within DM halos cannot be addressed adequately using perturbative quantum effects. This issue is less pronounced for fermionic DM but suffers from a higher degree of severity when considering bosonic DM. We find that halos made of bosons with notable quantum effects have sharp edges, while those made of fermions show more diffuse boundaries extending toward infinity. To present the potentials of the cross-model approach, we explore the broadest form of the effective free energy around a chosen mass profile. Next, as a case study, we employ a model where the deviation from the standard cold, collisionless DM model is represented by a two-body interaction in the effective free energy to show how to use observations to investigate universal classes of DM models.

Construction of infinitely many solutions for two-component Bose-Einstein condensates with nonlocal critical interaction

In this paper, we deal with the coupled Hartree system with axisymmetric potentials,{−Δu+P(|x′|,x″)u=α1(|x|−4⁎u2)u+β(|x|−4⁎v2)uinR6,−Δv+Q(|x′|,x″)v=α2(|x|−4⁎v2)v+β(|x|−4⁎u2)vinR6, where (x′,x″)∈R2×R4, β>max⁡{α1,α2}≥min⁡{α1,α2}>0, P(|x′|,x″) and Q(|x′|,x″) are bounded nonnegative functions in R+×R4. The system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. When the functions r2P(r,x″) and r2Q(r,x″) have a common topologically nontrivial critical point, using a finite dimensional reduction argument and developing new local Pohožaev identities, we construct infinitely many solutions of synchronized type, whose energy can be made arbitrary large. The main difficulty is caused by the non-local terms, since little is known about the nondegeneracy of the positive solutions of the limit system and the error estimates of the nonlocal parts in applying the reduction arguments and establishing the local Pohožaev identities.

Implementation of a ternary lattice Boltzmann model in LAMMPS

The properties of multicomponent fluids are governed by the interplay of phase behavior, fluid dynamics, and interfacial thermodynamics. A mixture formulation that leverages this interplay is an important aspect in many fabrication processes based on emulsion templating. The lattice Boltzmann method (LBM) has become a popular approach for simulating hydrodynamic effects in complex fluids and soft matter. Here we present an implementation of a ternary lattice Boltzmann model that allows to simulate a mixture of three immiscible fluids. We build on the LATBOLTZ extension of the open-source package LAMMPS and implement a ternary free energy model recently introduced by Semprebon et al. [Phys. Rev. E 93, 033305 (2016)]. We validate the static and dynamic properties by simulating liquid lenses, double emulsions, and ternary mixtures. From the simulations, we obtain the complete morphology diagram of the ternary mixture in composition space. We further discuss an application of the method to phase segregation of ternary films. The implementation of the ternary LBM in LAMMPS opens vast opportunities for mesoscale simulations of interfacial phenomena and non-equilibrium transport processes in multicomponent fluid mixtures. Program summaryProgram Title: fix lb/multicomponentCPC Library link to program files:https://doi.org/10.17632/gn8znmyxrh.1Developer's repository link:https://github.com/uschille/lammps/tree/ternary-lbmLicensing provisions: GPLv2Programming language: C++Journal reference of previous version: C. Denniston, N. Afrasiabian, M.G. Cole-André, F.E. Mackay, S.T.T Ollila, T. Whitehead. LAMMPS lb/fluid fix version 2: Improved hydrodynamics forces implemented into LAMMPS through a lattice-Boltzmann fluid. Comput. Phys. Commun.275, 108318 (2022).Does the new version supersede the previous version?: NoReasons for the new version: The lb/multicomponent fix provides new capabilities for simulating ternary fluid mixtures.Summary of revisions: An implementation of the ternary free-energy lattice Boltzmann model [1] has been added to the LATBOLTZ extension of LAMMPS. The new functionality is available through the script command fix lb/multicomponent.Nature of problem: Multicomponent fluids involve non-local interactions that give rise to fluid-fluid surface tension and a non-ideal pressure tensor. This requires a modification of the standard lattice Boltzmann method.Solution method: The implementation extends the existing lattice Boltzmann code in LAMMPS [2] and introduces data structures and methods for the ternary algorithm introduced in [1]. The code carries out the streaming and collision steps for three fluid components using the appropriate equilibrium distributions. The equilibrium distribution is calculated based on thermodynamic quantities derived from the free energy.Additional comments including restrictions and unusual features: The current version of the lb/multicomponent fix is restricted to the D3Q19 lattice.

