Nonlocal Phase Research Articles

Intrinsic translational symmetry breaking in a doped Mott insulator

Whether a doped hole propagates as a Bloch wave or not is an important issue of doped Mott physics. Here we examine this problem based on the quasiparticle spectral weight $Z$ distribution, calculated by density matrix renormalization group (DMRG). By tuning the anisotropy of a two-leg $t$-$J$ ladder without closing the background spin gap, the $Z$ distribution unambiguously reveals a transition of the single hole state from a Bloch wave to a novel one with spontaneous translational symmetry breaking. We further establish a direct connection of such a transition with a nonlocal phase string entanglement between the hole and quantum spins, which explains numerical observations.

Rigidity of minimizers in nonlocal phase transitions

We obtain the classification of certain global bounded solutions for semilinear nonlocal equations of the type $$\triangle^s u=W'(u)$$ in $\mathbb{R}^n$,with $s \in (1/2 ,1),$ where $W$ is a double well potential.

Planelike minimizers of nonlocal Ginzburg–Landau energies and fractional perimeters in periodic media

We consider here a nonlocal phase transition energy in a periodic medium and we construct solutions whose interfaces lie at a bounded distance from any given hyperplane.These solutions are either periodic or quasiperiodic, depending on the rational dependency of the normal direction to the reference hyperplane.Remarkably, the oscillations of the interfaces with respect to the reference hyperplane are bounded by a universal constant times the periodicity scale of the medium.This geometric property allows us to establish, in the limit, the existence of planelike nonlocal minimal surfaces in a periodic structure.The proofs rely on new optimal density and energy estimates. In particular, roughly speaking, the energy of phase transition minimizers is controlled, both from above and below, by the energy of one-dimensional transition layers.

Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation

Comparing with the well-known classic Cahn–Hilliard equation, the nonlocal Cahn–Hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials. On the other hand, it evidently brings more computational costs in numerical simulations, thus efficient and accurate time integration schemes are highly desired. In this paper, we propose two energy-stable linear semi-implicit methods with first and second order temporal accuracies respectively for solving the nonlocal Cahn–Hilliard equation. The temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly, while the spatial discretization is carried out by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are rigorously established for both methods in the fully discrete sense. Numerical experiments are conducted for a typical case involving Gaussian kernels. We test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one. In addition, long-time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay.

The Impact of Airborne Radio Occultation Observations on the Simulation of Hurricane Karl (2010)

Abstract This study evaluates, for the first time, the impact of airborne global positioning system radio occultation (ARO) observations on a hurricane forecast. A case study was conducted of Hurricane Karl during the Pre-Depression Investigation of Cloud-Systems in the Tropics (PREDICT) field campaign in 2010. The assimilation of ARO data was developed for the three-dimensional variational (3DVAR) analysis system of the Weather Research and Forecasting (WRF) Model version 3.2. The impact of ARO data on Karl forecasts was evaluated through data assimilation (DA) experiments of local refractivity and nonlocal excess phase (EPH), in which the latter accounts for the integrated horizontal sampling along the signal ray path. The tangent point positions (closest point of an RO ray path to Earth’s surface) drift horizontally, and the drifting distance of ARO data is about 2 to 3 times that of spaceborne RO, which was taken into account in these simulations. Results indicate that in the absence of other satellite observations, the assimilation of ARO EPH resulted in a larger impact on the analysis than local refractivity did. In particular, the assimilation of ARO observations at the actual tangent point locations resulted in more accurate forecasts of the rapid intensification of the storm. Among all experiments, the best forecast was obtained by assimilating ARO data with the most accurate geometric representation, that is, the use of nonlocal EPH operators with tangent point drift, which reduced the error in the storm’s predicted minimum sea level pressure (SLP) by 43% beyond that of the control experiment.

A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime

We consider bounded solutions of the nonlocal Allen–Cahn equation $$\begin{aligned} (-\Delta )^s u=u-u^3\qquad { \text{ in } }\mathbb {R}^3, \end{aligned}$$ under the monotonicity condition $$\partial _{x_3}u>0$$ and in the genuinely nonlocal regime in which $$s\in \left( 0,\frac{1}{2}\right) $$ . Under the limit assumptions $$\begin{aligned} \lim _{x_n\rightarrow -\infty } u(x',x_n)=-1\quad { \text{ and } }\quad \lim _{x_n\rightarrow +\infty } u(x',x_n)=1, \end{aligned}$$ it has been recently shown in Dipierro et al. (Improvement of flatness for nonlocal phase transitions, 2016) that u is necessarily 1D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by De Giorgi (Proceedings of the international meeting on recent methods in nonlinear analysis (Rome, 1978), Pitagora, Bologna, pp 131–188, 1979).

