Table of contents

We show that the Schrödinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics; we also discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorized form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarize recent progress in understanding the dynamics of condensation. We then turn to several generalizations which also, under certain specified conditions, share the property of a factorized steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.

A study is made of an anisotropic Potts model in three dimensions where the coupling depends on both the Potts state on each site and also the direction of the bond between them using both analytical and numerical methods. The phase diagram is mapped out for all values of the exchange interactions. Six distinct phases are identified. Monte Carlo simulations have been used to obtain the order parameter and the values for the energy and entropy in the ground state and also the transition temperatures. Excellent agreement is found between the simulated and analytic results. We find one region where there are two phase transitions with the lines meeting in a triple point. The orbital ordering that occurs in LaMnO₃ occurs as one of the ordered phases.

In this paper, we study the certain qualitative properties of a new anisotropic continuum traffic flow model in which the dimensionless parameter or anisotropic factor controls the non-isotropic character and diffusive influence. We discussed the travelling wave solution for our model and find out the condition for the shock wave. Shock and rarefaction waves are obtained from the new model and are consistent with the diverse nonlinear dynamical phenomena observed in a real traffic flow. However, our model for large values of anisotropic parameter removes the discontinuity as pointed out by Berg et al (2000 Phys. Rev. E 61 1056). The nonlinear theory of the cluster effect in a traffic flow i.e., the effect of appearance of a region of high density and low average velocity of vehicles in an initially homogeneous flow, is also discussed. It is shown that an appearance of a localized perturbation of finite amplitude in the stable homogeneous flow can lead to a self-formation of a local cluster of vehicles. It is also been observed that the cluster effect from our model shows a good agreement with the results of Kerner and Konhäuser (1994 Phys. Rev. E 50 54) and Jiang et al (2002 Trans. Res. B 36 405).

We develop techniques which allow us to calculate the spectra of pinned charge density waves with background current in one-dimensional space. We show that in such systems the low-lying modes are always localized, and we compute their spectral density and the localization length. We also show that as the energy is increased, the modes delocalize in a way similar to that in the Hatano–Nelson non-Hermitian quantum mechanics.

The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a Fokker–Planck technique. We obtain an exact expression for the area distribution for the zero drift case, and provide various asymptotic results for the nonzero drift case, emphasizing the critical nature of the behaviour in the limit of vanishing drift. The results offer important insights into the asymptotic behaviour of a number of discrete models. We also provide a succinct derivation for the distribution of the maximum displacement observed during a first passage.

We investigate the dynamics of two Bose–Einstein condensates (BECs) tunnel coupled by a double-well potential. In the case of attractive interatomic interaction, the effects of the three-body recombination losses and the feeding of the condensates from the thermal cloud are studied by using a two-mode approximation. It is shown that the stability region of the parameter space could be enlarged by increasing the three-body losses or the tunnelling coefficient. The numerical results also reveal some interesting motion characteristics of the BECs for different s-wave scattering lengths, including macroscopic quantum self-trapping of non-stationary state BECs, and periodic oscillation of the population whose phase orbit tends to a stable limit cycle.

The phase diagram of a model of rectangular vesicles in three dimensions is examined. The scaling of the generating function is determined in the area–perimeter, volume–area and volume–perimeter ensembles. The results are interpreted within the framework of tricritical scaling, and the crossover exponents associated with the transitions are determined. We identify three phases in the phase diagram, a needle phase, a disc phase and a cubical phase. These phases are separated by three curves of transitions that meet in a multicritical point.

We give the Fuchsian linear differential equation satisfied by χ⁽⁴⁾, the 'four-particle' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to χ⁽⁴⁾. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution χ⁽²⁾ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, χ⁽³⁾, leads us to a conjecture on the structure of all the n-particle contributions χ⁽ⁿ⁾.

In this paper, we present an elementary theory on the existence and robustness of horseshoes under perturbations in terms of stability of crossing. The framework is developed in the setting of 2D Euclidean space but can be generalized to metric spaces. As an application, we give a rigorous verification of the existence of a horseshoe in the Ikeda map.