Critical thresholds in the Euler-Poisson-alignment system

This paper is concerned with the global wellposedness of the Euler-Poisson-alignment (EPA) system. This system arises from collective dynamics, and features two types of nonlocal interactions: the repulsive electric force and the alignment force. It is known that the repulsive electric force generates oscillatory solutions, which is difficult to be controlled by the nonlocal alignment force using conventional comparison principles. We construct invariant regions such that the solution trajectories cannot exit, and therefore obtain global wellposedness for subcritical initial data that lie in the invariant regions. Supercritical regions of initial data are also derived which leads to finite-time singularity formations. To handle the oscillation and the nonlocality, we introduce a new way to construct invariant regions piece by piece in the phase plane of a reformulation of the EPA system. Our result is extended to the case when the alignment force is weakly singular. The singularity leads to the loss of a priori bounds crucial in our analysis. With the help of improved estimates on the nonlocal quantities, we design non-trivial invariant regions that guarantee global wellposedness of the EPA system with weakly singular alignment interactions.

Revised Tolmachev-Morel-Anderson pseudopotential for layered conventional superconductors with nonlocal Coulomb interaction

Revised Tolmachev-Morel-Anderson pseudopotential for layered conventional superconductors with nonlocal Coulomb interaction

A novel tuberculosis model incorporating a Caputo fractional derivative and treatment effect via the homotopy perturbation method

BackgroundTuberculosis (TB), caused by Mycobacterium tuberculosis, is a contagious infectious disease that primarily targets the lungs but can also impact other critical systems such as the bones, joints, and neurological system. Despite significant efforts to combat TB, it remains a major global health concern. To address this challenge, this study aims to explore and evaluate various tuberculosis control approaches using a mathematical modeling framework.ResultsThe study utilized a novel SEITR mathematical model to investigate the impact of treatment on physical limitations in tuberculosis. The model underwent qualitative analysis to validate key aspects, including positivity, existence, uniqueness, and boundedness. Disease-free and endemic equilibria were identified, and both local and global stability of the model was thoroughly examined using the derived reproduction number. To estimate the impact of each parameter on each compartment, sensitivity analysis was conducted, and numerical simulations were performed using Maple 18 software with the homotopy perturbation method. The obtained results are promising and highlight the potential of the proposed interventions to significantly reduce tuberculosis virus prevalence. The findings emphasize the significance of fractional-order analysis in understanding the effectiveness of treatment strategies for mitigating tuberculosis prevalence. The study suggests that the time fractional dynamics of TB treatment correspond to the treatment’s efficacy, as the conceptual results showed that non-local interactions between the disease and the treatment may lead to more accurate ways of eradicating tuberculosis in real-world scenarios. These insights contribute to a better understanding of effective treatment strategies and their potential impact on tuberculosis control and public health.ConclusionsIn conclusion, scientists, researchers, and healthcare personnel are urged to take action and utilize the discoveries from this research to facilitate the eradication of the hazardous tuberculosis bacteria.

