Table of contents

Ideas and techniques (asymptotic decoupling of single-trace subspace, asymptotic operator algebras, duality and role of supersymmetry) relevant in current Fock space investigations of quantum field theories have very simple roles in a class of toy models.

We calculate the exact asymptotic survival probability, Q, of a one-dimensional Brownian particle, initially located at the point x ∊ (−L, L), in the presence of two moving, absorbing boundaries located at ±(L + ct). The result is Q(y, λ) = ∑^∞_n=−∞(−1)ⁿcosh(ny)exp(−n²λ), where y = cx/D, λ = cL/D and D is the diffusion constant of the particle. The results may be extended to the case where the absorbing boundaries have different speeds. As an application, we compute the asymptotic survival probability for the trapping reaction A + B → B, for evanescent traps with a long decay time.

We formulate a linear Schrödinger equation with a temperature-dependent potential for the one-particle density matrix and obtain the condensation temperature of the Bose–Einstein condensate from a bound-state condition for the Schrödinger equation both with and without the confining trap. The results are in very good agreement with those of the full statistical physics treatment. This is an alternative to the Bose–Einstein condensation in the standard ideal Bose-gas treatment.

We analyse the Lévy processes produced by means of two interconnected classes of nonstable, infinitely divisible distribution: the variance gamma and the Student laws. While the variance gamma family is closed under convolution, the Student one is not: this makes its time evolution more complicated. We prove that—at least for one particular type of Student processes suggested by recent empirical results, and for integral times—the distribution of the process is a mixture of other types of Student distributions, randomized by means of a new probability distribution. The mixture is such that along the time the asymptotic behaviour of the probability density functions always coincide with that of the generating Student law. We put forward the conjecture that this can be a general feature of the Student processes. We finally analyse the Ornstein-Uhlenbeck process driven by our Lévy noises and show a few simulations of it.

We consider a planar Brownian motion starting from O at time t = 0 and stopped at t. Denoting by T the time spent in a wedge of apex O and angle Θ, we develop a method to compute systematically the moments of T for general Θ values. We apply it to obtain analytically the second and third moments for a general wedge angle and, also, the fourth moment for the quadrant (Θ = π/2). We compare our results with numerical simulations. Finally, with standard perturbation theory, we establish a general formula for the second moment of an orthant occupation time.

Adsorption at a one-dimensional planar substrate equipped with a localized chemical inhomogeneity is studied within the framework of a continuum interfacial model from the point of view of interfacial morphology and correlation function properties. Exact expressions for the one-point and two-point probability distribution functions P_Γ(l_Γ) and being the interface position above a fixed point Γ of the substrate, are derived for temperature corresponding to the inhomogeneity's wetting transition. It is demonstrated that in the limit of macroscopic inhomogeneity's size the net effect of the remaining homogeneous parts of the substrate on the interfacial morphology above the inhomogeneity is exactly equivalent to appropriate pinning of the interface at its boundaries. The structure of the average interfacial morphology and correlation function in this limit are discussed and compared to earlier results obtained for systems with homogeneous substrate.

Multi-particle non-equilibrium dynamics in two-channel asymmetric exclusion processes with narrow entrances is investigated theoretically. Particles move on two parallel lattices in opposite directions without changing them, while the channels are coupled only at the boundaries. A particle cannot enter the corresponding lane if the exit site of the other lane is occupied. Stationary phase diagrams, particle currents and densities are calculated in a mean-field approximation. It is shown that there are four stationary phases in the system, with two of them exhibiting spontaneous symmetry breaking phenomena. Extensive Monte Carlo computer simulations confirm qualitatively our predictions, although the phase boundaries and stationary properties deviate from the mean-field results. Computer simulations indicate that several dynamic and phase properties of the system have a strong size dependence, and one of the stationary phases predicted by the mean-field theory probably disappears in the thermodynamic limit.

