Class Of Two-dimensional Systems Research Articles

A novel two-dimensional exponential potential bi-stable stochastic resonance system and its application in bearing fault diagnosis

It is difficult for classical stochastic resonance systems to extract weak signals under strong noise environment, therefore a novel two-dimensional exponential potential bi-stable stochastic resonance system (NTBSR) is proposed. First, the equivalent potential function, the mean first-pass time (MFPT) and the output signal-to-noise ratio (SNR) of NTBSR are derived under the adiabatic approximation theory. At the same time, the influence of different system parameters on them is explored. Then, NTBSR, the one-dimensional bi-stable stochastic resonance system (OBSR) and the two-dimensional classical bi-stable stochastic resonance system (TCBSR) are respectively simulated numerically, based on the fourth-order Runge–Kutta algorithm. It is found that the output SNR of NTBSR is the best. Finally, the NTBSR is applied to the fault signal diagnosis of different types of bearings, and the parameters are optimized through the adaptive genetic algorithm (AGA). The test results are compared with wavelet transform method, and TCBSR. The detection results on two sets of bearing fault data show that the NTBSR system has better effects on the enhancement and detection of bearing fault signals, and it is verified that the stochastic resonance method is superior to the traditional wavelet transform method in terms of signal detection and noise utilization. This provides good theoretical support and application value for practical engineering application.

Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.

Global boundedness of solutions resulting from both the self-diffusion and the logistic-type source

This paper is concerned with the zero-flux logistic chemotaxis system with nonlinear signal production: \(u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u)\), \(v_t=\Delta v-v+g(u)\) in a bounded and smooth domain \(\Omega \subset {\mathbb {R}}^n\) (\(n\ge 1\)), where \(\chi >0\) is a constant, and \(f,g\in C^1({\mathbb {R}})\) generalize the prototypes \(f(s)=s-\mu s^{\alpha }\) and \(g(s)=s(s+1)^{\beta -1}\) with \(\alpha >1\) and \(\beta ,\mu >0\). The existing studies, for the case that the self-diffusion or the logistic-type source prevails over the cross-diffusion singly, have asserted the global boundedness of solutions under the assumption that \(\beta 0\) suitably large. In the present paper, we prove that if \(\alpha -1=\beta =\frac{2}{n}\), then due to the self-diffusion and the logistic kinetics working together, the solutions are globally bounded regardless of the size of \(\mu >0\). Note that \(\alpha -1=\beta =\frac{2}{n}\) reduces to \(\alpha =2\) and \(\beta =1\) if \(n=2\). Hence, our result covers that of the two-dimensional classical chemotaxis system with logistic source: \(u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+u-\mu u^2\), \(v_t=\Delta v-v+u\) (Osaki et al. in Nonlinear Anal 51:119–144, 2002).

Ground state of dipolar hard spheres confined in channels.

We investigate the ground state of a classical two-dimensional system of hard-sphere dipoles confined between two hard walls. Using lattice sum minimization techniques we reveal that at fixed wall separations, a first-order transition from a vacuum to a straight one-dimensional chain of dipoles occurs upon increasing the density. Further increase in the density yields the stability of an undulated chain as well as nontrivial buckling structures. We explore the close-packed configurations of dipoles in detail, and we find that, in general, the densest packings of dipoles possess complex magnetizations along the principal axis of the slit. Our predictions serve as a guideline for experiments with granular dipolar and magnetic colloidal suspensions confined in slitlike channel geometry.

Thermodynamics and dynamics of two-dimensional systems with dipolelike repulsive interactions.

Thermodynamics and dynamics of a classical two-dimensional system with dipolelike isotropic repulsive interactions are studied systematically using extensive molecular dynamics (MD) simulations supplemented by appropriate theoretical approximations. This interaction potential, which decays as an inverse cube of the interparticle distance, belongs to the class of very soft long-ranged interactions. As a result, the investigated system exhibits certain universal properties that are also shared by other related soft-interacting particle systems (like, for instance, the one-component plasma and weakly screened Coulomb systems). These universalities are explored in this article to construct a simple and reliable description of the system thermodynamics. In particular, Helmholtz free energies of the fluid and solid phases are derived, from which the location of the fluid-solid coexistence is determined. The quasicrystalline approximation is applied to the description of collective modes in dipole fluids. Its simplification, previously validated on strongly coupled plasma fluids, is used to derive explicit analytic dispersion relations for the longitudinal and transverse wave modes, which compare satisfactory with those obtained from direct MD simulations in the long-wavelength regime. Sound velocities of the dipole fluids and solids are derived and analyzed.