Evolution of entanglement under an Ising-like Hamiltonian with particle losses

We present analytical compact solution for the density matrix and correlation functions of two collective-macroscopic spins evolving via Ising-like Hamiltonian in the presence of particle losses. The losses introduce non-local phase noise which destroys highly entangled states arising in the evolution. On the other hand, the states appearing at relatively short timescales, possessing EPR-like entanglement will survive. Applying our solutions to the recently proposed scheme to entangle two Bose-Einstein condensates, we estimate the optimal number of atoms for EPR correlations.

Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams

Strain-driven and stress-driven integral elasticity models are formulated for the analysis of the structural behaviour of fuctionally graded nano-beams. An innovative stress-driven two-phases constitutive mixture defined by a convex combination of local and nonlocal phases is presented. The analysis reveals that the Eringen strain-driven fully nonlocal model cannot be used in Structural Mechanics since it is ill-posed and the local-nonlocal mixtures based on the Eringen integral model partially resolve the ill-posedeness of the model. In fact, a singular behaviour of continuous nano-structures appears if the local fraction tends to vanish so that the ill-posedness of the Eringen integral model is not eliminated. On the contrary, local-nonlocal mixtures based on the stress-driven theory are mathematically and mechanically appropriate for nanosystems. Exact solutions of inflected functionally graded nanobeams of technical interest are established by adopting the new local-nonlocal mixture stress-driven integral relation. Effectiveness of the new nonlocal approach is tested by comparing the contributed results with the ones corresponding to the mixture Eringen theory.

Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.

Phase-based, high spatial resolution and distributed, static and dynamic strain sensing using Brillouin dynamic gratings in optical fibers

A novel technique combining Brillouin phase-shift measurements with Brillouin dynamic gratings (BDGs) reflectometry in polarization-maintaining fibers is presented here for the first time. While a direct measurement of the optical phase in standard BDG setups is impractical due to non-local phase contributions, their detrimental effect is reduced by ~4 orders of magnitude through the coherent addition of Stokes and anti-Stokes reflections from two counter-propagating BDGs in the fiber. The technique advantageously combines the high-spatial-resolution of BDGs reflectometry with the increased tolerance to optical power fluctuations of phasorial measurements, to enhance the performance of fiber-optic strain sensors. We demonstrate a distributed measurement (20cm spatial-resolution) of both static and dynamic (5kHz of vibrations at a sampling rate of 1MHz) strain fields acting on the fiber, in good agreement with theory and (for the static case) with the results of commercial reflectometers.

A classical optical approach to the ‘non-local Pancharatnam-like phases’ in Hanbury-Brown–Twiss correlations

We examine a recent proposal to show the presence of nonlocal Pancharatnam type geometric phases in a quantum mechanical treatment of intensity interferometry measurements upon inclusion of polarizing elements in the setup. It is shown that a completely classical statistical treatment of such effects is adequate for practical purposes. Further we show that the phase angles that appear in the correlations, while at first sight appearing to resemble Pancharatnam phases in their mathematical structure, cannot actually be interpreted in that manner. We also describe a simpler Mach–Zehnder type setup where similar effects can be observed without use of the paraxial approximation.

Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors nonlocal model here studied consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called 'deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.

Elasto-viscoplastic phase field modelling of anisotropic cleavage fracture

A finite-strain anisotropic phase field method is developed to model the localisation of damage on a defined family of crystallographic planes, characteristic of cleavage fracture in metals. The approach is based on the introduction of an undamaged configuration, and the inelastic deformation gradient mapping this configuration to a damaged configuration is microstructurally represented by the opening of a set of cleavage planes in the three fracture modes. Crack opening is modelled as a dissipative process, and its evolution is thermodynamically derived. To couple this approach with a physically-based phase field method for brittle fracture, a scalar measure of the overall local damage is introduced, whose evolution is determined by the crack opening rates, and weakly coupled with the non-local phase field energy representing the crack opening resistance in the classical sense of Griffith. A finite-element implementation of the proposed model is employed to simulate the crack propagation path in a laminate and a polycrystalline microstructure. As shown in this work, it is able to predict the localisation of damage on the set of pre-defined cleavage planes, as well as the kinking and branching of the crack resulting from the crystallographic misorientation across the laminate boundary and the grain boundaries respectively.

