One-dimensional Subsystems Research Articles

Linear and Nonlinear Splitting Schemes Conserving Total Energy and Mass in the Shallow Water Model

Two linear and one nonlinear implicit unconditionally stable finite-difference schemes of the second-order approximation in all variables are given for a shallow-water model including the rotation and topography of the earth. The schemes are based on splitting the model equation into two one-dimensional subsystems. Each of the subsystems conserves the mass and total energy in both differential and discrete (in time and space) forms. One of the linear schemes contains a smoothing procedure not violating the conservation laws and suppressing spurious oscillations caused by the application of central-difference approximations of spatial derivatives. The unique solvability of the linear schemes and convergence of iterations used to find their solutions are proved.

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.

Stacking-induced symmetry-protected topological phase transitions

We study symmetry-protected topological (SPT) phase transitions induced by stacking two gapped one-dimensional subsystems in BDI symmetry class. The topological invariant of the entire system is a sum of three topological invariants: two from each subsystem and an emerging topological invariant from the stacking. We find that any symmetry-preserving stacking of topologically trivial subsystems can drive the entire system into a topologically nontrivial phase. We explain this intriguing SPT phase transitions by conditions set by orbital degrees of freedom and time-reversal symmetry. To exemplify the SPT transition, we provide a concrete model which consists of an atomic chain and a spinful nanowire with spin-orbit interaction and $s$-wave superconducting order. The stacking-induced SPT transition drives this heterostructure into a zero-field topological superconducting phase.

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type.

Calculation of steam drop mixture flows with boiling explosion mechanism using a multidimensional nodal method of characteristics

A hyperbolic model of a vapor-drop mixture is presented which takes into account the evaporation of drops by highlighting an explosive mechanism. The model is developed on the basis of the generalized-equilibrium model of a mixture previously proposed by the author. In this early model, the equations were hyperbolized by adding interfractional-interaction forces, and the liquid fraction was assumed to be incompressible. In the model proposed here, we assume that the phase transition at intense droplet evaporation occurs in an overheated state, when the fluid temperature exceeds the saturation temperature. A characteristic analysis of the model equations is carried out and their hyperbolicity is shown. An analytical formula is obtained for calculating the speed of sound in a vapor-drop mixture. It is noted that the speed of sound in the mixture in the presence of phase transformations turns out to be somewhat lower than that given by wood's formula. The multidimensional nodal method of characteristics for integrating hyperbolic systems, which is based on splitting the original system of equations into a number of one-dimensional subsystems, is described. Differential relations that hold true along the characteristic directions for each of the subsystems are derived. When calculating one-dimensional problems, an iterative algorithm of the inverse method of characteristics is applied. The computational method has been tested on a number of problems with self-similar solutions. Using the described method of integrating a multidimensional system of equations, the flow of a vapor-droplet flow near the disk was studied. It is shown that, in a number of cases, the explosive boiling of liquid droplets should be taken into account because it can significantly change the pattern of the flow around a disk.

A Quantum Mechanical Model for the Calculation of the Surface Free Energy and Surface Stress of Nanoparticles

As a model for a nanoparticle we use a nanocube. We assume that the potential of the cube is separable. Therefore, we can split Schrodinger’s equation into three one-dimensional ones. The potential in each of the one-dimensional sub-systems has a Meander-like run. For the calculation of the surface free energy of the nanoparticles we split a nanocube into smaller ones. We calculate the difference of the electron energies in the original nanocube and in the disintegrated cubes. If we divide the energy increment by the newly generated surface we have a very easy and correct procedure for the calculation of the surface free energy resulting from the cleavage of a nanocube into nanocubes. For an exact calculation of the surface free energy emerging in the disintegration of a macroscopic body into nanocubes we have to calculate the energy levels in an infinitely extended body with the same type of potential. Also a formula for the calculation of the surface stress for a nanocube has been derived. Approximating methods are given for the calculation of the surface free energy and the surface stress of nanocubes. In the case that the mean electron energy for a material with completely occupied energy bands is known it is possible to calculate, at least approximately, with this easy method the surface free energy and the surface stress. In this way, the surface free energy as a function of the chemical potential and temperature has been calculated. For a nanocube with 10 × 10 × 10 atoms with a completely occupied energy band the surface free energy amounts to 0.4 J/m2 and the surface stress is – 0.19 N/m.

Entanglement entropy for the valence bond solid phases of two-dimensional dimerized Heisenberg antiferromagnets

We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to $L' = 1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but do not diverge as the dimerization decreases and the system approaches the N\'eel--VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.

A subinterval decomposition analysis method for uncertain structures with large uncertainty parameters

A simple and efficient subinterval decomposition analysis method is proposed to evaluate the lower and upper bounds of structural responses with large uncertain parameters. The proposed method decomposes the original structural system with multi-dimensional interval parameters into multiple one-dimensional subsystems. Every subsystem has only one interval parameter and the other interval parameters are substituted by their midpoint values. By dividing the interval parameter of each subsystem into several subintervals with small uncertainty, the lower and upper bounds of the system are approximately calculated by only a few subinterval combinational analyses instead of all possible combinations of subintervals. Finally, the accuracy and efficiency of the proposed method compared with the first-order Taylor method, Chebyshev interval method and traditional subinterval method are verified by several numerical examples and applications.

