Nonlocal Variables Research Articles

A strange recursion operator demystified

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanli) et al. (arXiv: nlin.SI/0203036) is bihamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators. Using this, we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution, and the evolution systems corresponding to these symmetries are bihamiltonian as well. We also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form, with the integration operator $D^{-1}$ appearing in each term at most once.

Nonlocal symmetries and recursion operators: Partial differential and differential–difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential–difference equations with two independent variables is presented and shown that the Korteweg–de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses ( 2 × 2 ) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential–difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.

Quantum theory from quantum gravity

We provide a mechanism by which, from a background-independent model with no quantum mechanics, quantum theory arises in the same limit in which spatial properties appear. Starting with an arbitrary abstract graph as the microscopic model of spacetime, our ansatz is that the microscopic dynamics can be chosen so that (i) the model has a low-energy limit which reproduces the nonrelativistic classical dynamics of a system of $N$ particles in flat spacetime, (ii) there is a minimum length, and (iii) some of the particles are in a thermal bath or otherwise evolve stochastically. We then construct simple functions of the degrees of freedom of the theory and show that their probability distributions evolve according to the Schr\"odinger equation. The nonlocal hidden variables required to satisfy the conditions of Bell's theorem are the links in the fundamental graph that connect nodes adjacent in the graph but distant in the approximate metric of the low-energy limit. In the presence of these links, distant stochastic fluctuations are transferred into universal quantum fluctuations.

On Non-Abelian Coverings Over the Liouville Equation

Treating the hyperbolic Liouville equation as the flat connections equation on the semisimple Lie algebra A1, we investigate relationships between zero-curvature representations of the Liouville equation and its Backlund transformations provided by a special one-dimensional coverings. Formal deformations of these Backlund transformations and integration in nonlocal variables are studied.

Collective Mining of Bayesian Networks from Distributed Heterogeneous Data

We present a collective approach to learning a Bayesian network from distributed heterogeneous data. In this approach, we first learn a local Bayesian network at each site using the local data. Then each site identifies the observations that are most likely to be evidence of coupling between local and non-local variables and transmits a subset of these observations to a central site. Another Bayesian network is learnt at the central site using the data transmitted from the local site. The local and central Bayesian networks are combined to obtain a collective Bayesian network, which models the entire data. Experimental results and theoretical justification that demonstrate the feasibility of our approach are presented.

Instantaneous measurements of nonlocal variables

Relativistic causality imposes rigid restrictions on nondemolition (repeatable) measurements of nonlocal variables. We show that there are no causal restrictions on demolition (nonrepeatable) measurements: all Hermitian operators of multipartite quantum system can be measured instantaneously, provided unlimited supply of entanglement resources.

Instantaneous measurement of nonlocal variables.

It is shown, under the assumption of the possibility to perform an arbitrary local operation, that all nonlocal variables related to two or more separate sites can be measured instantaneously, except for a finite time required for bringing to one location the classical records from these sites which yield the result of the measurement. It is a verification measurement: it yields reliably the eigenvalues of the nonlocal variables, but it does not prepare the eigenstates of the system.

Instantaneous measurements of nonlocal variables

Abstract Relativistic causality imposes rigid restrictions on nondemolition (repeatable) measurements of nonlocal variables. We show that there are no causal restrictions on demolition (nonrepeatable) measurements: all Hermitian operators of multipartite quantum system can be measured instantaneously, provided unlimited supply of entanglement resources.

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3 D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations. A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.

1+3covariant dynamics of scalar perturbations in braneworlds

We discuss the dynamics of linear, scalar perturbations in an almost Friedmann-Robertson-Walker braneworld cosmology of Randall-Sundrum type II using the 1+3 covariant approach. We derive a complete set of frame-independent equations for the total matter variables, and a partial set of equations for the non-local variables which arise from the projection of the Weyl tensor in the bulk. The latter equations are incomplete since there is no propagation equation for the non-local anisotropic stress. We supplement the equations for the total matter variables with equations for the independent constituents in a cold dark matter cosmology, and provide solutions in the high and low-energy radiation-dominated phase under the assumption that the non-local anisotropic stress vanishes. These solutions reveal the existence of new modes arising from the two additional non-local degrees of freedom. Our solutions should prove useful in setting up initial conditions for numerical codes aimed at exploring the effect of braneworld corrections on the cosmic microwave background (CMB) power spectrum. As a first step in this direction, we derive the covariant form of the line of sight solution for the CMB temperature anisotropies in braneworld cosmologies, and discuss possible mechanisms by which braneworld effects may remain in the low-energy universe.

Nonlocal variables with product-state eigenstates

An alternative proof for existence of ``quantum nonlocality without entanglement'', i.e. existence of variables with product-state eigenstates which cannot be measured locally, is presented. A simple``nonlocal'' variable for the case of one-way communication is given and the limit for its approximate measurability is found.

Quantum non-demolition measurement of nonlocal variables and its application in quantum authentication

Quantum non-demolition (QND) variables are generalized to the nonlocal ones by proposing QND measurement networks of Bell states and multi-partite GHZ states, which means that we can generate and measure them without any destruction. One of its prospective applications in the quantum authentication (QA) system of the quantum security automatic Teller machine (QSATM) which is much more reliable than the classical ones is also presented.

Reconstruction of potential evaporation for water balance studies

A large number of environmental management decisions such as the remediation of a mine site require the best possible knowledge of the water balance of the site or the catchment area. For precipitation a large number of observations exists and usually some data are available which are more or less representative for the precipitation conditions in the catchment. However, to close the water balance knowledge about evaporation is also required. This is usually not available from observations. Also bulk formulas cannot be used to reconstruct evaporation for a sufficiently long time, since observations of the input variables are limited as well. In this article an approach is presented showing how local potential evaporation may be reconstructed from other local and non-local atmospheric variables for which long-term observations are available. The method is demonstrated for the planned artificial lakes in the closed down mine sites in the Geiseltal (Germany) and aims at the estimation of the probability distribution of the difference between precipitation and evaporation for the surface water balance.

New τ-dependent correlation functional combined with a modified Becke exchange

A new correlation functional is derived within the Kohn–Sham (KS) Density Functional Theory (DFT) involving the electron kinetic energy density τ and the Laplacian of the electron density as key nonlocal variables. The derivation is based on a direct resolution of the adiabatic connection formula and using an analogy with the local thermodynamic approach in DFT, following the Lap3 theory developed previously. Compared to the latter, the new functional involves higher order τ-dependent energy terms in a form suggesting a possible resummation procedure that could be used for further development. It is combined with the nonlocal exchange functional of Becke, by modifying the latter in an empirical fashion to achieve better synchronization between the two energy components. The resulting exchange-correlation scheme (named “Bmτ1”) is validated on several test systems known as difficult for DFT, at least at the Local Spin Density and Generalized Gradient Approximation levels. The recent nonempirical hybrid scheme PBE1PBE (“PBE0”) is included in the comparative tests as a parameter-free benchmark for the hybrid HF-KS DFT approach. Improved results for relative energies, activation barriers and equilibrium geometries are obtained with the Bmτ1 functional, particularly concerning aromatic compounds, systems with weak hydrogen bonds, proton transfer processes and transition-metal carbonyls.

Numerical aspects of non-local modeling of the damage evolution in elastic–plastic materials

The design of mechanical systems in modern industrial plants requires reliable and efficient methods to predict the behavior of structural materials. For complex loading conditions, the behavior of the structural materials is determined by damage evolution, strain rate and temperature. The subject of the article is the modeling of the damage evolution in elastic–plastic materials of structural components, which are utilized at various temperatures. To achieve this goal, a hybrid model of steel cracking is applied. The hybrid model uses a finite element simulation combined with an experimental test realized in the macroscale. By using the hybrid model, the modeling of the damage evolution affords possibilities of determining macroscopic effects of the steel micro-defects. An essence of solving the predicting behavior of structural materials with micro-defects consists in time integration procedures for constitutive equations. In the article a semi-implicit time integration procedure is presented. The semi-implicit time integration procedure is suitable for the inelastic materials (compressible or incompressible) with the combined kinematic–isotropic hardening behavior. Its numerical solutions are stable, namely without the oscillatory behavior. By spatial averaging over a representative volume (RV), the homogenization technique (HT) is used for the defining of non-local variables in the constitutive equations. Evolutionary algorithms (EAs) based on local selections are applied to perform the homogenization technique. Within the framework of the large strain theory, the non-local continuum satisfies the objectivity requirements. Limitations on applicability of the J -integral approach to construct crack growth resistance curves are also presented.

State Vectors and Physical States

Causality and the relativity of simultaneity seem at odds with the apparently sudden, acausal state-vector changes (collapses) characteristic of quantum phenomena. The problem of how physical phenomena can be causally determined, have the probabilities predicted by quantum theory, and be consistent with special relativity appears to be solved by the assumption, essentially the same as one first used by Aharonov, Bergmann, and Lebowitz to address a different problem, that the initial and final state vectors of a phenomenon or observation, along with certain other state vectors, all represent the system's state at all times. Each member of such an aggregate of state vectors is postulated to represent a different aspect of a physical state rather than a state, so that most of the state vectors in effect constitute a set of nonlocal hidden variables. Various implications of this assumption are illustrated through several physical situations. Among the results is a somewhat surprising resolution of a paradox in a three-spin Greenberger-Horne-Zeilinger experiment (a three-particle EPR experiment), which involves a logical consequence that differs from familiar ideas of quantum physics that has no practical experimental consequence, and the prediction of new experimental phenomena related to the Zou-Wang-Mandel superposed-idler system. The potential value of the concepts described as heuristics for new predictions and for developing physical intuition by clarifying the interrelation and coherence of physical principles that would otherwise seem contradictory is briefly discussed.

Large quantum gravity effects and nonlocal variables

We reconsider here the model where large quantum gravity effects were first found, but now in its null surface formulation (NSF). We find that although the set of coherent states for Z, the basic variable of NSF, is as restricted as is the one for the metric, while some type of small deviations from these states may cause huge fluctuations on the metric, the corresponding fluctuations on Z remain small.

REDUCED PHASE SPACE QUANTIZATION OF MASSIVE VECTOR THEORY

We quantize massive vector theory in such a way that it has a well-defined massless limit. In contrast to the approach by Stückelberg where ghost fields are introduced to maintain manifest Lorentz covariance, we use reduced phase space quantization with nonlocal dynamical variables which in the massless limit smoothly turn into the photons, and check explicitly that the Poincaré algebra is fulfilled. In contrast to conventional covariant quantization our approach leads to a propagator which has no singularity in the massless limit and is well behaved for large momenta. For massive QED, where the vector field is coupled to a conserved fermion current, the quantum theory of the nonlocal vector fields is shown to be equivalent to that of the standard local vector fields. An inequivalent theory, however, is obtained when the reduced nonlocal massive vector field is coupled to a nonconserved classical current.

A gradient thermodynamic theory of self-organization

In this paper we present a gradient theory of internal variables in a thermodynamic context of the Gibbs free energy density o. A fundamental point of the theory is that o is a function of three different classes of internal variables: (a) tensorial dissipative and local; (b) vectorial dissipative and non-local and (c) vectorialinviscid and non-local. These classes obey different types of evolution equations. The ones pertaining to the non-local variables are partial differential equations of the diffusion-reaction type. We associate the inviscid non-local variables with energy release mechanisms and show that they lead to patterned deformation, otherwise known as self-organization. We conclude by giving a solution to the problem of a flat plate in nominal axial tension and derive various types of deformation patterns that result from small but unavoidable experimental deviations from this loading.

On nonlocal rate-independent plasticity

Abstract A nonlocal rate-independent large strain theory for elastic-plastic bodies consistent with thermodynamic theory is derived. The theory is based on a strain space formulation, where plastic strain is regarded as a primitive variable, characterised by an appropriate constitutive equation for its rate. Stress and free energy are assumed to be functions of a set of nonlocal variables, constructed from a collection of basic state functions, constituted by strain, plastic strain and a scalar measure of strain hardening. A yield function is introduced depending on the same set of independent, nonlocal variables. Yield criteria, flow rules, and loading conditions are formulated. The consistency condition is not, as in local theory, expressed by an algebraic equation, but by an integral equation defined throughout the region of plastic loading.