Nonlocal Forces Research Articles

Minimal Quantization Model in Pilot-Wave Hydrodynamics.

Investigating how classical systems may manifest dynamics analogous to those of quantum systems is a broad subject of fundamental interest. Walking droplets, which self-propel through a resonant interaction with their own wave field, provide a unique macroscopic realization of wave-particle duality that exhibits behaviors previously thought exclusive to quantum particles. Despite significant efforts, elucidating the precise origin and form of the wave-mediated forces responsible for the walker's quantumlike behavior remained elusive. Here, we demonstrate that, owing to wave interference, the force responsible for orbital quantization originates from waves excited near stationary points on the walker's past trajectory. Moreover, we derive a minimal model with the essential ingredients to capture quantized orbital dynamics, including quasiperiodic and chaotic orbits. Notably, this minimal model provides an explicit distinction between local forces, which account for the walker's preferred speed and wave-induced added mass, and spatiotemporal nonlocal forces responsible for quantization. The quantization mechanism revealed here is generic, and will thus play a role in other hydrodynamic quantum analogs.

Second Order Two-Species Systems with Nonlocal Interactions: Existence and Large Damping Limits

We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the compressible Euler system with linear damping. Our results are limited to the 1 space dimensional case but allow for initial data taken in a Wasserstein space of probability measures. We first consider the case of smooth nonlocal interaction potentials, not subject to any symmetry condition, and prove existence and uniqueness. The concept of solutions relies on a stickiness condition in case of collisions, in the spirit of previous works in the literature. The result uses concepts from classical Hilbert space theory of gradient flows (cf. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973) and a trick used in Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013). We then consider a large-time and large-damping scaled version of our system and prove convergence to solutions to the corresponding first order system. Finally, we consider the case of Newtonian potentials - subject to symmetry of the cross-interaction potentials - and external convex potentials. After showing existence in the sticky particles framework in the spirit of Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013), we prove convergence for large times towards Dirac delta solutions for the two densities. All the results share a common technical framework in that solutions are considered in a Lagrangian framework, which allows to estimate the behavior of solutions via L^{2} estimates of the pseudo-inverse variables corresponding to the two densities. In particular, due to this technique, the large-damping result holds under a rather weak condition on the initial data, which does not require well-prepared initial velocities. We complement the results with numerical simulations.

Small Municipalities Attracting Rural Newcomers and Fostering Local Cohesion: Innovative Approaches for Rural Regeneration in Italy

The phenomenon of population decline—with all its associated economic, environmental and social consequences—characterizes many small rural municipalities. This study aims to discuss the role that small rural municipalities can have in addressing this phenomenon by promoting rural regeneration and attracting newcomers. The framework of the analysis is based on the neo-endogenous theories of rural development, which consider the interaction between endogenous and exogenous resources and forces. To analyse the role of small rural municipalities, this paper refers to the findings of a case study on the Italian rural village of Castel del Giudice and of the “confrontation” activities carried out in another village, Malito. The research suggests that a participatory process activated by a municipality can produce positive social and economic benefits for the local community. This process can enhance territorial cohesion, contribute to halting emigration, and favour the establishment of newcomers. Nevertheless, the research highlights that this approach requires adequate organizational knowledge, including political and administrative functionaries with appropriate skills who are also capable of experimenting with new modalities of fundraising. Moreover, it emerges that non-local resources and forces can support a solid participatory process focused on valorising local resources rather than having an extractive approach.