Nonlocal phase transitions in homogeneous and periodic media

We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as $\Gamma$-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media. Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.

Distributed optimal control of a nonstandard nonlocal phase field system

We investigate a distributed optimal control problem for a nonlocal phase field modelof viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-speciesphase segregation on an atomic lattice under the presence of di usion that has been studied in a seriesof papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinearparabolic equation coupled to an ordinary di erential equation. The latter equation contains bothnonlocal and singular terms that render the analysis di cult. Standard arguments of optimal controltheory do not apply directly, although the control constraints and the cost functional are of standardtype. We show that the problem admits a solution, and we derive the first-order necessary conditionsof optimality.

Morphological evolution and migration of void in bi-piezoelectric interface based on nonlocal phase field method

This paper reports the result of investigation into the morphological evolution and migration of void in bi-piezoelectric material interface by utilizing nonlocal phase field model and finite element method (FEM), where the small scale effect containing the long-range forces among atoms is considered. The nonlocal elastic strain energy and the nonlocal electric energy around the void are firstly calculated by the finite element method. Then based on the finite difference method (FDM), the thermodynamic equilibrium equation containing the surface energy and anisotropic diffusivity is solved to simulate the morphological evolution and migration of elliptical void in bi-piezoelectric films interface. Results show that the way of load condition plays a significant role in the evolution process, and the boundary of void's long axis gradually collapses toward the center of ellipse. In addition, the evolutionary speed of left boundary gradually decreases with scale effect coefficient growth. This work can provide references for the safety evaluation of piezoelectric materials in micro electro mechanical system.

Evidence of enhanced self-organized criticality (SOC) dynamics during the radially non-local transient transport in the HL-2A tokamak

Self-organized criticality (SOC) dynamics have been investigated in non-local transport regimes induced by the supersonic molecular beam injection in the HL-2A tokamak. Experimental evidence shows that the SOC or avalanche behaviors, such as the Hurst parameter, self-similarity and large-scale radial correlations in turbulence, are remarkably enhanced during the prompt non-local phase, together with an increase of inward propagation of turbulent events. These results highlight the importance of SOC paradigm during the transient non-local thermal transport in magnetically confined plasmas.

Canonical proper time quantum gravitation

At the root of the tensions involved in modeling the quantum dynamics of gravitating systems are the subtleties of quantum locality. Quantum mechanics describes physical phenomena using a theory of non-local phase relationships (non-local in the sense that quantum states maintain a space-like coherence that is acausal). However, the principle of equivalence in general relativity asserts that freely falling frames are locally inertial frames of reference. Thus, gravitating systems are often described using constituents that are freely falling, undergoing geodesic motion defining well localized trajectories. The canonical proper time formulation of relativistic dynamics is particularly useful for describing such inertial constituents using the coordinates of non-inertial observers. The physics of the simplest of gravitating inertial quantum systems, consistent with presented experimental evidence, will be examined. Subsequently, descriptions of both weakly and strongly gravitating quantum systems will be developed using canonical proper gravitation.

Nonlocal SAR Interferometric Phase Filtering Through Higher Order Singular Value Decomposition

Interferometric phase filtering is an indispensable step to obtain accurate measurement of digital elevation model and surface displacement. In the case of low-correlation or complicated topography, traditional phase filtering methods fail in balancing noise elimination and phase preservation, which leads to inaccurate interferometric phase. A new nonlocal interferometric phase filtering method taking advantage of higher order singular value decomposition (HOSVD) is proposed in this letter. For each pixel of the interferometric phase, a 3-D data array is established, and shrinkage is applied after HOSVD. A Wiener filter is used to improve the denoising performance in the end. Simulated and real data are employed to validate that the proposed method outperforms other traditional methods and some of the state-of-the-art nonlocal methods.

Nonlocal mode-coupling interactions and phase transition near tricriticality

Employing Wilson's renormalization group scheme, we investigate the critical behaviour of a modified Ginzburg-Landau model with a nonlocal mode-coupling interaction in the quartic term. Carrying out the calculations at one-loop order, we obtain the critical exponents in the leading order of , where ρ is an exponent occurring in the nonlocal interaction term and d is the space dimension. Interestingly, the correlation exponent η is found to be non-zero at one-loop order and the ϵ expansion corresponds to an expansion about the tricritical mean-field theory in three dimensions, unlike the conventional theory. The ensuing critical exponents are in good agreement with experimental values for samples close to tricriticality. Our analysis indicates that tricriticality is a feature only in three dimensions.