Quantization on a two-dimensional phase space with a constant curvature tensor

Some properties of the star product of the Weyl type (i.e., associated with the Weyl ordering) are proved. Fedosov construction of the *-product on a two-dimensional phase space with a constant curvature tensor is presented. Eigenvalue equations for momentum p and position q on a two-dimensional phase space with constant curvature tensors are solved.     

q-deformed phase space and its lattice structure

A q-deformed two-dimensional phase space is studied as a model for a noncommutative phase space. A lattice structure arises that can be interpreted as a spontaneous breaking of a continuous symmetry. The eigenfunctions of a Hamiltonian that lives on such a lattice are derived as wavefunctions in ordinaryx-space.     

q-deformed phase space and its lattice structure

A q-deformed two-dimensional phase space is studied as a model for a noncommutative phase space. A lattice structure arises that can be interpreted as a spontaneous breaking of a continuous symmetry. The eigenfunctions of a Hamiltonian that lives on such a lattice are derived as wavefunctions in ordinaryx-space.     

Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps

Bifurcation structures for nonlinear dynamical systems in a space of two parameters often display geometric shapes resembling shrimps. For one-dimensional maps with two parameters and multiple extrema, the underlying structure of the shrimps can be elucidated by computing the locus of superstable cycles which form a “skeleton” that supports the shrimps. Here we use continuation methods to identify and compute structures in two-dimensional maps that play the same role as the skeleton in one-dimensional maps. This facilitates determining the complex geometries for situations in which there is multistability, and for which the regions of parameter space supporting stable orbits get vanishingly small.     

Automated spatiotemporal diffraction of ultrashort laser pulses

We exploit the close similarities between time-frequency and position-wave-vector correspondences to control the spatiotemporal diffraction pattern of ultrashort laser pulses. This approach permits novel, automated generation of sophisticated two-dimensional femtosecond waveforms. A two-dimensional space-time version of a Gerchberg-Saxton algorithm is used to iteratively determine the phase pattern in position-frequency space that produces a user-defined intensity profile in wave-vector-time space.     

A New Type of Seiberg-Witten Map and Its Application

Deformation quantization is a powerful tool to deal with systems in noncommutative space to get their energy spectra and corresponding Wigner functions, especially for the case of both coordinates and momenta being noncommutative. In order to simplify solutions of the relevant *-genvalue equation, we introduce a new kind of Seiberg-Witten-like map to change the variables of the noncommutative phase space into ones of a commutative phase space, and demonstrate its role via an example of two-dimensional oscillator with both kinetic and elastic couplings in the noncommutative phase space.     

Numerical Solution of the Two-Dimensional Vlasov Equation

The two-dimensional Vlasov equation is solved by direct integration in phase space. Two problems, namely the nonlinear evolution of the two-dimensional electrostatic two-stream instability, and the nonlinear evolution of a monochromatic wave in a two-dimensional Vlasov plasma, are studied. Comparison with previously available results is given.     

Two-Dimensional Linear Dependencies on the Coordinate Time-Dependent Interaction in Relativistic Non-Commutative Phase Space

In this paper, the Non-Commutative phase space and Dirac equation, time-dependent Dirac oscillator are introduced. After presenting the desire general form of a two-dimensional linear dependency on the coordinate time-dependent potential, the Dirac equation is written in terms of Non-Commutative phase space parameters and solved in a general form by using Lewis-Riesenfield invariant method and the time-dependent invariant of Dirac equation with two-dimensional linear dependency on the coordinate time-dependent potential in Non-Commutative phase space has been constructed, then such latter operations are done for time-dependent Dirac oscillator. In order to solve the differential equation of wave function time evolution for Dirac equation and time-dependent Dirac oscillator which are partial differential equation some appropriate ordinary physical problems have been studied and at the end the interesting result has been achieved.     

Classification of Phase Transitions and Ensemble Inequivalence, in Systems with Long Range Interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.     

FitzHugh-Nagumo系统中螺旋波的控制

采用FitzHugh-Nagumo方程,研究了二维时空系统中螺旋波的控制问题,利用相空间压缩方法对部分系统变量的振幅进行限制从而影响螺旋波的稳定性.研究表明,控制过程可分为三个不同的阶段:在较小压缩限条件下螺旋波可以被完全消除,系统进入均匀定态;在较大的压缩限条件下螺旋波能够稳定存在,而且其振荡频率不随控制参数的改变而发生变化;当压缩限介于上述两者之间时,系统表现为时空混沌态.对上述控制过程进行了进一步的讨论,研究了不同控制参数条件下的系统斑图、变量的演化、相空间轨道等性质,并且对振幅函数和振荡频率特征进 关键词： 螺旋波 相空间压缩 FitzHugh-Nagumo方程     

A maximum-entropy principle for two-dimensional perfect fluid dynamics

We use Kullback entropy for Young measures to define statistical equilibrium states for a two-dimensional incompressible flow of a perfect fluid. This approach is justified, as it gives a concentration property about the equilibrium state in the phase space. It might give a statistical understanding of the appearance of coherent structures in two-dimensional turbulence.     

Generic Two-Phase Coexistence and Nonequilibrium Criticality in a Lattice Version of Schlögl’s Second Model for Autocatalysis

A two-dimensional atomistic realization of Schlögl’s second model for autocatalysis is implemented and studied on a square lattice as a prototypical nonequilibrium model with first-order transition. The model has no explicit symmetry and its phase transition can be viewed as the nonequilibrium counterpart of liquid-vapor phase separations. We show some familiar concepts from study of equilibrium systems need to be modified. Most importantly, phase coexistence can be a generic feature of the model, occurring over a finite region of the parameter space. The first-order transition becomes continuous as a temperature-like variable increases. The associated critical behavior is studied through Monte Carlo simulations and shown to be in the two-dimensional Ising universality class. However, some common expectations regarding finite-size corrections and fractal properties of geometric clusters for equilibrium systems seems to be inapplicable.     

Non-equilibrium agglomeration of helium-vacancy clusters in irradiated materials

Abstract

Time evolution of non-equilibrium systems, where the probability density is described by a continuum Fokker-Planck (F-P) equation, is a central area of interest in stochastic processes. In this paper, a numerical solution of a two-dimensional (2-D) F-P equation describing the growth of helium-vacancy clusters (HeVCs) in metals under irradiation is given. First, nucleation rates and regions of stability of HeVCs in the appropriate phase space for fission and fusion devices are established. This is accomplished by solving a detailed set of cluster kinetic rate equations. A nodal line analysis is used to map spontaneous and stochastic nucleation regimes in the helium-vacancy (h-v) phase space. Growth trajectories of HeVCs are then used to evaluate the average HeVC size and helium content during the growth phase of HeVCs in typical growth instability regions.

The growth phase of HeVCs is modeled by a continuum 2-D, time-dependent F-P equation. Growth trajectories are used to define a finite solution space in the h-v phase space. A highly efficient dynamic remeshing scheme is developed to solve the F-P equation. As a demonstration, typical HFIR irradiation conditions are chosen. Good agreement between the computed size distributions and those measured experimentally are obtained.     

Scaling invariance for the escape of particles from a periodically corrugated waveguide

The escape dynamics of a classical light ray inside a corrugated waveguide is characterised by the use of scaling arguments. The model is described via a two-dimensional nonlinear and area preserving mapping. The phase space of the mapping contains a set of periodic islands surrounded by a large chaotic sea that is confined by a set of invariant tori. When a hole is introduced in the chaotic sea, letting the ray escape, the histogram of frequency of the number of escaping particles exhibits rapid growth, reaching a maximum value at np and later decaying asymptotically to zero. The behaviour of the histogram of escape frequency is characterised using scaling arguments. The scaling formalism is widely applicable to critical phenomena and useful in characterisation of phase transitions, including transitions from limited to unlimited energy growth in two-dimensional time varying billiard problems.     

Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.     

Direct estimation of aberrating delays in pulse-echo imaging systems

Nearfield fluctuations in wave propagation velocity and system timing errors are among the sources of focusing aberrations in pulse-echo imaging systems. For situations in which the source of these errors can be modeled by a stationary phase aberrator placed in front of the transmitter and receiver aperture, appropriate electronic delays might be applied to the signals associated with each array element in order to restore the system to focus. A method is described and evaluated for estimating the set of aberrating delays in a linear array utilizing data from a single two-dimensional scan. The underlying principle is analogous to that of phase closure used for one-way passive interferometry and readily generalizes to two-dimensional arrays. Although the following theory is developed in the context of acoustic imaging, the general approach is applicable to other pulse-echo systems, such as radar.     

Corrugated waveguide under scaling investigation

Some scaling properties for classical light ray dynamics inside a periodically corrugated waveguide are studied by use of a simplified two-dimensional nonlinear area-preserving map. It is shown that the phase space is mixed. The chaotic sea is characterized using scaling arguments revealing critical exponents connected by an analytic relationship. The formalism is widely applicable to systems with mixed phase space, and especially in studies of the transition from integrability to nonintegrability, including that in classical billiard problems.     

Phase transition in a self-gravitating planar gas

We consider a gas of Newtonian self-gravitating particles in two-dimensional space, finding a phase transition, with a high temperature homogeneous phase and a low temperature clumped one. We argue that the system is described in terms of a gas with fractal behaviour.     

Statistical thermodynamics of the formation of a new phase in the boiling up of volatile liquids

The fluctuation growth of a macroscopic bubble containing a vapor in a moderately superheated or tensile-stressed volatile liquid is treated as the two-dimensional diffusion of a nucleus of a new phase in the space of variables made up of its volumev and the pressure of the vapor in itp. The shape of the free energy surface of the system liquid plus bubble with vapor in the plane (V; p) in the neighborhood of the labile equilibrium of the system is examined, and a two-dimensional nucleus distribution function given with respect to its variables is derived. Close to the pass in the surface a nondiagonal diffusion tensor in the space (V, p) is also calculated. A two-dimensional stationary equation of the kinetics of the formation of a new phase of Kramers type is solved, and an expression is derived for the probability of homogeneous nucleation for an arbitrary viscosity and volatility of a liquid far from its critical point. Various limiting cases are examined.