Multiscale nonlocal elasticity: A distributed order fractional formulation

This study presents a generalized multiscale nonlocal elasticity theory that leverages distributed order fractional calculus to accurately capture coexisting multiscale and nonlocal effects within a macroscopic continuum. The nonlocal multiscale behavior is captured via distributed order fractional constitutive relations derived from a nonlocal thermodynamic formulation. The governing equations of the inhomogeneous continuum are obtained via the Hamilton principle. As a generalization of the constant order fractional continuum theory, the distributed order theory can model complex media characterized by inhomogeneous nonlocality and multiscale effects. In order to understand the correspondence between microscopic effects and the properties of the continuum, an equivalent mass–spring lattice model is also developed by direct discretization of the distributed order elastic continuum. Detailed theoretical arguments are provided to show the equivalence between the discrete and the continuum distributed order models in terms of internal nonlocal forces, potential energy distribution, and boundary conditions. These theoretical arguments facilitate the physical interpretation of the role played by the distributed order framework within nonlocal elasticity theories. They also highlight the outstanding potential and opportunities offered by this methodology to account for multiscale nonlocal effects. The capabilities of the methodology are also illustrated via a numerical study that highlights the excellent agreement between the displacement profiles and the total potential energy predicted by the two models under various order distributions. Remarkably, multiscale effects such as displacement distortion, material softening, and energy concentration are well captured at the continuum level by the distributed order theory.

Quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with singular interaction forces

We establish a quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean–Vlasov equation. Introducing an intermediate system via a coarse-graining map, we quantitatively estimate the error between the spatial densities of the Vlasov–Fokker–Planck equation and the intermediate system in the Wasserstein distance of order 2. We then derive an evolution-variational-like inequality for Wasserstein gradient flows which allows us to quantify the error between the intermediate system and the corresponding limiting equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov–Poisson–Fokker–Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.

Improved peridynamics approach for the progressive fracture of marine concrete

Crack initiation and propagation in marine concrete play a significant role in the security and durability of concrete in ocean construction. Peridynamics (PD), which reformulates classical continuum mechanics in terms of spatial integral equations instead of partial differential equations, has been used successfully to match the fracture patterns observed in experiments. However, the classical bond-based peridynamics (BPD) describe brittle fracture instead of quasi-brittle fracture; this includes the failure of quasi-brittle solid marine concrete. Further, the models have significant surface effects, and the effect of the internal length on nonlocal long-range forces is not reflected. The current work proposes an improved BPD approach to overcome the three drawbacks of fracture analyses of marine concrete. A novel kernel function is incorporated, an effective PD surface effect correction method is used, and the BPD bond stiffness at the boundary is modified. Then, the constitutive force function of the BPD is established based on the linear and nonlinear mechanical behaviours of the progressive fracture of ocean concrete materials, and a corresponding failure criterion is given. The validity and accuracy of the proposed method are demonstrated via numerical analyses and experiments. The proposed method can fully capture the nonlinear deformation and progressive fracture of marine concrete.

Global existence for a highly nonlinear temperature-dependent system modeling nonlocal adhesive contact

In this paper we analyze a new temperature-dependent model for adhesive contact that encompasses nonlocal adhesive forces and damage effects, as well as nonlocal heat flux contributions on the contact surface. The related PDE system combines heat equations, in the bulk domain and on the contact surface, with mechanical force balances, including micro-forces, that result in the equation for the displacements and in the flow rule for the damage-type internal variable describing the state of the adhesive bonds. Nonlocal effects are accounted for by terms featuring integral operators on the contact surface.The analysis of this system poses several difficulties due to its overall highly nonlinear character, and in particular to the presence of quadratic terms, in the rates of the strain tensor and of the internal variable, that appear in the bulk and surface heat equations. Another major challenge is related to proving strict positivity for the bulk and surface temperatures.We tackle these issues by very careful estimates that enable us to prove the existence of global-in-time solutions and could be useful in other contexts. All calculations are rigorously rendered on an accurately devised time discretization scheme in which the limit passage is carried out via variational techniques.

Optical van-der-Waals forces in molecules: from electronic Bethe-Salpeter calculations to the many-body dispersion model

Molecular forces induced by optical excitations are connected to a wide range of phenomena, from chemical bond dissociation to intricate biological processes that underpin vision. Commonly, the description of optical excitations requires the solution of computationally demanding electronic Bethe-Salpeter equation (BSE). However, when studying non-covalent interactions in large-scale systems, more efficient methods are desirable. Here we introduce an effective approach based on coupled quantum Drude oscillators (cQDO) as represented by the many-body dispersion model. We find that the cQDO Hamiltonian yields semi-quantitative agreement with BSE calculations and that both attractive and repulsive optical van der Waals (vdW) forces can be induced by light. These optical-vdW interactions dominate over vdW dispersion in the long-distance regime, showing a complexity that grows with system size. Evidence of highly non-local forces in the human formaldehyde dehydrogenase 1MC5 protein suggests the ability to selectively activate collective molecular vibrations by photoabsorption, in agreement with recent experiments.