We introduce a model for a population on a lattice with diffusion and birth/death according to and for a particle A. We find that the model displays a phase transition from an active to an absorbing state which is continuous in 1 + 1 dimensions and of first-order in higher dimensions in agreement with the mean field equation. For the (1 + 1)-dimensional case, we examine the critical exponents and a scaling function for the survival probability and show that it belongs to the universality class of directed percolation. In higher dimensions, we look at the first-order phase transition by plotting a histogram of the population density and use the presence of phase coexistence to find an accurate value for the critical point in 2 + 1 dimensions.

We consider the dynamics of a quantum particle in a one-dimensional periodic potential (lattice) under the action of a static and time-periodic field. The analysis is based on a nearest-neighbour tight-binding model which allows a convenient closed form description of the transport properties in terms of generalized Bessel functions. The case of bichromatic driving is analysed in detail and the intricate transport and localization phenomena depending on the communicability of the two excitation frequencies and the Bloch frequency are discussed. The case of polychromatic driving is also discussed, in particular for flipped static fields, i.e. rectangular pulses, which can support an almost dispersionless transport with a velocity independent of the field amplitude.

A method for studying isochronous oscillations in some systems of ODE reducible to the equation is described. It is applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two-dimensional autonomous system depending on six parameters. For all isochronous systems in this family the Urabe function is explicitly constructed.

We determine the location of the expansion points with respect to which the two Maxwell's multipole vectors of the quadrupole moment and the dipole vector of a distribution of charge form an orthogonal trihedron. We find that with respect to these 'orthogonality centres' both the dipole and the quadrupole moments are each characterized by a single real parameter. We further show that the orthogonality centres coincide with the stationary points of the magnitude of the quadrupole moment and, therefore, they can be seen as an extension of the concept of centre of the dipole moment of a neutral system introduced previously in the literature. The nature of the stationary points then provides the means for the classification of a distribution of charge in two different categories.

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg–Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram–Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.

The standard block orthogonal (SBO) polynomials P_i;n(x), 0 ⩽ i ⩽ n are real polynomials of degree n which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than i. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, which we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an Euclidean orthogonality relation.

The Adler–Kostant–Symes scheme is used to describe mechanical systems for quadratic Hamiltonians of on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that on . This system is a Lax pair equation whose solution can be computed with the help of the adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in is related to the commutativity of a family of derivations of the (2n + 1)-dimensional Heisenberg Lie algebra . Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in . For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.

Using several kinds of coordinate transformations, we standardize the noncanonical symplectic structure of the Ablowitz–Ladik model (A–L model) of nonlinear Schrödinger equation (NLSE), then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the A–L model and also the conserved quantities of the original NLSE. In comparison with a higher order non-symplectic scheme applied directly to the A–L model, we show the overwhelming superiorities of the symplectic method. We also compare the implementation of the same symplectic scheme to different standardized Hamiltonian systems resulting from different coordinate transformations, and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one, in preserving the invariants of the A–L model and the original NLSE.

Decompositions of Lie groups are used in systems and control, both to analyse dynamics and to design control algorithms for systems with state varying on a Lie group. In this paper, we investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition, there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd–even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in [6–8] in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of N qubits to the case where the component subsystems have arbitrary dimensions. We present an example of application of the results to control design for quantum systems.

We investigate decoherence of a central quantum system uniformly coupled to an XY spin-1/2 bath in a transverse field. Through explicitly calculating the Loschmidt echo (LE) used to characterize decoherence quantitatively we find that the anisotropy parameter γ sensitively affects the decoherence of the central system when γ ∊ [0, 1]. Interestingly, the LE becomes unit under the condition that the initial state of the environment is a product state. Although it is difficult to control the environment to be product states in reality, our findings may provide a new understanding of mechanism of the decoherence.