Equilibrium and uniform charge distribution of a classical two-dimensional system of point charges with hard-wall confinement

We study the minimum energy equilibrium configurations of a classical two-dimensional system of point charges confined by a triangular, square and disk region with a hard-wall boundary. It is assumed that the point charges interact via a repulsive Coulomb interaction potential. Monte Carlo simulations with the annealing algorithm suggest that the equilibrium configurations of a given system are strongly influenced by the external (isotropic/anisotropic) geometry of the hard-wall boundary. The numerically obtained energies extrapolated in the bulk limit converge to the expected continuum equilibrium values (when known). It is found that the equilibrium charge distribution is non-uniform in the continuum limit for all the hard-wall confining regions considered in this work. Since the continuum equilibrium charge distribution is not known for the case of an equilateral triangle or a square domain we choose to compare the numerically obtained bulk energy results to corresponding values for a uniformly charged system. We calculated exactly the electrostatic energy of various uniformly charged planar objects and used the results to assess the discrepancy between such results and the numerically obtained equilibrium bulk energy values for the cases of equilateral triangle and square hard-wall boundaries. These estimates help us understand how an anisotropic boundary with the shape of an equilateral triangle or square influences the energy of an equilibrium charge distribution. The results indicate that the energy discrepancy between equilibrium and uniform charge distributions in the continuum limit is not very large. It is found that the order of magnitude of the relative deviation of the energy for all three different planar domains considered here is approximately the same.

Formation probabilities in quantum critical chains and Casimir effect

We find a connection between logarithmic formation probabilities of two disjoint intervals of quantum critical spin chains and the Casimir energy of two aligned needles in two-dimensional classical critical systems. Using this connection we provide a formula for the logarithmic formation probability of two disjoint intervals in generic (1 + 1)-dimensional critical systems. The quantity is depenedent on the full structure of the underlying conformal field theory and so useful to find the universality class of the critical system. The connection that we find also provides a very efficient numerical method to calculate the Casimir energy between needles using quantum critical chains. The agreement between numerical results performed on the critical transverse-field Ising model and the XX chain with our exact results is very good. We also comment on the mutual Rényi information of two disjoint intervals.

On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.

Snapshot Spectrum and Critical Phenomenon for Two-Dimensional Classical Spin Systems

We investigate the eigenvalue distribution of the snapshot density matrix (SDM) generated by Monte Carlo simulation for two-dimensional classical spin systems. We find that the distribution in the high-temperature limit is well explained by the random-matrix theory, while that in the low-temperature limit can be characterized by the zero-eigenvalue condensation. At the critical point, we obtain the power-law distribution with a nontrivial exponent $\alpha\equiv(2-\eta)/(1-\eta)$ and the asymptotic form of the snapshot entropy, on the basis of the relationship of the SDM with the correlation function matrix. The aspect-ratio dependence of the SDM spectrum is also mentioned.

Stability and Stabilization of Differential Nonlinear Repetitive Processes with Applications

Repetitive processes are a class of two-dimensional systems that arise in the modeling of physical examples and also the control systems theory developed for them has, in the case of linear dynamics, been applied to design iterative learning control laws with experimental verification. This paper gives new results on the stability of nonlinear differential repetitive processes for applications where a linearized model is either very limited or not applicable. The stability results are then applied to the design of iterative learning control laws in the presence of uncertain parameters and to the same problem when random failures occur that are modeled by a homogeneous Markov chain with a finite set of states. In both cases the computations required are expressed as a finite set of linear matrix inequalities.

KYP lemma based stability and control law design for differential linear repetitive processes with applications

Repetitive processes are a class of two-dimensional systems that have physical applications, including the design of iterative learning control laws where experimental validation results have been reported. This paper uses the Kalman–Yakubovich–Popov lemma to develop new stability tests for differential linear repetitive processes that are computationally less intensive than those currently available. These tests are then extended to allow control law design for stability and performance.

Multi-GPU-based Swendsen–Wang multi-cluster algorithm for the simulation of two-dimensional [formula omitted]-state Potts model

We present multiple GPU computing with the common unified device architecture (CUDA) for the Swendsen–Wang multi-cluster algorithm of two-dimensional (2D) q-state Potts model. Extending our algorithm for single GPU computing [Y. Komura, Y. Okabe, GPU-based Swendsen–Wang multi-cluster algorithm for the simulation of two-dimensional classical spin systems, Comput. Phys. Comm. 183 (2012) 1155–1161], we realize the GPU computation of the Swendsen–Wang multi-cluster algorithm for multiple GPUs. We implement our code on the large-scale open science supercomputer TSUBAME 2.0, and test the performance and the scalability of the simulation of the 2D Potts model. The performance on Tesla M2050 using 256 GPUs is obtained as 37.3 spin flips per a nano second for the q=2 Potts model (Ising model) at the critical temperature with the linear system size L=65536.

GPU-based Swendsen–Wang multi-cluster algorithm for the simulation of two-dimensional classical spin systems

We present the GPU calculation with the common unified device architecture (CUDA) for the Swendsen–Wang multi-cluster algorithm of two-dimensional classical spin systems. We adjust the two connected component labeling algorithms recently proposed with CUDA for the assignment of the cluster in the Swendsen–Wang algorithm. Starting with the q-state Potts model, we extend our implementation to the system of vector spins, the q-state clock model, with the idea of embedded cluster. We test the performance, and the calculation time on GTX580 is obtained as 2.51 nsec per a spin flip for the q=2 Potts model (Ising model) and 2.42 nsec per a spin flip for the q=6 clock model with the linear size L=4096 at the critical temperature, respectively. The computational speed for the q=2 Potts model on GTX580 is 12.4 times as fast as the calculation speed on a current CPU core. That for the q=6 clock model on GTX580 is 35.6 times as fast as the calculation speed on a current CPU core.

Melting properties of two-dimensional multi-species colloidal systems in a parabolic trap

The angular and radial melting properties of two-dimensional classical systems consisting of different types of particles confined in a parabolic trap are studied through modified Monte Carlo simulations. A universal behavior of the angular melting process is found, which occurs in multiple steps due to shell depended melting temperatures. The melting sequence of the different shells is determined by two major factors: (1) the confinement strength which each shell is subjected to, and (2) the specific structure of each shell. Further, a continuous radial disordering of the particle types forming a single circular shell is found and analyzed. This phenomenon has never been observed before in two-dimensional mono-dispersive systems. This continuous radial disordering results from the high energy barrier between different particle types in multi-species systems.

Periodic orbits for an infinite family of classical superintegrable systems

We show that all bounded trajectories in the two-dimensional classical system with the potential are closed for all integer and rational values of k. The period is and does not depend on k. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

Fluctuation relations for a classical harmonic oscillator in an electromagnetic field

In this work, we establish some fluctuation relations for a classical two-dimensional system of independent charged harmonic oscillators in the presence of an electromagnetic field. The main fluctuation relation quantifies irreversible behavior by comparing probabilities of observing particular trajectories during forward and backward processes and is expressed in terms of the work performed by the externally time-dependent electric field when the system is driven away from equilibrium. In the absence of a harmonic force and assuming a constant electric field, our theoretical results reduce to the fluctuation relations for a classical two-dimensional system of noninteracting electrons under the influence of externally crossed electric and magnetic fields, which were recently studied [D. Roy and N. Kumar, Phys. Rev. E 78, 052102 (2008)]. Finally, by considering the dragging of the center of the harmonic trap potential given by the presence of the arbitrary time-dependent electric field, the work-fluctuation theorem and the Jarzynski equality are verified.

Infinite time-evolving block decimation algorithm beyond unitary evolution

The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.

Construction of exact complex dynamical invariant of a two-dimensional classical system

We present the construction of exact complex dynamical invariant of a two-dimensional classical dynamical system on an extended complex space utilizing Lie algebraic approach. These invariants are expected to play a vital role in understanding the complex trajectories of both classical and quantum systems.

Construction of exact dynamical invariants of two-dimensional classical system

A general method is used for the construction of second constant of motion of fourth order in momenta using the complex coordinates (z, z - ). A fourth-order potential equation is obtained whose solutions directly provide a large class of integrable systems. The potential equation is tested with an interesting example which admits second constants of motion.

Finite-size corrections for Coulomb systems in the Debye–Hückel regime

It has been argued that for a finite two-dimensional classical Coulomb system of characteristic size $R$, in its conducting phase, as $R\to \infty $ the total free energy (times the inverse temperature $\beta$) admits an expansion of the form: $\beta F=AR^{2}+BR+{1/6}\chi \ln R,$ where $\chi $ is the Euler characteristic of the manifold where the system lives. The first two terms represent the bulk free energy and the surface free energy respectively. The coefficients $A$ and $B$ are non-universal but the coefficient of $\ln R$ is universal: it does not depend on the detail of the microscopic constitution of the system (particle densities, temperature, etc...). By doing the usual Legendre transform this universal finite-size correction is also present in the grand potential. The explicit form of the expansion has been checked for some exactly solvable models for a special value of the coulombic coupling. In this paper we present a method to obtain these finite-size corrections at the Debye--H\"uckel regime. It is based on the sine-Gordon field theory to find an expression for the grand canonical partition function in terms of the spectrum of the Laplace operator. As examples we find explicitly the grand potential expansion for a Coulomb system confined in a disk and in an annulus with ideal conductor walls.