Mechanics of Tunable Adhesion With Surface Wrinkles

Abstract Surface wrinkles have emerged as a promising avenue for the development of smart adhesives with dynamically tunable adhesion, finding applications in diverse fields, such as soft robots and medical devices. Despite intensive studies and great achievements, it is still challenging to model and simulate the tunable adhesion with surface wrinkles due to roughened surface topologies and pre-stress inside the materials. The lack of a mechanistic understanding hinders the rational design of these smart adhesives. Here, we integrate a lattice model for nonlinear deformations of solids and nonlocal interaction potentials for adhesion in the framework of molecular dynamics to explore the roles of surface wrinkles on adhesion behaviors. We validate the proposed model by comparing wrinkles in a neo-Hookean bilayer with benchmarked results and reproducing the analytical solution for cylindrical adhesion. We then systematically study the pull-off force of the wrinkled surface with varied compressive strains and adhesion energies. Our results reveal the competing effect between the adhesion-induced contact and the roughness due to wrinkles on enhancing or weakening the adhesion. Such understanding provides guidance for tailoring material and geometry as well as loading wrinkled surfaces for different applications.

Nonlocal interactions on dynamic growth of an inclusion in locally resonant elastic wave metamaterials

Nonlocal interactions on dynamic growth of an inclusion in locally resonant elastic wave metamaterials

A numerical method for a nonlocal form of Richards' equation based on peridynamic theory

Forecasting water content dynamics in heterogeneous porous media has significant interest in hydrological applications; in particular, the treatment of infiltration when in presence of cracks and fractures can be accomplished resorting to peridynamic theory, which allows a proper modeling of non localities in space. In particular, in this framework, we introduce a suitable influence function able to provide a nonlocal interaction between material particles; moreover, we make use of Chebyshev transform on the diffusive component of the equation and then we integrate forward in time using an explicit method. We prove that the proposed spectral numerical scheme provides a solution converging to the unique solution in some appropriate Sobolev space. We finally exemplify on several different soils, also considering a sink term representing the root water uptake.

On the onset of long-wavelength three-dimensional instability in the cylinder wake

We study the onset of the three-dimensional mode A instability in the near wake behind a circular cylinder. We show that long-wavelength perturbations organise in a time-shifting pattern such that the in-plane velocity in each streamwise slice corresponds to the base flow solution at shifted times. This observation introduces an additional unifying characteristic for certain mode A type instabilities. We then analyse the mechanisms which control the growth or decay of these perturbations and highlight the crucial role played by the tilting mechanism which operates via non-local interactions in a manner similar to Biot–Savart induction. We characterise its domain of influence using a Green's function-based approach which allows us to rationalise the non-trivial dependence of the growth rate on the spanwise wavenumber. We connect this behaviour to the subtle balance between the local growth of the perturbations as they are swept along by the flow and the feedback on the perturbations that are generated during the next period of the time-periodic base flow. Finally, we discuss generalisations of our findings to other types of flows.

Nonlocal topological states in elastic wave metamaterials with active feedback control

Topological elastic wave metamaterials provide a new mechanism to tune wave energy and provide engineering applications which are difficult to realize by traditional structures. This work reports topological characteristics of nonlocal elastic wave metamaterials with active feedback control. The effects of nonlocal connection on band structures and dynamic behaviors of rectilinear lattices are studied. Then, the honeycomb lattice with nonlocal interactions is investigated, which realizes interface transmission and hybrid order topological states. The introduction of active feedback control systems can change the Young’s modulus of nonlocal metamaterials, which flexibly tune the topological characteristic of elastic waves. According to finite element simulation and experiment, the interface transmission and hybrid order can be generated by active feedback control, which agrees with theoretical predictions.

Porous medium equation and cross-diffusion systems as limit of nonlocal interaction

This paper studies the derivation of the quadratic porous medium equation and a class of cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a nonlocal interaction equation, resp. system, to solutions of the quadratic porous medium equation, resp. cross-diffusion system, in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use gradient flow techniques to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme, which can be slightly modified in order to extend our result to PDEs without gradient flow structure. In particular, it does not require convexity of the associated energies. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy.

Nonlinear semigroups for nonlocal conservation laws

We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall–Liggett Theorem. We also show that the unique mild solution satisfies a Kružkov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution.