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Published: 14 March 2023

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1

Eringen, A. Cemal, ed. Nonlocal Continuum Field Theories. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/b97697.

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2

Nonlocal continuum field theories. New York: Springer, 2002.

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3

Chen, Jingkai. Nonlocal Euler–Bernoulli Beam Theories. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-69788-4.

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4

S, Ilʹi͡ashenko I͡U. Nonlocal bifurcations. Providence, R.I: American Mathematical Society, 1999.

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5

Ilyashenko, Yu S. Nonlocal bifurcations. Providence, R.I: American Mathematical Society, 1999.

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6

Nonlocal quantum field theory and stochastic quantum mechanics. Dordrecht: D. Reidel pub. Co., 1986.

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7

Boyd, J. P. Weakly nonlocal solitary waves and beyond-all-orders asymptotics: Generalized solitons and hyperasymptotic perturbation theory. Dordrecht: Kluwer Academic Publishers, 1998.

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8

Ordinary differential equations: Qualitative theory. Providence, R.I: American Mathematical Society, 2010.

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9

Kloeden, Peter E. Nonautonomous dynamical systems. Providence, R.I: American Mathematical Society, 2011.

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10

Eringen, A. Cemal. Nonlocal Continuum Field Theories. Springer, 2002.

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11

Eringen, A. Cemal. Nonlocal Continuum Field Theories. Springer, 2010.

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12

Suhubi. Nonlocal Field Theories of Mechanics. Pitman Pub Ltd, 1986.

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13

Mashhoon, Bahram. Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.001.0001.

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A postulate of locality permeates through the special and general theories of relativity. First, Lorentz invariance is extended in a pointwise manner to actual, namely, accelerated observers in Minkowski spacetime. This hypothesis of locality is then employed crucially in Einstein’s local principle of equivalence to render observers pointwise inertial in a gravitational field. Field measurements are intrinsically nonlocal, however. To go beyond the locality postulate in Minkowski spacetime, the past history of the accelerated observer must be taken into account in accordance with the Bohr-Rosenfeld principle. The observer in general carries the memory of its past acceleration. The deep connection between inertia and gravitation suggests that gravity could be nonlocal as well and in nonlocal gravity the fading gravitational memory of past events must then be taken into account. Along this line of thought, a classical nonlocal generalization of Einstein’s theory of gravitation has recently been developed. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. A significant observational consequence of this theory is that the nonlocal aspect of gravity appears to simulate dark matter. The implications of NLG are explored in this book for gravitational lensing, gravitational radiation, the gravitational physics of the Solar System and the internal dynamics of nearby galaxies as well as clusters of galaxies. This approach is extended to nonlocal Newtonian cosmology, where the attraction of gravity fades with the expansion of the universe. Thus far only some of the consequences of NLG have been compared with observation.
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14

Morawetz, Klaus. Nonlocal Collision Integral. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0013.

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The kinetic equation with the nonlocal shifts is presented as the final result on the way to derive the kinetic equation with nonlocal corrections. The exclusive dependence of the nonlocal and non-instant corrections on the scattering phase shift confirms the results from the theory of gases. With the approximation on the level of the Brueckner reaction matrix, the corresponding non-instant and nonlocal scattering integral in parallel with the classical Enskog’s equation, can be treated with Monte-Carlo simulation techniques. Neglecting the shifts, the Landau theory of quasiparticle transport appears. In this sense the presented kinetic theory unifies both approaches. An intrinsic symmetry is found from the optical theorem which allows for representing the collision integral equivalently either in particle-hole symmetric or space-time symmetric form.
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15

McArthur, Randall Douglas. Higher-dimensional chiral anomalies and nonlocal theories. 1991.

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16

Chen, Jingkai. Nonlocal Euler-Bernoulli Beam Theories: A Comparative Study. Springer International Publishing AG, 2021.

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17

Mashhoon, Bahram. Linearized Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0007.

