Nonlocal Interactions Research Articles

Nonlocal anisotropic interactions of Coulomb type

In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

Weakly nonlinear analysis of a two-species non-local advection–diffusion system

Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg–Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg–Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.

Large‐Area, High‐Numerical‐Aperture, Freeform Metasurfaces

AbstractNanophotonic devices are optical platforms capable of unprecedented wavefront control. To push the limits of experimental device performance, scalable design methodologies that combine the simplicity and fabricability of conventional design paradigms with the extended capabilities of freeform optimization are required. A novel gradient‐based design framework for large‐area freeform metasurfaces is introduced in which nonlocal interactions between simply shaped nanostructures, placed on an irregular lattice, are tailored to produce high‐order hybridized modes that support customizable large‐angle scattering profiles. Utilizing this approach, multifunctional super‐dispersive metalenses are designed and experimentally demonstrated. The approach to high‐numerical‐aperture radial metalenses capable of diffraction limited focusing and the generation of donut‐shaped point spread functions is also extended. It is anticipated that these concepts will have utility in super‐resolution microscopy, particle trapping, additive manufacturing, and metrology applications that require ultra‐high numerical apertures.

Anomalous phase diagram of the elastic interface with nonlocal hydrodynamic interactions in the presence of quenched disorder.

We investigate the influence of quenched disorder on the steady states of driven systems of the elastic interface with nonlocal hydrodynamic interactions. The generalized elastic model (GEM), which has been used to characterize numerous physical systems such as polymers, membranes, single-file systems, rough interfaces, and fluctuating surfaces, is a standard approach to studying the dynamics of elastic interfaces with nonlocal hydrodynamic interactions. The criticality and phase transition of the quenched generalized elastic model are investigated numerically and the results are presented in a phase diagram spanned by two tuning parameters. We demonstrate that in the one-dimensional disordered driven GEM, three qualitatively different behavior regimes are possible with a proper specification of the order parameter (mean velocity) for this system. In the vanishing order parameter regime, the steady-state order parameter approaches zero in the thermodynamic limit. A system with a nonzero mean velocity can be in either the continuous regime, which is characterized by a second-order phase transition, or the discontinuous regime, which is characterized by a first-order phase transition. The focus of this research is to investigate the critical scaling features near the pinning-depinning threshold. The behavior of the quenched generalized elastic model at the critical depinning force is explored. Near the depinning threshold, the critical exponent is obtained numerically.

On a mathematical model for cancer invasion with repellent pH-taxis and nonlocal intraspecific interaction

Starting from a mesoscopic description of cell migration and intraspecific interactions, we obtain by upscaling an effective reaction–diffusion–taxis equation for the cell population density involving spatial nonlocalities in the source term and biasing its motility and growth behavior according to environmental acidity. We prove global existence, uniqueness, and boundedness of a nonnegative solution to a simplified version of the coupled system describing cell and acidity dynamics. A 1D study of pattern formation is performed. Numerical simulations illustrate the qualitative behavior of solutions.

Phase chimera states on nonlocal hyperrings.

Chimera states are dynamical states where regions of synchronous trajectories coexist with incoherent ones. A significant amount of research has been devoted to studying chimera states in systems of identical oscillators, nonlocally coupled through pairwise interactions. Nevertheless, there is increasing evidence, also supported by available data, that complex systems are composed of multiple units experiencing many-body interactions that can be modeled by using higher-order structures beyond the paradigm of classic pairwise networks. In this work we investigate whether phase chimera states appear in this framework, by focusing on a topology solely involving many-body, nonlocal, and nonregular interactions, hereby named nonlocal d-hyperring, (d+1) being the order of the interactions. We present the theory by using the paradigmatic Stuart-Landau oscillators as node dynamics, and we show that phase chimera states emerge in a variety of structures and with different coupling functions. For comparison, we show that, when higher-order interactions are "flattened" to pairwise ones, the chimera behavior is weaker and more elusive.

Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

This study proposes continuous density-based three-dimensional topology optimization (TO) approaches developed by coupling the peridynamic theory (PD) with optimality criteria (OC) and proportional approach (PROP). These frameworks, abbreviated as PD-OC-TO and PD-PROP-TO, can be practically utilized to enhance the fracture toughness of the structures during the optimization process by taking critical regions into account as pre-defined cracks. Breaking the non-local interactions (bonds) between relevant PD particles enables us to readily model cracks. Utilizing this advantage, we solve several benchmark optimization problems including different numbers, positions, and alignments of the cracks. The major differences between the proposed methods are examined by comparing optimum topologies for various cracked scenarios. Moreover, the mechanical behaviour of the optimized structures is investigated under dynamic loads to prove the significant improvements achieved by the present approach in the final designs. The results of dynamic analyses reveal the viability of both PD-TO methods for increasing the fracture toughness of the structure in the optimization stage. Overall, the proposed approach is confirmed as a superior design and optimization tool for future engineering structures.Graphical abstract

Computational multiphase micro‐periporomechanics for dynamic shear banding and fracturing of unsaturated porous media

AbstractDynamic shearing banding and fracturing in unsaturated porous media are significant problems in engineering and science. This article proposes a multiphase micro‐periporomechanics (PPM) paradigm for modeling dynamic shear banding and fracturing in unsaturated porous media. Periporomechanics (PPM) is a nonlocal reformulation of classical poromechanics to model continuous and discontinuous deformation/fracture and fluid flow in porous media through a single framework. In PPM, a multiphase porous material is postulated as a collection of a finite number of mixed material points. The length scale in PPM that dictates the nonlocal interaction between material points is a mathematical object that lacks a direct physical meaning. As a novelty, in the coupled PPM, a microstructure‐based material length scale is incorporated by considering micro‐rotations of the solid skeleton following the Cosserat continuum theory for solids. As a new contribution, we reformulate the second‐order work for detecting material instability and the energy‐based crack criterion and J‐integral for modeling fracturing in the PPM paradigm. The stabilized Cosserat PPM correspondence principle that mitigates the multiphase zero‐energy mode instability is augmented to include unsaturated fluid flow. We have numerically implemented the novel PPM paradigm through a dual‐way fractional‐step algorithm in time and a hybrid Lagrangian–Eulerian meshfree method in space. Numerical examples are presented to demonstrate the robustness and efficacy of the proposed PPM paradigm for modeling shear banding and fracturing in unsaturated porous media.

Abstract A023: Integrating mathematical modeling and in vitro experiments to measure ecological interactions in cancer

Abstract Integrating mathematical modeling and in vitro experiments to measure ecological interactions in cancer Maximilian Strobl, Dagim Tadele, Jeffrey Maltas, Rowan Barker-Clarke, Mina Dinh, Jacob Scott Over the past two decades it has become clear that tumours are complex and evolving ecosystems, in which different cancer cell populations interact with each other and with their non-tumour microenvironment. This understanding has given rise to the intriguing idea that we might be able to leverage these interactions to slow, or even revert, the expansion of resistant cells. In so-called “Adaptive Therapy” (AT) treatment is dynamically reduced to maintain the tumour burden tolerable whilst also preserving a pool of drug-sensitive cells which competitively suppress resistant cells. However, how strong and how common is this suppression, and how do we stratify patients who might benefit from AT from those who should receive standard-of-care continuous therapy? In this contribution, we present development of methodology and initial experimental results aimed at measuring the distribution of ecological interactions between drug-sensitive and resistant cells. Previously, we pioneered an assay with which to quantify ecological interactions in cancer in vitro. In the “Game Assay”, the two populations are co-cultured at a range of initial ratios in 2-D, and we record how the growth rate of each population is modulated by the relative abundance of the other. In the current study, we begin by using an agent-based mathematical model to characterize the accuracy and precision of this assay and how we may improve these by modifying different aspects of the experimental design (number of replicates, number of proportions) and analysis (regression technique, regression window). This shows that the choice of seeding ratios is important in being able to reliably identify interactions and we discuss guidelines for assay design. Upon identifying an interaction, a key next question is the underlying mechanism. Next, we thus use the model to investigate the degree to which we can use the assay to distinguish between local (e.g. contact inhibition) and non-local interactions (e.g. paracrine signaling). To conclude, we present preliminary experimental data in which we have used our optimized protocol to quantify the distribution of interactions between Osimertinib sensitive and resistant NSCLC cells (PC9). Our data show that while suppressive interactions do occur, they vary in their strength between different resistant lineages and can be attenuated in a frequency-dependent manner. Overall, we demonstrate how mathematical models can help in the design and interpretation of experimental assays and we contribute towards a more systematic and quantitative understanding of tumor ecology as a foundation for adaptive therapy. Citation Format: Maximilian A. R. Strobl, Dagim S. Tadele, Jeffrey Maltas, Rowan Barker-Clarke, Mina Dinh, Jacob Scott. Integrating mathematical modeling and in vitro experiments to measure ecological interactions in cancer [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Translating Cancer Evolution and Data Science: The Next Frontier; 2023 Dec 3-6; Boston, Massachusetts. Philadelphia (PA): AACR; Cancer Res 2024;84(3 Suppl_2):Abstract nr A023.

