Nonlocal Field Research Articles

Nonlocal statistical field theory of dipolar particles forming chain-like clusters

We present a nonlocal statistical field theory of a diluted solution of dipolar particles which are capable of forming chain-like clusters in accordance with the ’head-to-tail’ mechanism. As in our previous study [Yu.A. Budkov 2018 J. Phys.: Condens. Matter 30 344001], we model dipolar particles as dimers comprised of oppositely charged point-like groups, separated by fluctuating distance. For the special case of the Yukawa-type distribution function of distance between the charged groups of dipolar particles we obtain an analytical expression for the electrostatic free energy of solution within the random phase approximation. We show that an increase in the association constant leads to a decrease in the absolute value of the electrostatic free energy of solution, preventing its phase separation which is in agreement with the former computer simulations and theoretical results. We obtain a non-linear integro-differential equation for the self-consistent field potential created by the fixed external charges in a solution medium, taking into account the association of dipolar particles. As a consequence of the derived self-consistent field equation, in regime of weak electrostatic interactions, we obtain an analytical expression for the electrostatic potential of the point-like test ion, surrounded by the chain-like clusters of the dipolar particles. We show that in the mean-field approximation the association does not change the bulk dielectric permittivity of the solution, but increases the solvation radius of the point-like charge, relative to the theory of non-associating dipolar particles.

Longitudinal magnetic field effect on torsional vibration of carbon nanotubes

Torsional dynamic analysis of carbon nanotubes under the effect of longitudinal magnetic field is carried out in the present study. Torque effect of an axial magnetic field on a carbon nanotube has been defined using Maxwell’s relation. Nonlocal governing equation and boundary conditions for carbon nanotubes are obtained by using Hamilton’s minimum energy principle. Eringen’s nonlocal stress gradient elasticity theory is used in the formulation. Fourth order nonlocal equation of motion is solved by utilizing differential quadrature method. Clamped-clamped and clamped-free nonlocal boundary conditions are considered. Nonlocal and axial magnetic field effects on torsional vibration of carbon nanotubes are investigated. The magnetic field has significant effects on the dynamics of carbon nanotubes and may lead to torsional buckling. Critical torsional buckling load reduces with nonlocal effects. Nonlocality shows softening effect on carbon nanotube’s lattice structure. Present results can be used in the design and analysis of nanoelectromechanical products like nano-motors.

Gradient‐Enhancement of a Rock Model Applied to Numerical Simulations of Tunnel Advance

AbstractRock is classified as a frictional cohesive material characterized by nonlinear stress‐strain relations including softening leading to structural failure. To describe the mechanical behavior of rock comprising the accumulation of irreversible strains, strain hardening, and the degradation of stiffness and strength, an isotropic damage plasticity model for intact rock and rock mass was proposed in [1]. In the context of numerical simulations of tunneling, considering strain softening of rock allows to predict the formation of shear bands emanating from the tunnel surface due to the excavation process, which is crucial for engineers to forecast potentially dangerous situations related to failure. To ensure objective results in the softening regime upon mesh refinement, in the aforementioned rock model the concept of the crack band approach by Bažant and Oh was adopted. As a remedy of known deficiencies related to this regularization approach, the rock model is extended by the over‐nonlocal implicit gradient‐enhancement proposed in [2]. Accordingly, a nonlocal field of the damage‐driving variable, implicitly defined as the solution of a Helmholtz‐like partial differential equation, is introduced yielding a fully coupled system of second‐order PDEs. In the present contribution, the over‐nonlocal gradient enhancement of the rock model is presented and applied together with a linear‐elastic material model for shotcrete to finite element simulations of a deep tunnel constructed by the New Austrian Tunneling Method. Based on this real example, objectivity of the results with respect to the finite element discretization is discussed.

Influence of long-range interatomic and interlayer interactions on dispersion of acoustic waves by multilayer graphene

In this study, effects of the long-range interatomic and interlayer interactions on the dispersion of acoustic waves by multilayer graphene are investigated. A hybrid model that accounts for the long-range interatomic and interlayer interactions is developed. In the context of this model, the interatomic interactions are modeled based on the general nonlocal theory. The general nonlocal theory effectively models the dispersions of acoustic waves where it defines two independent nonlocal fields for the longitudinal and transverse acoustic waves. Interlayer interactions are defined to model the van der Waals interaction between the graphene layers. A new approach is proposed to determine the mechanical properties and the equivalent thickness of a single-layer graphene. The Young's modulus and Poisson's ratio of a single-layer graphene is obtained by ∼345MPa and ∼0.08, respectively. The equivalent thickness of a graphene sheet is determined by ∼0.057nm. An experimental validation for the dispersions of four-layer graphene is carried out where a good assent is observed. A parametric study is then presented to demonstrate the significant influence of the long-rang interatomic and interlayer interactions on the dispersion of incident acoustic waves by multilayer graphene.