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The only known exact solution of the field equation of nonlocal gravity (NLG) is the trivial solution involving Minkowski spacetime that indicates the absence of a gravitational field. Therefore, this chapter is devoted to a thorough examination of NLG in the linear approximation beyond Minkowski spacetime. Moreover, the solutions of the linearized field equation of NLG are discussed in detail. We adopt the view that the kernel of the theory must be determined from observation. In the Newtonian regime of NLG, we recover the phenomenological Tohline-Kuhn approach to modified gravity. A simple generalization of the Kuhn kernel leads to a three-parameter modified Newtonian force law that is always attractive. Gravitational lensing is discussed. It is shown that nonlocal gravity (NLG), with a characteristic galactic lengthscale of order 1 kpc, simulates dark matter in the linear regime while preserving causality.
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18

Mashhoon, Bahram. Nonlocal Newtonian Cosmology. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0010.

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We explore some of the cosmological implications of nonlocal gravity (NLG) theory, in which nonlocality is due to the gravitational memory of past events. Memory dies out in space and time. The fading of memory in time implies that in NLG the strength of the gravitational interaction must decrease with cosmic time. In the Newtonian regime of NLG, the nonlocal character of gravity simulates dark matter in spiral galaxies and clusters of galaxies. However, dark matter is considered indispensable as well for structure formation in standard models of cosmology. Can nonlocal gravity solve the problem of structure formation in cosmology without recourse to dark matter? In this chapter, a beginning is made in this direction by extending nonlocal gravity in the Newtonian regime to the cosmological domain. The nonlocal analog of the Zel’dovich solution is formulated and the consequences of the resulting nonlocal Zel’dovich model are investigated in detail.
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19

Mashhoon, Bahram. Nonlocal Gravity and Dark Matter. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0008.

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The implications of linearized NLG for the gravitational physics of the Solar System, spiral galaxies and nearby clusters of galaxies are critically examined in this chapter. In the Newtonian regime, NLG involves a reciprocal kernel with three length parameters. We discuss the determination of these parameters by comparing the predictions of the theory with observational data. Furthermore, the virial theorem for the Newtonian regime of NLG is derived and its consequences for nearby “isolated” astronomical systems in virial equilibrium are investigated. For such a galaxy, in particular, the galaxy’s baryonic diameter namely, the diameter of the smallest sphere that completely surrounds the baryonic system at the present time, is predicted to be larger than the basic nonlocality lengthscale, which is about 3 kpc, times the effective dark matter fraction of the galaxy.
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20

John Lee Kenneth.* Terning. Nonlocal models of goldstone bosons in asymptotically free gauge theories. 1990.

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21

Mashhoon, Bahram. Field Equation of Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0006.

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In extended general relativity (GR), Einstein’s field equation of GR can be expressed in terms of torsion and this leads to the teleparallel equivalent of GR, namely, GR||, which turns out to be the gauge theory of the Abelian group of spacetime translations. The structure of this theory resembles Maxwell’s electrodynamics. We use this analogy and the world function to develop a nonlocal GR|| via the introduction of a causal scalar constitutive kernel. It is possible to express the nonlocal gravitational field equation as modified Einstein’s equation. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. The field equation of NLG can be expressed as Einstein’s field equation with an extra source that has the interpretation of the effective dark matter. It is possible that the kernel of NLG, which is largely undetermined, could be derived from a more general future theory.
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22

Rogula, D. Nonlocal Theory of Material Media. Springer London, Limited, 2014.

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23

Mashhoon, Bahram. Linearized Gravitational Waves in Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0009.

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Gravitational radiation is investigated within the framework of linearized nonlocal gravity. In this theory, linearized gravitational waves are damped as they travel from the source to the receiver. This gravitational memory drag leads to the exponential decay of the wave amplitude. The damping effect could be significant for waves with very long wavelegths comparable to galactic distances. More generally, for gravitational waves with wavelengths comparable to the basic nonlocality lengthscale of order 1 kpc, the nonlocal deviations from general relativity can be significant. However, gravitational waves of current observational interest have wavelengths that are very small in comparison with 1 kpc; in this case, the nonlocal deviations from general relativity essentially average out and can be safely neglected in practice.
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24

Sagi, Maria, and Ervin Laszlo. Remote Healing: Nonlocal Information Medicine and the Akashic Field. Inner Traditions International, Limited, 2020.

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25

Sagi, Maria, and Ervin Laszlo. Remote Healing: Nonlocal Information Medicine and the Akashic Field. Inner Traditions International, Limited, 2020.

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26

Morawetz, Klaus. Nonequilibrium Quantum Hydrodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0015.

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The balance equations resulting from the nonlocal kinetic equation are derived. They show besides the Landau-like quasiparticle contributions explicit two-particle correlated parts which can be interpreted as molecular contributions. It looks like as if two particles form a short-living molecule. All observables like density, momentum and energy are found as a conserving system of balance equations where the correlated parts are in agreement with the forms obtained when calculating the reduced density matrix with the extended quasiparticle functional. Therefore the nonlocal kinetic equation for the quasiparticle distribution forms a consistent theory. The entropy is shown to consist also of a quasiparticle part and a correlated part. The explicit entropy gain is proved to complete the H-theorem even for nonlocal collision events. The limit of Landau theory is explored when neglecting the delay time. The rearrangement energy is found to mediate between the spectral quasiparticle energy and the Landau variational quasiparticle energy.
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27

Petrosyan, Arshak, Camelia A. Pop, and Donatella Danielli. New Developments in the Analysis of Nonlocal Operators: AMS Special Session on New Developments in the Analysis of Nonlocal Operators, October 28-30, 2016, University of St. Thomas, Minneapolis, Minnesota. American Mathematical Society, 2019.

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28

Morawetz, Klaus. Historical Background. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0001.

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The historical development of kinetic theory is reviewed with respect to the inclusion of virial corrections. Here the theory of dense gases differs from quantum liquids. While the first one leads to Enskog-type of corrections to the kinetic theory, the latter ones are described by quasiparticle concepts of Landau-type theories. A unifying kinetic theory is envisaged by the nonlocal quantum kinetic theory. Nonequilibrium phenomena are the essential processes which occur in nature. Any evolution is built up of involved causal networks which may render a new state of quality in the course of time evolution. The steady state or equilibrium is rather the exception in nature, if not a theoretical abstraction at all.
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29

Mashhoon, Bahram. Acceleration-Induced Nonlocality. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0002.

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The locality postulate of the standard relativity theory is exact when dealing with phenomena involving classical point particles and rays of radiation, but breaks down for electromagnetic fields, as field properties cannot be measured instantaneously. Furthermore, Bohr and Rosenfeld pointed out in 1933 that only spacetime averages of the classical electric and magnetic fields have immediate physical significance. This assertion acquires the status of a physical principle when the intrinsic acceleration scales of observers are taken into account. To incorporate acceleration-induced nonlocality into relativity theory, a general integral relation is postulated between the field as measured by an accelerated observer and the instantaneous field measurements of the momentarily comoving inertial observers along the past world line of the observer. This nonlocal ansatz involves an acceleration kernel and leads to nonlocal special relativity once the kernel is determined.
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30

Mashhoon, Bahram. Extension of General Relativity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0005.

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Nonlocal general relativity (GR) requires an extension of the mathematical framework of GR. Nonlocal GR is a tetrad theory such that the orthonormal tetrad frame field of a preferred set of observers carries the sixteen gravitational degrees of freedom. The spacetime metric is then defined via the orthonormality condition. The preferred frame field is used to define a new linear Weitzenböck connection in spacetime. The non-symmetric Weitzenböck connection is metric compatible, curvature-free and renders the preferred (fundamental) frame field parallel. This circumstance leads to teleparallelism. The fundamental parallel frame field defined by the Weitzenböck connection is the natural generalization of the parallel frame fields of the static inertial observers in a global inertial frame in Minkowski spacetime. The Riemannian curvature of the Levi-Civita connection and the torsion of the Weitzenböck connection are complementary aspects of the gravitational field in extended GR.
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31

Morawetz, Klaus. Systems with Condensates and Pairing. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0012.

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The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown to correct the T-matrix from unphysical multiple-scattering events. The resulting generalised Soven scheme provides the Beliaev equations for Boson’s and the Nambu–Gorkov equations for fermions without the usage of anomalous and non-conserving propagators. This systematic theory allows calculating the fluctuations above and below the critical parameters. Gap equations and Bogoliubov–DeGennes equations are derived from this theory. Interacting Bose systems with finite temperatures are discussed with successively better approximations ranging from Bogoliubov and Popov up to corrected T-matrices. For superconductivity, the asymmetric theory leading to the corrected T-matrix allows for establishing the stability of the condensate and decides correctly about the pair-breaking mechanisms in contrast to conventional approaches. The relation between the correlated density from nonlocal kinetic theory and the density of Cooper pairs is shown.
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32

Morawetz, Klaus. Deep Impurities with Collision Delay. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0017.