In-Gap Edge and Domain-Wall States in Largely Perturbed Phononic Su–Schrieffer–Heeger Lattices

Topological states of matter have attracted significant attention due to their intrinsic wave-guiding and localization capabilities robust against disorders and defects in electronic, photonic, and phononic systems. Despite the above topological features that phononic crystals share with their electronic and photonic counterparts, finite-frequency topological states in phononic crystals may not always survive. In this work, we discuss the survivability of topological states in Su–Schrieffer–Heeger models with both local and non-local interactions and larger symmetry perturbation. Although such a discussion is still about ideal mass-spring models, the insights from this study set the expectations for continuum phononic crystals, which can further instruct the application of phononic crystals for practical purposes.

Compact bright pulse in inhomogeneous and nonlinear medium: Case of the Bose–Einstein Condensate

The extended one-dimensional Gross–Pitaevskii equation which describes the dynamics of localized wave in nonlocal nonlinear media, in particular the Bose–Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms, is investigated. Through analytical methods invoking a modified lens-type transformation, an equivalent nonlocal GP equation with constant coefficients is derived as well as the integrability conditions. In the limit of zero transverse kinetic energy, we show that the nonlocal GP equation exhibits a stationary bright pulse with strictly localized support and with the width independent of the amplitude. However due to the property of conservation of the norm, the BEC amplitude will be a function of both the pulse width and the number of trap atoms. Similarly, the obtained integrability conditions also appear as the conditions under which the compact bright waves describing the BECs can be managed by controlling the parameters of the external potential. Thus, the number of trap atoms can be managed through the functional gain or loss which is not sensitive to the strength of the magnetic trap. It also appears that, in the presence of the nonlocal interaction, the shape of the BEC interpolates between the shape of the compact bright pulse and that of the NLS bright pulse, and the number of condensed atoms participating to the formation of the BEC increases with the strength of the nonlocal interactions.

A novel fractional-order stochastic epidemic model to analyze the role of media awareness in the spread of conjunctivitis

This study introduces a novel fractional-order stochastic epidemic model to analyze the spread of conjunctivitis, a prevalent ocular infection, while accounting for the influence of media awareness on disease transmission. The model incorporates fractional derivatives to capture memory effects and non-local interactions inherent in epidemic processes, allowing for a more accurate representation of disease dynamics. The stability analysis of equilibrium points is carried out based on the basic reproduction number ℛ0 and fractional-order α. Further, the Hopf bifurcation phenomenon is discussed in this paper. Stochasticity accounts for the randomness in transmission events. The findings of this study provide insights into the complex interrelationship between disease dynamics and media influence, shedding light on the role of public awareness in mitigating or exacerbating conjunctivitis outbreaks. The implications of this work extend to public health policy formulation, highlighting the importance of targeted communication strategies in controlling and preventing the spread of conjunctivitis and similar infectious diseases.