Signal propagation on κ -Minkowski spacetime and nonlocal two-point functions

We study the propagation of quantum fields on $\kappa$-Minkowsi spacetime. Starting from the non-commutative partition function for a free field written in momentum space we derive the Feynman propagator and analyze the non-trivial singularity structure determined by the group manifold geometry of momentum space. The additional contributions due to such singularity structure result in a deformed field propagation which can be alternatively described in terms of an ordinary field propagation determined by a source with a blurred spacetime profile. We show that the $\kappa$-deformed Feynman propagator can be written in terms of vacuum expectation values of a commutative non-local quantum field. For sub-Planckian modes the $\kappa$-deformed propagator corresponds to the vacuum expectation value of the time-ordered product of non-local field operators while for trans-Plankian modes this is replaced by the Hadamard two-point function, the vacuum expectation value of the anti-commutator of non-local field operators.

An incompatibility tensor-based gradient plasticity formulation—Theory and numerics

In this work we discuss a gradient plasticity formulation which relies on the introduction of higher-order gradient contributions as additional arguments of the free energy function. These gradients can be interpreted in terms of the local geometrically necessary dislocation density. This gives rise to physically well-motivated kinematic-type hardening models in contrast to purely phenomenological approaches. At the same time, the framework results into a regularisation of the formulation such that localised plastic deformation processes in softening materials, e.g. the formation of shear bands, can be simulated. The employed theory is based on an extended nonlocal form of the Clausius–Duhem inequality, motivated by a possible energy exchange between particles at the microstructural level. This gives rise to the balance equation of a nonlocal stress tensor that is found to be work-conjugated to the plastic velocity gradient. The solution of the governing system of partial differential equations is approached by means of a multi-field finite element formulation with the solution of the Karush–Kuhn–Tucker conditions being addressed on a global level by means of Fischer–Burmeister complementary functions. We discuss a specific quadratic energy contribution in terms of the incompatibility, respectively dislocation density tensor that results into well-interpretable contributions to the nonlocal stress field and study the formation of shear bands induced by geometric imperfections as well as the constitutive response at a material interface where the yield limit exhibits a jump discontinuity.

Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu=|u|p−2u−|u|q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p<2⁎, the ground state solution asymptotically coincides with unique positive ground state solution of (−Δ)su+u=up, whereas for p=2⁎ the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical Emden–Fowler type equation. For p>2⁎, the solution asymptotically coincides with a ground-state solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.

Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio

In this paper, a novel lattice Boltzmann (LB) model based on the Allen-Cahn phase-field theory is proposed for simulating axisymmetric multiphase flows. The most striking feature of the model is that it enables to handle multiphase flows with large density ratio, which are unavailable in all previous axisymmetric LB models. The present model utilizes two LB evolution equations, one of which is used to solve fluid interface, and another is adopted to solve hydrodynamic properties. To simulate axisymmetric multiphase flows effectively, the appropriate source term and equilibrium distribution function are introduced into the LB equation for interface tracking, and simultaneously, a simple and efficient forcing distribution function is also delicately designed in the LB equation for hydrodynamic properties. Unlike many existing LB models, the source and forcing terms of the model arising from the axisymmetric effect include no additional gradients, and consequently, the present model contains only one non-local phase field variable, which in this regard is much simpler. In addition, to enhance the model’s numerical stability, an advanced multiple-relaxation-time (MRT) model is also applied for the collision operator. We further conducted the Chapman-Enskog analysis to demonstrate the consistencies of our present MRT-LB model with the axisymmetric Allen-Cahn equation and hydrodynamic equations. A series of numerical examples, including static droplet, oscillation of a viscous droplet, breakup of a liquid thread, and bubble rising in a continuous phase, are used to test the performance of the proposed model. It is found that the present model can generate relatively small spurious velocities and can capture interfacial dynamics with higher accuracy than the previously improved axisymmetric LB model. Besides, it is also found that our present numerical results show excellent agreement with analytical solutions or available experimental data for a wide range of density ratios, which highlights the strengths of the proposed model.

