Arbitrary Spatial Dimension Research Articles

Entropy bound with generalized uncertainty principle in general dimensions

In this letter, the entropy bound for local quantum field theories (LQFT) is studied in a class of models of the generalized uncertainty principle (GUP) which predicts a minimal length as a reflection of the quantum gravity effects. Both bosonic and fermionic fields confined in an arbitrary spatial dimension d ⩾ 4 ball are investigated. It is found that the GUP leads to the same scaling A(d−3)/(d−2)d−2 correction to the entropy bound for bosons and fermions, although the coefficients of this correction are different for each case. Based on our calculation, we conclude that the GUP effects can become manifest at the short distance scale. Some further implications and speculations of our results are also discussed.

Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions

We propose a method of simulating efficiently many-body interacting fermion lattice models in trapped ions, including highly nonlinear interactions in arbitrary spatial dimensions and for arbitrarily distant couplings. We map products of fermionic operators onto nonlocal spin operators and decompose the resulting dynamics in efficient steps with Trotter methods, yielding an overall protocol that employs only polynomial resources. The proposed scheme can be relevant in a variety of fields such as condensed-matter or high-energy physics, where quantum simulations may solve problems intractable for classical computers.

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle‐node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution‐bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schrödinger equations. Another shows a power loop phenomenon which contains several saddle‐node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.

Growth Inside a Corner: The Limiting Interface Shape

We investigate the growth of a crystal that is built by depositing cubes inside a corner. The interface of this crystal approaches a deterministic growing limiting shape in the long-time limit. Building on known results for the corresponding two-dimensional system and accounting for basic three-dimensional symmetries, we conjecture a governing equation for the evolution of the interface profile. We solve this equation analytically and find excellent agreement with simulations of the growth process. We also present a generalization to arbitrary spatial dimension.

Interface Conditions for a Phase Field Model with Anisotropic and Non-Local Interactions

An alternative formulation of the phase field method is utilized from an integral equation perspective. The technique allows one to derive macroscopic conditions at the interface from the microscopic potentials. Differential geometry and asymptotic analysis yield interface conditions, in arbitrary spatial dimension, for interactions that may include anisotropy as well as non-local potentials. The interface conditions can be expressed in various formulations, for example, in terms of the principal curvature directions of the interface, or the second order directional derivatives of the (signed) distance function and the Hessian of the surface tension.

GLOBAL EXISTENCE AND DECAY OF SOLUTIONS FOR A QUASI-LINEAR DISSIPATIVE PLATE EQUATION

In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2energy method combined with the optimal L2decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is the fundamental solution of the corresponding fourth-order linear parabolic equation.

Kinetics of first passage in a cone

We study statistics of first passage inside a cone in an arbitrary spatial dimension. The probability that a diffusing particle avoids the cone boundary decays algebraically with time. The decay exponent depends on two variables: the opening angle of the cone and the spatial dimension. In four dimensions, we find an explicit expression for the exponent, and in general, we obtain it as a root of a transcendental equation involving associated Legendre functions. At large dimensions, the decay exponent depends on a single scaling variable, while roots of the parabolic cylinder function specify the scaling function. Consequently, the exponent is of order one only if the cone surface is very close to a plane. We also perform asymptotic analysis for extremely thin and extremely wide cones.

Macroscopic description of arbitrary Knudsen number flow using Boltzmann–BGK kinetic theory. Part 2

We extend our previous analysis of closed-form equations for finite Knudsen number flow and scalar transport that result from the Boltzmann–Bhatnagar–Gross–Krook (BGK) kinetic theory with constant relaxation time. Without approximation, we obtain closed-form equations for arbitrary spatial dimension and flow directionality which are local differential equations in space and integral equations in time. These equations are further simplified for incompressible flow and scalars. The particular case of no-flow scalar transport admits analytical solutions that exhibit ballistic behaviour at short times while behaving diffusively at long times. It is noteworthy that, even with constant relaxation time BGK microphysics, quite complex macroscopic descriptions result that would be difficult to obtain using classical constitutive models or continuum averaging.

Topological insulators and superconductors: tenfold way and dimensional hierarchy

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a or a topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via ‘dimensional reduction’ by compactifying one or more spatial dimensions (in ‘Kaluza–Klein’-like fashion). For -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -topological insulators in the same class, from which they inherit their topological properties. The eightfold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle–hole symmetries) is a reflection of the eightfold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern–Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization (d=1 spatial dimensions) or to the magnetoelectric polarizability (d=3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).

Fermionic projected entangled pair states

We introduce a family of states, the fermionic projected entangled pair states (fPEPS), which describe fermionic systems on lattices in arbitrary spatial dimensions. It constitutes the natural extension of another family of states, the PEPS, which efficiently approximate ground and thermal states of spin systems with short-range interactions. We give an explicit mapping between those families, which allows us to extend previous simulation methods to fermionic systems. We also show that fPEPS naturally arise as exact ground states of certain fermionic Hamiltonians. We give an example of such a Hamiltonian, exhibiting criticality while obeying an area law.

Splitting Integrators for Nonlinear Schrödinger Equations Over Long Times

Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie–Trotter splitting in time for cubic Schrodinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schrodinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results are valid in arbitrary spatial dimension.

Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times

Cubic Schrodinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both the solution of the nonlinear Schrodinger equation as well as the solution of the semi-discretized equation the actions of the linear Schrodinger equation are approximately conserved over long times. This also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary spatial dimension.