Processes of excitation energy transport in EuPO4 and EuP3O9 nanocrystals

Processes of excitation energy transport were investigated for EuPO4 and EuP3O9 nanocrystals, in which Eu3+ ions form three- and one-dimensional subsystems, respectively. It was determined that for EuPO4 nanocrystals 5D0 luminescence quenching manifested itself even at low temperatures due to effective phonon-assisted energy migration. At the same time, for EuP3O9 nanocrystals no influence of energy migration on the luminescence quenching was observed in the whole temperature range from 10 to 293 K that can be ascribed to modification of phonon spectra for EuP3O9 nanocrystals as compared to their bulk counterparts.

Quantum Monte Carlo detection of SU(2) symmetry breaking in the participation entropies of line subsystems

Using quantum Monte Carlo simulations, we compute the participation (Shannon-Rényi) entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length LL embedded in two-dimensional (L\times LL×L) square lattices. We also study the line entropy at finite temperature, i.e. of the diagonal elements of the density matrix, for three-dimensional (L\times L\times LL×L×L) cubic lattices. The breaking of SU(2) symmetry is clearly captured by a universal logarithmic scaling term l_q\ln LlqlnL in the Rényi entropies, in good agreement with the recent field-theory results of Misguish, Pasquier and Oshikawa . We also study the dependence of the log prefactor l_qlq on the Rényi index qq for which a transition is detected at q_c\simeq 1qc≃1.

A very simple quantum mechanical model for the calculation of the surface free energy and surface stress of solids

For a model like that of Tamm, a linear periodical arrangement of a limited number of potential thresholds and wells is considered which is bordered by two higher potential steps at the ends representing the surfaces. Supposing the potential energy of the electrons in a solid is the sum of three functions each of which depends only on one of the co-ordinates x, y and z, the Schrodinger Equation is separable and a model for a three-dimensional solid can be assembled easily by the one-dimensional sub-systems. By adding the contributions of the electron states in the one-dimensional sub-systems, the surface energy and the surface stress have been calculated for a three-dimensional body.

Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries

We explore the R\'enyi entanglement entropies of a one-dimensional (line) subsystem of length $L$ embedded in two-dimensional $L\times L$ square lattice for quantum spin models whose ground-state breaks a continuous symmetry in the thermodynamic limit. Using quantum Monte Carlo simulations, we first study the $J_1 - J_2$ Heisenberg model with antiferromagnetic nearest-neighbor $J_1>0$ and ferromagnetic second-neighbor couplings $J_2\le 0$. The signature of SU(2) symmetry breaking on finite size systems, ranging from $L=4$ up to $L=40$ clearly appears as a universal additive logarithmic correction to the R\'enyi entanglement entropies: $l_q \ln L$ with $l_q\simeq 1$, independent of the R\'enyi index and values of $J_2$. We confirm this result using a high precision spin-wave analysis (with restored spin rotational symmetry) on finite lattices up to $10^5\times 10^5$ sites, allowing to explore further non-universal finite size corrections and study in addition the case of U(1) symmetry breaking. Our results fully agree with the prediction $l_q=n_G/2$ where $n_G$ is the number of Goldstone modes, by Metlitski and Grover [arXiv:1112.5166].

Global adaptive stabilisation of high‐order uncertain non‐linear systems with double control input channels

This study considers the global adaptive control design for a class of high-order uncertain non-linear systems. Different from the existing results, the systems under investigation have serious unknowns which cannot be bounded by any known constants, and particularly have double control input channels, that is, the scalar control input directly affects two one-dimensional subsystems. Inspired by the author's recent results, by skillfully combing the method of adding a power integrator and some adaptive techniques, an adaptive controller is successfully constructed which can guarantee the asymptotical convergence of the system states for any given initial conditions. A simulation example is given to demonstrate the validity of the obtained theoretical results.

具有双控制输入通道高阶不确定非线性系统控制设计

A problem of globally asymptotically stable control design is investigated for a class of high-order uncertain nonlinear systems. These systems, unlike the existing results, have double control input channels; that is, the control input directly affects two one-dimensional subsystems. This makes the existing methods of control design inapplicable or the control design problem difficult to solve, and therefore, it is of interest to explore a new control design method to solve the control design of high-order nonlinear systems with multiple control input channels. First, an effective feedback transformation is introduced. Hence, this control design problem under appropriate assumptions can be transformed into one with a single control input channel. Then, using existing results, particularly, the method of adding a power integrator, a methodology is proposed to design the globally asymptotically stable controller for the nonlinear systems under investigation. Although the systems studied in the paper are a special case of high-order nonlinear systems with multiple control input channels, the results obtained in the paper, while unavoidably having certain limitations, will provide a reference and guidance in tackling the general case. Finally, a simulation is provided to verify the effectiveness of the theoretical results obtained.