Disclination based model of grain boundary in crystalline materials with microstructural defects

This paper deals with continuous description of elasto-plastic materials with lattice defects such as dislocations and disclinations, and grain boundaries. The behaviour of crystalline solids containing defects is described by non-local fields that are smooth over an interatomic length scale and at times of micro-seconds. The dislocation and disclination densities are appropriate measures of incompatibilities of the plastic distortion and of the plastic connection relative to plastic distortion, respectively. The grain boundaries will be simulated in terms of an array of disclination dipoles. The heterogeneous initial disclinations are a source of dislocations. These lattice defects, stress, strain and the displacement vector will be provided by solving initial and boundary value problems in elasto-plastic bodies, which involve dissipative evolution equations for plastic distortion and disclination tensor. The numerical procedure and corresponding algorithms to solve the variational formulation of the initial and boundary value problems are provided. The results of numerical simulations are analyzed and a number of graphs illustrate the elasto-plastic behaviour of the sheet when a shear stress is applied on one of its edge.

Critical behavior of a nonlocalϕ4field theory and asymptotic freedom

The critical behavior of a nonlocal scalar field theory is studied. This theory has a nonlocal kinetic term which involves a real power $1\ensuremath{-}2\ensuremath{\alpha}$ of the Laplacian. The interaction term is the usual local ${\ensuremath{\phi}}^{4}$ interaction. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. Particular features appearing in the renormalization of this nonlocal theory that differ from the case of local theories are studied. The previous calculations lead to the perturbative computation of the coupling constant beta function and critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$. In four dimensions for $\ensuremath{\alpha}&lt;0$ this beta function presents asymptotic freedom in the UV. This is remarkable since no non-Abelian vector fields are included. However, this comes at the expense of losing reflection positivity.

Nonlocal Damage Mechanics for Quantification of Health for Piezoelectric Sensor

In this paper, a novel method to quantify the incubation of damage on piezoelectric crystal is presented. An intrinsic length scale parameter obtained from nonlocal field theory is used as a novel measure for quantification of damage precursor. Features such as amplitude decay, attenuation, frequency shifts and higher harmonics of guided waves are commonly-used damage features. Quantification of the precursors to damage by considering the mentioned features in a single framework is a difficult proposition. Therefore, a nonlocal field theory is formulated and a nonlocal damage index is proposed. The underlying idea of the paper is that inception of the damage at the micro scale manifests the evolution of damage at the macro scale. In this paper, we proposed a nonlocal field theory, which can efficiently quantify the inception of damage on piezoelectric crystals. The strength of the method is demonstrated by employing the surface acoustic waves (SAWs) and longitudinal bulk waves in Lithium Niobate (LiNbO3) single crystal. A control damage was introduced and its manifestation was expressed using the intrinsic dominant length scale. The SAWs were excited and detected using interdigital transducers (IDT) for healthy and damage state. The acoustic imaging of microscale damage in piezoelectric crystal was conducted using scanning acoustic microscopy (SAM). The intrinsic damage state was then quantified by overlaying changes in time of flight (TOF) and frequency shift on the angular dispersion relationship.

Nonlinear performance analysis of forced carbon nanotube-based bio-mass sensors

In this effort, an analytical solution is proposed for the large amplitude nonlinear vibrations of doubly clamped carbon nanotube (CNT)-based nano-scale bio-mass sensors. The single walled CNT is modeled as an elastic Euler–Bernoulli nano-scale beam and the size effects are introduced into the mathematical model of the system through Eringen’s nonlocal elastic field theory. The nonlinearity arises due to mid-plane stretching of the bridged CNT, and is accounted for as the von Karman nonlinearity. The impacts of deposited nano-scale bio-object, its geometrical properties, and its landing position along the longitudinal axis of the CNT-based resonator are considered. The nonlinear equations of motion are derived based on Hamilton’s principle and then the Method of Multiple Scales is employed to derive an analytical approximate solution for the system’s response. To verify the analytical solution and show its limits of applicability, the equations of motion are discretized by multi-mode Galerkin’s method and then the obtained set of equations are numerically solved by Runge–Kutta method and compared with those obtained by analytical solution. The potential applications of the CNT-based resonators for both of nonlinear frequency-/amplitude-based mass sensing are investigated and discussed. The obtained results show that the amplitude-based mass sensing has higher performance than frequency-based one in high quality-factor environments, such as in vitro biological mass sensing in the air or vacuum and inversely the frequency-based mass sensing method has higher mass sensibility in low quality factor environments, such as in vivo biological mass sensing in the liquid solution samples.