The PDE framework Peano applied to fluid dynamics: an efficient implementation of a parallel multiscale fluid dynamics solver on octree-like adaptive Cartesian grids

This paper presents the general purpose framework Peano for the solution of partial differential equations (PDE) on adaptive Cartesian grids. The strict structuredness and inherent multilevel property of these grids allows for very low memory requirements, efficient (in terms of hardware performance) implementations of parallel multigrid solvers on dynamically adaptive grids, and arbitrary spatial dimensions. This combination of advantages distinguishes Peano from other PDE frameworks. We describe shortly the underlying octree-like grid type and its most important properties. The main part of the paper shows the framework concept of Peano and the implementation of a Navier–Stokes solver as one of the main currently implemented application examples. Various results ranging from hardware and numerical performance to concrete application scenarios close the contribution.

On an expansion method for black hole quasinormal modes and Regge poles

We present a new method for determining the frequencies and wavefunctions of black hole quasinormal modes (QNMs) and Regge poles. The key idea is a novel ansatz for the wavefunction, which relates the high-l wavefunctions to null geodesics which start at infinity and end in perpetual orbit on the photon sphere. Our ansatz leads naturally to the expansion of QNMs in inverse powers of L = l + 1/2 (in 4D), and to the expansion of Regge poles in inverse powers of ω. The expansions can be taken to high orders. We begin by applying the method to the Schwarzschild spacetime, and validate our results against existing numerical and Wentzel–Kramers–Brillouin methods. Next, we generalize the method to treat static spherically symmetric spacetimes of arbitrary spatial dimension. We confirm that, at lowest order, the real and imaginary components of the QNM frequency are related to the orbital frequency and the Lyapunov exponent for geodesics at the unstable orbit. We apply the method to five spacetimes of current interest, and conclude with a discussion of the advantages and limitations of the new approach, and its practical applications.

Two efficient localization algorithms for multilateration

Two localization algorithms for multilateration systems are derived and analyzed. Instead of the classical time difference of arrival (TDOA), a direct use of the time of arrival (TOA) is made. The algorithms work for arbitrary spatial dimensions and overdetermined systems. These derivations are tested in a real-case implementation with simulated data (in particular, the multilateration (MLAT) system installed on the Malpensa Airport in Milan was considered for the MLAT simulation and its possible extension to wide area multilateration (WAM) system was considered for WAM trials). The results are also compared with the present-day algorithms performance, mostly based on TDOA.

AN ALTERNATING EVOLUTION APPROXIMATION TO SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

In this paper, we present an alternating evolution (AE) approximation [Formula: see text] to systems of hyperbolic conservation laws [Formula: see text] in arbitrary spatial dimension. We prove the convergence of the approximate solutions towards an entropy solution of scalar multi-D conservation laws, and the L1 contraction property for the approximate solution is established as well. It is also shown that such an approximation is extremely accurate in the sense that if initial data is prepared such that u0 = v0 = U0, then no method error is induced as time evolves, and the exact entropy solution is precisely captured. Furthermore, in the approximation system time evolution of one variable is associated with spatial redistribution in another variable. These features render such an approximation ideal to be used for construction of high resolution numerical schemes to solve hyperbolic conservation laws. The usual obstacles caused by jumps crossing computational cell interfaces are not felt when both u and v are sampled alternatively, and reconstructed independently. Herewith we discuss the designing principle for constructing AE schemes, with illustration of two preliminary schemes for systems of conservation laws in one dimension. Both l∞ monotonicity and the TVD (Total Variational Diminishing) property are established for these schemes when applied to the scalar laws.

Accelerated Imaginary‐time Evolution Methods for the Computation of Solitary Waves

Two accelerated imaginary‐time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as conditions for optimal accelerations are derived. In addition, it is shown that for nodeless solitary waves, this method converges if and only if the solitary wave is linearly stable. The second method is similar to the first method except that it uses a novel amplitude normalization. The performance of these methods is illustrated on various examples. It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.

Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms

We consider the unconditional stability of second-order ADI schemes in the numerical solution of finite difference discretizations of multi-dimensional diffusion problems containing mixed spatial-derivative terms. We investigate an ADI scheme proposed by Craig and Sneyd, an ADI scheme that is a modified version thereof, and an ADI scheme introduced by Hundsdorfer and Verwer. Both sufficient and necessary conditions are derived on the parameters of each of these schemes for unconditional stability in the presence of mixed derivative terms. Our main result is that, under a mild condition on its parameter θ, the second-order Hundsdorfer and Verwer scheme is unconditionally stable when applied to semi-discretized diffusion problems with mixed derivative terms in arbitrary spatial dimensions k ⩾ 2 .

A variational method based on weighted graph states

In a recent article Anders et al 2006 Phys. Rev. Lett. 97 107206, we have presented a class of states which is suitable as a variational set to find ground states in spin systems of arbitrary spatial dimension and with long-range entanglement. Here, we continue the exposition of our technique, extend from spin 1/2 to higher spins and use the boson Hubbard model as a non-trivial example to demonstrate our scheme.

Optimal blended spectral-element operators for acoustic wave modeling

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D factors. We apply standard lumping to the factor associated with the 1D mass matrix. A simplified numerical dispersion analysis of arbitrary order and spatial dimension provides a practical criterion for weighting consistent and lumped matrices. The approach is general and is valid for solving both the time-dependent and the stationary (Helmholtz) wave equations.