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The linearised nonlocal kinetic equation is solved analytically for impurity scattering. The resulting response function provides the conductivity, plasma oscillation and Fermi momentum. It is found that virial corrections nearly compensate the wave-function renormalizations rendering the conductivity and plasma mode unchanged. Due to the appearance of the correlated density, the Luttinger theorem does not hold and the screening length is influenced. Explicit results are given for a typical semiconductor. Elastic scattering of electrons by impurities is the simplest but still very interesting dissipative mechanism in semiconductors. Its simplicity follows from the absence of the impurity dynamics, so that individual collisions are described by the motion of an electron in a fixed potential.
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33

Mashhoon, Bahram. Acceleration Kernel. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0003.

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The phenomenon of spin-rotation coupling provides the key to the determination of the kernel. Imagine an observer rotating in the positive sense about the direction of propagation of an incident plane monochromatic electromagnetic wave of positive helicity. Using the locality postulate, the field as measured by the rotating observer can be determined. If the observer rotates with the same frequency as the wave, the measured radiation field loses its temporal dependence. By a mere rotation, observers could in principle stay at rest with respect to an incident positive-helicity wave. To avoid this possibility, we assume that a basic radiation field cannot stand completely still with respect to an accelerated observer. This basic principle eventually leads to the determination of the kernel and a nonlocal theory of accelerated systems that is in better agreement with quantum mechanics than the standard theory based on the hypothesis of locality.
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34

Morawetz, Klaus. Elementary Principles. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0002.

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The many-body theory combines ideas of thermodynamics with ideas of mechanics. In this introductory chapter, the symbiosis of these two different fields of physics is demonstrated on overly simplified models. We explore the principles of finite-range forces to show the twofold nature of virial corrections. Infrequent collisions with a large deflection angle lead to collision integrals and rather frequent encounters with deflections on small angles act as a mean field. The (mean-field) corrections to drift result in the internal pressure and the nonlocal correction to the collisions results in the effect of the molecular volumes. The concept of distribution functions is introduced and the measure of information as entropy. The binary correlation allows one to distinguish tails and cores of the interaction potential. The concept of binary correlation is thus behind the intuitive picture of the kinetic equation.
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35

Horing, Norman J. Morgenstern. Quantum Statistical Field Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.001.0001.

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The methods of coupled quantum field theory, which had great initial success in relativistic elementary particle physics and have subsequently played a major role in the extensive development of non-relativistic quantum many-particle theory and condensed matter physics, are at the core of this book. As an introduction to the subject, this presentation is intended to facilitate delivery of the material in an easily digestible form to students at a relatively early stage of their scientific development, specifically advanced undergraduates (rather than second or third year graduate students), who are mathematically strong physics majors. The mechanism to accomplish this is the early introduction of variational calculus with particle sources and the Schwinger Action Principle, accompanied by Green’s functions, and, in addition, a brief derivation of quantum mechanical ensemble theory introducing statistical thermodynamics. Important achievements of the theory in condensed matter and quantum statistical physics are reviewed in detail to help develop research capability. These include the derivation of coupled field Green’s function equations of motion for a model electron-hole-phonon system, extensive discussions of retarded, thermodynamic and non-equilibrium Green’s functions, and their associated spectral representations and approximation procedures. Phenomenology emerging in these discussions includes quantum plasma dynamic, nonlocal screening, plasmons, polaritons, linear electromagnetic response, excitons, polarons, phonons, magnetic Landau quantization, van der Waals interactions, chemisorption, etc. Considerable attention is also given to low-dimensional and nanostructured systems, including quantum wells, wires, dots and superlattices, as well as materials having exceptional conduction properties such as superconductors, superfluids and graphene.
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36

Cardaliaguet, Pierre, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions. The Master Equation and the Convergence Problem in Mean Field Games. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691190716.001.0001.

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This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.
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