Unruh effect of nonlocal field theories with a minimal length

The nonlocal field theory commonly requires a minimal length, and so it appears to formulate the nonlocal theory in terms of the doubly special relativity which makes the speed of light and the minimal length invariant simultaneously. We set up a generic nonlocal model having the same set of solutions as the local theory but allowing Lorentz violations due to the minimal length. It is exactly corresponding to the model with the modified dispersion relation in the doubly special relativity. For this model, we calculate the modified Wightman function and the rate of response function by using the Unruh–DeWitt detector method. It turns out that the Unruh effect should be corrected by the minimal length related to the nonlocality in the regime of the doubly special relativity. However, for the Lorentz-invariant limit, it is shown that the Wightman function and the Unruh effect remain the same as those of the local theory.

A nonlocal strain gradient theory for scale-dependent wave dispersion analysis of rotating nanobeams considering physical field effects

This paper is concerned with the wave propagation behavior of rotating functionally graded temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field. Uniform, linear and nonlinear temperature distributions across the thickness are investigated. Thermo-elastic properties of FG beam change gradually according to the Mori-Tanaka distribution model in the spatial coordinate. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The governing equations are derived by Hamilton\'s principle as a function of axial force due to centrifugal stiffening and displacement. By applying an analytical solution and solving an eigenvalue problem, the dispersion relations of rotating FG nanobeam are obtained. Numerical results illustrate that various parameters including temperature change, angular velocity, nonlocality parameter, wave number and gradient index have significant effect on the wave dispersion characteristics of the understudy nanobeam. The outcome of this study can provide beneficial information for the next generation researches and exact design of nano-machines including nanoscale molecular bearings and nanogears, etc.

Higgs mechanism in nonlocal field theories

We study spontaneous gauge symmetry breaking and the Higgs mechanism in nonlocal field theories. Motivated by the level truncated action of string field theory, we consider a class of nonlocal field theories with an exponential factor of the d’Alembertian attached to the kinetic and mass terms. Modifications of this kind are known to make mild the UV behavior of loop diagrams and thus have been studied not only in the context of string theory but also as an alternative approach to quantum gravity. In this paper we argue that such a nonlocal theory potentially includes a ghost mode near the nonlocal scale in the particle spectrum of the symmetry broken phase. This is in sharp contrast to local field theories and would be an obstruction to making a simple nonlocal model a UV complete theory. We then discuss a possible way out by studying nonlocal theories with extra symmetries such as gauge symmetries in higher spacetime dimensions.

Nonlocal statistical field theory of dipolar particles in electrolyte solutions

We present a nonlocal statistical field theory of a dilute electrolyte solution with a small additive of dipolar particles. We postulate that every dipolar particle is associated with an arbitrary probability distribution function (PDF) of distance between its charge centers. Using the standard Hubbard–Stratonovich transformation, we represent the configuration integral of the system in the functional integral form. We show that in the limit of a small permanent dipole moment, the functional in integrand exponent takes the well known form of the Poisson–Boltzmann–Langevin (PBL) functional. In the mean-field approximation we obtain a non-linear integro-differential equation with respect to the mean-field electrostatic potential, generalizing the PBL equation for the point-like dipoles obtained first by Abrashkin et al. We apply the obtained equation in its linearized form to derivation of the expressions for the mean-field electrostatic potential of the point-like test ion and its solvation free energy in salt-free solution, as well as in solution with salt ions. For the ‘Yukawa’-type PDF we obtain analytic relations for both the electrostatic potential and the solvation free energy of the point-like test ion. We obtain a general expression for the bulk electrostatic free energy of the solution within the Random phase approximation (RPA). For the salt-free solution of the dipolar particles for the Yukawa-type PDF we obtain an analytic relation for the electrostatic free energy, resulting in two limiting regimes. Finally, we analyze the limiting laws, following from the general relation for the electrostatic free energy of solution in presence of both the ions and the dipolar particles for the case of Yukawa-type PDF.