Fermat’s principle and the effect of jerk on light motion near the Sun

In classical mechanics, in the case of gravitational and electromagnetic interactions, the force on a particle is usually proportional to its acceleration: The force acts locally on the particle. However, there are situations possible-if the particle moves through a suitable medium, for example, in which the force depends also on the first-time derivative of its acceleration, the jerk, and on its second-time derivative, the snap, and possibly also on higher-time derivatives. Such forces are called nonlocal, and this work investigates such nonlocal forces, mainly those depending on the jerk. In particular, we implement jerk and acceleration in geodesics by means of the nonlocal-in-time kinetic energy approach to spacetime physics. We describe a framework that can be used to estimate the quantum nonlocal time parameter by studying the deflection of light around the Sun. Comparing our results with long baseline interferometry (VLBI) observations, we concluded that the nonlocal time parameter [Formula: see text] s.

Large friction limit of pressureless Euler equations with nonlocal forces

We rigorously show a large friction limit of hydrodynamic models with alignment, attractive, and repulsive effects. More precisely, we consider pressureless Euler equations with nonlocal forces and provide a quantitative estimate of large friction limit to a continuity equation with nonlocal velocity fields, which is often called an aggregation equation. Our main strategy relies on the relative entropy argument combined with the estimate of p -Wasserstein distance between densities.

The neuronal calcium sensor Synaptotagmin-1 and SNARE proteins cooperate to dilate fusion pores.

All membrane fusion reactions proceed through an initial fusion pore, including calcium-triggered release of neurotransmitters and hormones. Expansion of this small pore to release cargo is energetically costly and regulated by cells, but the mechanisms are poorly understood. Here, we show that the neuronal/exocytic calcium sensor Synaptotagmin-1 (Syt1) promotes expansion of fusion pores induced by SNARE proteins. Pore dilation relied on calcium-induced insertion of the tandem C2 domain hydrophobic loops of Syt1 into the membrane, previously shown to reorient the C2 domain. Mathematical modelling suggests that C2B reorientation rotates a bound SNARE complex so that it exerts force on the membranes in a mechanical lever action that increases the height of the fusion pore, provoking pore dilation to offset the bending energy penalty. We conclude that Syt1 exerts novel non-local calcium-dependent mechanical forces on fusion pores that dilate pores and assist neurotransmitter and hormone release.

An improved unibond dual-parameter peridynamic model for fracture analysis of quasi-brittle materials

Peridynamics (PD) is a new nonlocal form of classical continuum mechanics, which is very suitable for investigating material failure. However, the application of classical bond-based PD (BPD) to brittle fracture is restricted because the failure models proposed in BPD mainly describe brittle fracture rather than quasi-brittle fracture. Moreover, the models have a limitation of fixed Poisson's ratio, and the effect of internal length on nonlocal long-range forces cannot be reflected. The present study proposes an improved unibond dual-parameter PD (UDPD) model to overcome the three drawbacks in the damage and fracture analyses of quasi-brittle materials. The conversion formulations of the normal stiffness and tangential stiffness between the elasticity modulus and Poisson's ratio were derived first. Then, a continuous function was introduced in the UDPD model by considering the internal length effect of long-range forces, and the corresponding expressions of the micromodulus functions were deduced. Furthermore, the basic form of the constitutive force function of the UDPD was constructed based on the linear and nonlinear mechanical behaviors of the failure process of quasi-brittle materials, and a corresponding failure criterion was used to evaluate the damage. The validity and accuracy of the proposed model were demonstrated by conducting numerical tests on four specimens, namely, a rectangular plate, double-edge notched concrete specimen, unilaterally notched gray cast iron, and concrete L-specimen panel under quasi-static loading. The model results were compared with those of experiments and numerical studies in the literature. It was found that the proposed model has the same simplicity and stability of the BPD model and can fully capture the nonlinear deformation and progressive failure of quasi-brittle materials with different Poisson's ratios.