A method is given to determine the Casimir operators of the perfect Lie algebras and the inhomogeneous Lie algebras in terms of polynomials associated with a parametrized (2N + 1) × (2N + 1)-matrix. For the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation. We further analyse the invariants for the extended Schrödinger algebra in (N + 1) dimensions, which arises naturally as a subalgebra of . The method is extended to other classes of Lie algebras, and some applications to the missing label problem are given.

An algebraic criterion that is sufficient to establish the existence of certain a priori estimates for the solution of first-order homogeneous linear characteristic problems is derived. Estimates of such kind ensure the stability of the solutions under small variations of the data. Characteristic problems that satisfy this criterion are, in a sense, manifestly well posed.

An iterative method due to Voslamber is reconsidered. It provides successive approximations for the logarithm of the time–displacement operator in quantum mechanics. The procedure may be interpreted, a posteriori, as an infinite re-summation of terms in the so-called Magnus expansion. A recursive generator for higher terms is obtained. From two illustrative examples, a detailed comparative study is carried out between the results of the iterative method and those of the Magnus expansion.

Decoherence of entanglement of qubits is investigated which is caused by the phenomenological quantum channel (the Bloch channel) equivalent to the Bloch equations. It is shown how the decoherence of entanglement depends on the longitudinal and transverse relaxation times and the equilibrium value of the qubit. The quantum dense coding system under the influence of the Bloch channel is also investigated. The Shannon mutual information obtained by the Bell measurement and the Holevo capacity is calculated. Furthermore, the microscopic system-reservoir model which yields the Bloch equations is considered. The result shows that the temperature of the thermal reservoir significantly affects the decoherence of entanglement.

We study Bertrand's duopoly of incomplete information. It is found that the effect of quantum entanglement on the outcome of the game is dramatically changed by the uncertainty of information. In contrast with the case of complete information where the outcome increases with entanglement, when information is incomplete the outcome is maximized at some finite entanglement. As a consequence, information and entanglement are both crucial factors that determine the properties of a quantum oligopoly.

In this paper, we clarify the relation between Manin's quantum theta function and Schwarz's theta vector. We do this in comparison with the relation between the kq representation, which is equivalent to the classical theta function, and the corresponding coordinate space wavefunction. We first explain the equivalence relation between the classical theta function and the kq representation in which the translation operators of the phase space are commuting. When the translation operators of the phase space are not commuting, then the kq representation is no longer meaningful. We explain why Manin's quantum theta function, obtained via algebra (quantum torus) valued inner product of the theta vector, is a natural choice for the quantum version of the classical theta function. We then show that this approach holds for a more general theta vector containing an extra linear term in the exponent obtained from a holomorphic connection of constant curvature than the simple Gaussian one used in Manin's construction.

We consider radiation transport theory applied to non-dispersive but refractive media. This setting is used to discuss Minkowski's and Abraham's electromagnetic momentum, and to derive conservation equations independent of the choice of momentum definition. Using general relativistic kinetic theory, we derive and discuss a radiation gas energy–momentum conservation equation valid in arbitrary curved spacetime with diffractive media.

The radiation emitted by a charged particle moving along a helical orbit inside a dielectric cylinder immersed in a homogeneous medium is investigated. Expressions are derived for the electromagnetic potentials, electric and magnetic fields, and for the spectral–angular distribution of radiation in the exterior medium. It is shown that under the Cherenkov condition for dielectric permittivity of the cylinder and the velocity of the particle image on the cylinder surface, strong narrow peaks are present in the angular distribution for the number of radiated quanta. At these peaks the radiated energy exceeds the corresponding quantity for a homogeneous medium by some orders of magnitude. The results of numerical calculations for the angular distribution of radiated quanta are presented and they are compared with the corresponding quantities for radiation in a homogeneous medium. The special case of relativistic charged particle motion along the direction of the cylinder axis with non-relativistic transverse velocity (helical undulator) is considered in detail. Various regimes for the undulator parameter are discussed. It is shown that the presence of the cylinder can increase essentially the radiation intensity.

Because of its non-Hermitian property, the stability analysis of a shear-flow system is rather complex. While the Kelvin–Helmholtz instability, being a spatially global and temporally exponential mode, can be detected by a standard analysis of eigenvalues, there may exist a variety of different instabilities. Invoking time asymptotic analysis, the existence of spatially localized and temporally algebraic instability in non-monotonic shear-flows with multiple stationary points is shown.