TWO-DIMENSIONAL COUPLED PARAMETRICALLY FORCED MAP

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is proposed. We investigate bifurcations and basins in the parametrically forced system when logistic map is used for the one-dimensional subsystem. On a parameter plane, crossroad areas centered at fold cusp points for several orders are detected. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling intensity is analyzed. Basin bifurcations and preimages with respect to critical curves are described. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity regarding parameters.

Magnetic ordering of weakly coupled frustrated quantum spin chains

The ordering temperature of a quasi-one-dimensional system, consisting of weakly interacting quantum spin-1/2 chains with antiferromagnetic spin-frustrating couplings (or zigzag ladder), is calculated. The results show that a quantum critical point between two phases of the one-dimensional subsystem plays a crucial role. If the one-dimensional subsystem is in the antiferromagneticlike phase in the ground state, similar to the phase of a spin chain without frustration, weak couplings yield magnetic ordering of the N\'eel type. For intrachain spin-frustrating interactions larger than the critical one (at which the quantum phase transition takes place), the quasi-one-dimensional spin system manifests a spiral magnetic incommensurate ordering. The obtained results of our quantum theory are compared with the quasiclassical approximations. The calculated features of magnetic ordering are expected to be generic for weakly-coupled quantum spin chains with gapless excitations and spin-frustrating nearest- and next-nearest-neighbor interactions.

A Model System for Dimensional Competition in Nanostructures: A Quantum Wire on a Surface

The retarded Green’s function (E−H + iε)−1is given for a dimensionally hybrid Hamiltonian which interpolates between one and two dimensions. This is used as a model for dimensional competition in propagation effects in the presence of one-dimensional subsystems on a surface. The presence of a quantum wire generates additional exponential terms in the Green’s function. The result shows how the location of the one-dimensional subsystem affects propagation of particles.

Emergence of diffusion in finite quantum systems

We study the emergence of diffusion for a quantum particle moving in a finite and translationally invariant one-dimensional subsystem described by a tight-binding Hamiltonian with a single energy band and interacting with its environment by an interaction energy proportional to some coupling parameter. We show that there exists a crossover between a nondiffusive relaxation regime for small sizes or low values of the coupling parameter and a diffusive regime above a critical size or for higher values of the coupling parameter. In the nondiffusive regime, the relaxation is characterized by oscillations decaying at rates independent of the size and proportional to the square of the coupling parameter and the temperature of the environment. In the diffusive regime, the damped oscillations have disappeared and the relaxation rate is inversely proportional to the square of the size. The diffusion coefficient is proportional to the square of the energy bandwidth of the subsystem and inversely proportional to the temperature of the environment and the square of the coupling parameter. The critical size where the crossover happens is obtained analytically.

Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas

This paper addresses digital communication in a Rayleigh fading environment when the channel characteristic is unknown at the transmitter but is known (tracked) at the receiver. Inventing a codec architecture that can realize a significant portion of the great capacity promised by information theory is essential to a standout long-term position in highly competitive arenas like fixed and indoor wireless. Use (n <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> , n <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</inf> ) to express the number of antenna elements at the transmitter and receiver. An (n, n) analysis shows that despite the n received waves interfering randomly, capacity grows linearly with n and is enormous. With n = 8 at 1% outage and 21-dB average SNR at each receiving element, 42 b/s/Hz is achieved. The capacity is more than 40 times that of a (1, 1) system at the same total radiated transmitter power and bandwidth. Moreover, in some applications, n could be much larger than 8. In striving for significant fractions of such huge capacities, the question arises: Can one construct an (n, n) system whose capacity scales linearly with n, using as building blocks n separately coded one-dimensional (1-D) subsystems of equal capacity? With the aim of leveraging the already highly developed 1-D codec technology, this paper reports just such an invention. In this new architecture, signals are layered in space and time as suggested by a tight capacity bound.

Three-dimensional ordering in weakly coupled antiferromagnetic ladders and chains

A theoretical description is presented for low-temperature magnetic-field induced three-dimensional (3D) ordering transitions in strongly anisotropic quantum antiferromagnets, consisting of weakly coupled antiferromagnetic spin-1/2 chains and ladders. First, effective continuum field theories are derived for the one-dimensional subsystems. Then the Luttinger parameters, which determine the low-temperature susceptibilities of the chains and ladders, are calculated from the Bethe ansatz solution for these effective models. The 3D ordering transition line is obtained using a random phase approximation for the weak inter-chain (inter-ladder) coupling. Finally, considering a Ginzburg criterion, the fluctuation corrections to this approach are shown to be small. The nature of the 3D ordered phase resembles a Bose condensate of integer-spin magnons. It is proposed that for systems with higher spin degrees of freedom, e.g. N-leg spin-1/2 ladders, multi-component condensates can occur at high magnetic fields.