Propagation of waves in nonlocal-periodic systems

This paper is concerned with emergence of novel effects in wave propagation in one-dimensional waveguides, when integrated with periodic nonlocalities. The nonlocalities are introduced by a connectivity superimposed to a conventional waveguide and depicted as a graph with trees and leaves, each with its own periodicity. Merging nonlocality and periodicity notions induces a distinction between homogenous and non-homogenous periodic configurations. Specifically, various unconventional phenomena linked to the presence of nonlocalities result in disruption of the energy transmission in such systems, disclosing new opportunities for vibration isolation applications. To demonstrate these effects, simple models of propagation of plane extension/compression waves in a uniform infinite rod equipped with co-axial spring-like elements is used. The homogenous case is analysed by a direct double, space and time, Fourier transform, leading to discussion of unusual dispersion effects, including vanishing and negative group velocity. In the non-homogeneous case, the canonical Floquet theory is used to identify stopbands and control their positions in the frequency domain. The results are compared with eigenfrequency analysis of unit periodicity cells and finite structures. Next, the forcing problem is considered and the insertion losses in a semi-infinite rod with nonlocal spring effects are computed to corroborate predictions of Floquet theory, providing physical explanations of the obtained results. Finally, possibilities to employ the non-local interaction forces in an active control format to generate stopbands at arbitrarily low frequencies are highlighted.

Unit cell interactions of non-local active acoustic media

Active acoustic metamaterials have greatly expanded upon the unique and unusual ways that passive metamaterials manipulate sound. Previously, we have shown how spatially non-local forces in an acoustic system can create band gaps and break acoustic reciprocity, and how such effects can be realized in practice using a non-local active acoustic metamaterial (NAM). We demonstrated experimentally the remarkable ability of a single NAM unit cell to generate large, subwavelength, broadband nonreciprocity using an open loop feedforward control mechanism where a measured local pressure is transmitted by an electronic controller downstream to actuate an acoustic source. In NAM systems with multiple unit cells, this control scheme strongly couples each cell with every other cell, and thus requires special design considerations. Here, we discuss the advantages and challenges of developing NAM systems with multiple unit cells by first considering a system with two cells, detailing the impacts of their complex interaction on both performance and stability. We validate model predictions with experiments and highlight the exciting prospects offered by larger multi-cell NAM arrays.

Understanding urban centers in Shanghai with big data: Local and non-local function perspectives

Along with the rising space-of-flow, non-local functions are identified as a comparable force to local urban-hinterland relations that determine urban development. However, little evidence and explanation have been presented for how urban centers (UCs) are made of local and non-local functions. By examining the case of Shanghai, this paper endeavors to fill the gap with multiple big/open datasets. We first identify UCs using a new method as well as employment/activity densities inferred from cell phone data. Then, local and non-local forces are measured using functional and interaction metrics derived from theoretical deduction based on classic economic-space theories. Our result demonstrates that non-local functions have become a comparable force as local functions. Spatial analyses show that non-local functions generally agglomerate in about 30 UCs (27%) in the inner-ring Puxi area, presumably indicating a locational preference for local services and demand; local functions are distributed more like the central places diagram in Isard (1956). Additionally, specialization analysis of UCs shows clear evidence that non-local functions cause both centralization and decentralization in UCs, which can be well explained by the deductive theoretical patterns. This paper contributes to the literature of urban spatial structure with new evidence as well as a generic theoretical explanation and transferable methodologies.

Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces

In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker–Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulomb interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. The existence of weak and strong solutions to the kinetic and fluid equations is also obtained. We emphasize that the existence of global weak solution with the needed free energy dissipation for the kinetic model is established.