On the time scales of spectral evolution of nonlinear waves

As presented in Annenkov & Shrira (Phys. Rev. Lett., vol. 102, 2009, 024502), when a surface gravity wave field is subjected to an abrupt perturbation of external forcing, its spectrum evolves on a ‘fast’ dynamic time scale of $O(\varepsilon ^{-2})$ , with $\varepsilon$ a measure of wave steepness. This observation poses a challenge to wave turbulence theory that predicts an evolution with a kinetic time scale of $O(\varepsilon ^{-4})$ . We revisit this unresolved problem by studying the same situation in the context of a one-dimensional Majda–McLaughlin–Tabak equation with gravity wave dispersion relation. Our results show that the kinetic and dynamic time scales can both be realised, with the former and latter occurring for weaker and stronger forcing perturbations, respectively. The transition between the two regimes corresponds to a critical forcing perturbation, with which the spectral evolution time scale drops to the same order as the linear wave period (of some representative mode). Such fast spectral evolution is mainly induced by a far-from-stationary state after a sufficiently strong forcing perturbation is applied. We further develop a set-based interaction analysis to show that the inertial-range modal evolution in the studied cases is dominated by their (mostly non-local) interactions with the low-wavenumber ‘condensate’ induced by the forcing perturbation. The results obtained in this work should be considered to provide significant insight into the original gravity wave problem.

Mechanisms of Sarcomere Assembly in Muscle Cells Inferred from Sequential Ordering of Myofibril Components

Voluntary movement in animals is driven by the contraction of myofibrils in striated muscle cells, which are linear chains of regular cytoskeletal units termed sarcomeres. While the ordered structure of mature sarcomeres, key to muscle contraction, is well established, the physical mechanism governing sarcomere assembly is still under debate. Here we put forward a theory of sarcomere self-assembly based on data. We measured the sequential ordering of sarcomere components in developing flight muscles, using a novel tracking-free algorithm. We find that myosin molecular motors, α-actinin cross-linkers, and the titin homologue Sallimus form periodic patterns before actin filaments become polarity sorted. Based on these, we propose that myosin, Sallimus, and sarcomere Z-disk proteins including α-actinin dynamically bind and unbind to an unordered bundle of actin to establish an initial periodic pattern. Periodicity of actin filaments is established only later. Our model proposes that nonlocal interactions between spatially extended myosin and titin/Sallimus containing complexes, and possibly tension-dependent feedback mediated by an α-actinin catch bond, drive this ordering process. We probe this hypothesis using mathematical models and derive predictive conditions for sarcomere pattern formation to guide future experiments. Published by the American Physical Society 2024

On the complex version of the Cahn–Hilliard–Oono type equation for long interactions phase separation

Abstract This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation.

Surface effect on the phase separation in thin liquid crystal film as seen from a mean-field approach

The effect of the interplay between attractive nonlocal surface interactions and attractive pair long-range intermolecular couplings on molecular structures of liquid crystals confined in thin cells with flat solid surfaces has been studied. It has been shown, in the framework of the extended McMillan′s mean field theory, that in the confined LC system a complex orientational and translational ordering of molecules can be induced. Several scenarios of the formation of local smectic A, nematic, and isotropic phases have been shown to coexist in a certain temperature range. Due to the nonlocality of the surface interaction, an unexpected complexity of the spatial arrangement of these local phases along the direction of the normal to the surface may exhibit itself.

Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model.

The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation.

A physics-aware neural network for protein-ligand interactions with quantum chemical accuracy.

Quantifying intermolecular interactions with quantum chemistry (QC) is useful for many chemical problems, including understanding the nature of protein-ligand interactions. Unfortunately, QC computations on protein-ligand systems are too computationally expensive for most use cases. The flourishing field of machine-learned (ML) potentials is a promising solution, but it is limited by an inability to easily capture long range, non-local interactions. In this work we develop an atomic-pairwise neural network (AP-Net) specialized for modeling intermolecular interactions. This model benefits from a number of physical constraints, including a two-component equivariant message passing neural network architecture that predicts interaction energies via an intermediate prediction of monomer electron densities. The AP-Net model is trained on a comprehensive dataset composed of paired ligand and protein fragments. This model accurately predicts QC-quality interaction energies of protein-ligand systems at a computational cost reduced by orders of magnitude. Applications of the AP-Net model to molecular crystal structure prediction are explored, as well as limitations in modeling highly polarizable systems.

Pattern alternations induced by nonlocal interactions

Pattern alternations induced by nonlocal interactions

The Sharp Interface Limit of an Ising Game

The Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, and many other phenomena and mathematical methods. We introduce and explore an Ising game, a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions, we consider a mean-field limit resulting in a nonlocal potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using a Γ-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied. We build on that literature with new techniques to handle a mixture of local derivative terms and nonlocal interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.