Relativistic corrections to nonrelativistic effective field theories

In this paper we develop a formalism for studying the nonrelativistic limit of relativistic field theories in a systematic way. By introducing a simple, nonlocal field redefinition, we transform a given relativistic theory, describing a real, self-interacting scalar field, into an equivalent theory, describing a complex scalar field that encodes at each time both the original field and its conjugate momentum. Our low-energy effective theory incorporates relativistic corrections to the kinetic energy as well as the backreaction of fast-oscillating terms on the behavior of the dominant, slowly varying component of the field. Possible applications of our new approach include axion dark matter, though the methods developed here should be applicable to the low-energy limits of other field theories as well.

Nonlocal vibrations and potential instability of monolayers from double-walled carbon nanotubes subjected to temperature gradients

Using nonlocal elasticity theory of Eringen, free transverse thermo-elastic vibrations of vertically aligned double-walled carbon nanotubes (DWCNTs) with a membrane configuration is going to be explored methodically. Accounting for nonlocal heat conduction along the side walls of DWCNTs with allowance of heat dissipation from the outer surfaces, the nonlocal temperature fields within the constitutive tubes are assessed for a steady-state regime. The van der Waals interactional forces between atoms of each pair of DWCNTs as well as the innermost and outermost tubes are displayed by laterally continuous springs whose constants are appropriately evaluated. By establishing suitable discrete and continuous models on the basis of the Rayleigh and higher-order beam theories, nonlocal in-plane and out-of-plane vibrations of thermally affected nanosystems of monolayers of DWCNTs are examined and discussed. The critical values of the temperature change are stated explicitly and the role of influential factors on this crucial factor as well as the free vibration behavior are investigated in some detail. Through conducting a fairly comprehensive numerical study, the influences of the slenderness ratio, temperature change in the low and high temperatures, number of nanotubes, small-scale parameter, intertube distance, and stiffness of the surrounding environment on the nonlocal-fundamental frequency are explained. The obtained results from this work could provide crucial guidelines for the design of membranes or even jungles of vertically aligned DWCNTs as thermal interface nanostructures.

Nonlocal vertices and analyticity: Landau equations and general Cutkosky rule

We study the analyticity properties of amplitudes in theories with nonlocal vertices of the type occurring in string field theory and a wide class of nonlocal field theory models. Such vertices are given in momentum space by entire functions of rapid decay in certain (including Euclidean) directions ensuring UV finiteness but are necessarily of rapid increase in others. A parametric representation is obtained by integrating out the loop (Euclidean) momenta after the introduction of generalized Schwinger parameters. Either in the original or parametric representation, the well-defined resulting amplitudesare then continued in the complex space of the external momenta invariants. We obtain the alternative forms of the Landau equations determining the singularity surfaces showing that the nonlocal vertices serve as UV regulators but do not affect the local singularity structure. As a result the full set of singularities known to occur in local field theory also occurs here: normal and anomalous thresholds as well as acnodes, crunodes, and cusps that may under certain circumstances appear even on the physical sheet. Singularities of the second type also appear as shown from the parametric representation. We obtain the general Cutkosky discontinuity rule for encircling a singularity by employing contour deformations only in the finite plane. The unitarity condition (optical theorem) is thendiscussed as a special application of the rule across normal thresholds and the hermitian analyticity property of amplitudes.

Electrovacuum solutions in nonlocal gravity

We consider the coupling of the electromagnetic field to a non-local gravity theory comprising of the Einstein-Hilbert action in addition to a non-local $R\, {\Box}^{-2} R$ term associated with a mass scale $m$. We demonstrate that in the case of the minimally coupled electromagnetic field, real corrections about the Reissner-Nordstr\"om background only exist between the inner Cauchy horizon and the event horizon of the black hole. This motivates us to consider the modified coupling of electromagnetism to this theory via the Kaluza ansatz. The Kaluza reduction introduces non-local terms involving the electromagnetic field to the pure gravitational non-local theory. An iterative approach is provided to perturbatively solve the equations of motion to arbitrary order in $m^2$ about any known solution of General Relativity. We derive the first-order corrections and demonstrate that the higher order corrections are real and perturbative about the external background of a Reissner-Nordstr\"om black hole. We also discuss how the Kaluza reduced action, through the inclusion of non-local electromagnetic fields, could also be relevant in quantum effects on curved backgrounds with horizons.