Multiple patterns formation for an aggregation/diffusion predator-prey system

<p style='text-indent:20px;'>We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant <inline-formula><tex-math id="M1">\begin{document}$ -\alpha $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ \alpha>0 $\end{document}</tex-math></inline-formula>. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.</p>

An Innovative Application of Generative Adversarial Networks for Physically Accurate Rock Images With an Unprecedented Field of View

AbstractHigh‐resolution X‐ray microcomputed tomography (micro‐CT) data are used for the accurate determination of rock petrophysical properties. High‐resolution data, however, result in a small field of view, and thus, the representativeness of a simulation domain can be brought into question when dealing with geophysical applications. This paper applies a cycle‐in‐cycle generative adversarial network (CinCGAN) to improve the resolution of 3‐D micro‐CT data and create a super‐resolution image using unpaired training images. Effective porosity, Euler characteristic, pore size distribution, and absolute permeability are measured on super‐resolution and high‐resolution ground‐truth images to evaluate the physical accuracy of the proposed CinCGAN. The results demonstrate that CinCGAN provides physically accurate images with an order of magnitude larger field of view when compared to typical micro‐CT methods. This unlocks new pathways for the geophysical characterization of subsurface rocks with broad implications for flow modeling in highly heterogeneous rocks or fundamental studies on nonlocal forces that extend beyond domain sizes typically used for pore‐scale simulation.

Nonlocal elastic metasurfaces: Enabling broadband wave control via intentional nonlocality

While elastic metasurfaces offer a remarkable and very effective approach to the subwavelength control of stress waves, their use in practical applications is severely hindered by intrinsically narrow band performance. In applications to electromagnetic and photonic metamaterials, some success in extending the operating dynamic range was obtained by using nonlocality. However, while electronic properties in natural materials can show significant nonlocal effects, even at the macroscales, in mechanics, nonlocality is a higher-order effect that becomes appreciable only at the microscales. This study introduces the concept of intentional nonlocality as a fundamental mechanism to design passive elastic metasurfaces capable of an exceptionally broadband operating range. The nonlocal behavior is achieved by exploiting nonlocal forces, conceptually akin to long-range interactions in nonlocal material microstructures, between subsets of resonant unit cells forming the metasurface. These long-range forces are obtained via carefully crafted flexible elements, whose specific geometry and local dynamics are designed to create remarkably complex transfer functions between multiple units. The resulting nonlocal coupling forces enable achieving phase-gradient profiles that are functions of the wavenumber of the incident wave. The identification of relevant design parameters and the assessment of their impact on performance are explored via a combination of semianalytical and numerical models. The nonlocal metasurface concept is tested, both numerically and experimentally, by embedding a total-internal-reflection design in a thin-plate waveguide. Results confirm the feasibility of the intentionally nonlocal design concept and its ability to achieve a fully passive and broadband wave control.

Effect of boundary conditions and constitutive relations on the free vibration of nonlocal beams

The free vibrations of nonlocal Euler and Timoshenko beams have been studied extensively, but there still remain some problems concerning boundary conditions and constitutive relations. For cantilever beams, a counterintuitive stiffening phenomenon is widely observed. And to the best of the authors’ knowledge, there is no work concerning about the Timoshenko beam with nonlocal shear force and local bending moment. This work reconsiders and explains these problems based on nonlocal differential theory. After obtaining the governing equation for nonlocal Euler beam, it is observed that the governing equation has the same form as that for the vibration of local beam with axial compressive force. Thus by comparing the nonlocal effects to the effects of end axial force, the stiffening phenomenon of cantilever beams is found to be caused by a meaningless term in boundary bending moment. This explanation is also suitable for nonlocal Timoshenko cantilever beams. Besides, for Timoshenko beams, the nonlocal effects on tensile and shear stress constitutive relations are considered separately. Three kinds of nonlocal Timoshenko beam models are solved. Numerical comparison shows that the effects of nonlocal tensile are bigger than that of nonlocal shear stress. Especially, for Timoshenko beams with nonlocal shear force only, the nonlocal effects lower the natural frequency for all boundary conditions. This finding can verify our explanation for the stiffening phenomenon of cantilever beams. In addition, it is also observed that, for both nonlocal beam theories, if all nonlocal internal forces are included in the governing equations, the eigenvalue equations and natural frequencies corresponding to local and nonlocal boundary conditions are the same.