We point out that the celebrated GRW master equation is invariant under translations, reflecting the homogeneity of space, thus providing a particular realization of a general class of translation-covariant Markovian master equations. Such master equations are typically used for the description of decoherence due to momentum transfers between the system and environment. Building on this analogy we show the exact relationship between the GRW master equation and decoherence master equations, further providing a collisional decoherence model formally equivalent to the GRW master equation. This allows for a direct comparison of order of magnitudes of relevant parameters. This formal analogy should not lead to confusion on the utterly different spirit of the two research fields, in particular it has to be stressed that the decoherence approach does not lead to a solution of the measurement problem. Building on this analogy however the feasibility of the extension of spontaneous localization models in order to avoid the infinite energy growth is discussed. Apart from a particular case considered in the paper, it appears that the amplification mechanism is generally spoiled by such modifications.

We replace time-averaged entanglement by ensemble-averaged entanglement and derive a simple expression for the latter. We show how to calculate the ensemble average for a two-spin system and for the Jaynes–Cummings model. In both cases the time-dependent entanglement is known as well so that one can verify that the time average coincides with the ensemble average.

We develop a systematic approach to bosonization and vertex algebras on quantum wires in the form of star graphs. The related bosonic fields propagate freely in the bulk of the graph, but interact with its vertex. Our framework covers all possible interactions preserving unitarity. Special attention is devoted to the scale-invariant interactions, which determine the critical properties of the system. Using the associated scattering matrices, we give a complete classification of the critical points on a star graph with any number of edges. Critical points where the system is not invariant under wire permutations are discovered. By means of an appropriate vertex algebra we perform the bosonization of fermions and solve the massless Thirring model. In this context we derive an explicit expression for the conductance and investigate its behaviour at the critical points. A simple relation between the conductance and the Casimir energy density is pointed out.

We construct certain eigenfunctions of the Calogero–Sutherland Hamiltonian for particles on a circle, with mixed boundary conditions. That is, the behaviour of the eigenfunction, as neighbouring particles collide, depend on the pair of colliding particles. This behaviour is generically a linear combination of two types of power laws, depending on the statistics of the particles involved. For fixed ratio of each type at each pair of neighbouring particles, there is an eigenfunction, the ground state, with lowest energy, and there is a discrete set of eigenstates and eigenvalues, the excited states and the energies above this ground state. We find the ground state and special excited states along with their energies in a certain class of mixed boundary conditions, interpreted as having pairs of neighbouring bosons and other particles being fermions. These particular eigenfunctions are characterized by the fact that they are in direct correspondence with correlation functions in boundary conformal field theory. We expect that they have applications to measures on certain configurations of curves in the statistical O(n) loop model. The derivation, although completely independent of results of conformal field theory, uses ideas from the 'Coulomb gas' formulation.

The kinematical formalism for describing spinning particles developed by the author is based upon the idea that an elementary particle is a physical system with no excited states. It can be annihilated by the interaction with its antiparticle but, if not destroyed, its internal structure can never be modified. All possible states of the particle are just kinematical modifications of any one of them. The kinematical state space of the variational formalism of an elementary particle is necessarily a homogeneous space of the kinematical group of spacetime symmetries. By assuming Poincaré invariance we have already described a model of a classical spinning particle which satisfies Dirac's equation when quantized. We have recently shown that the spacetime symmetry group of this Dirac particle is larger than the Poincaré group. It also contains spacetime dilations and local rotations. In this work we obtain an interaction Lagrangian for two Dirac particles, which is invariant under this enlarged spacetime group. It describes a short- and long-range interaction such that when averaged, to suppress the spin content of the particles, describes the instantaneous Coulomb interaction between them. As an application, we analyse the interaction between two spinning particles, and show that the existence is possible of metastable bound states for two particles of the same charge, when the spins are parallel and provided some initial conditions are fulfilled. The possibility of formation of bound pairs is due to the zitterbewegung spin structure of the particles because when the spin is neglected, the bound states vanish.

In this present paper, a quaternary gaseous reactive mixture, for which the chemical reaction is close to its final stage and the elastic and reactive frequencies are comparable, is modelled within the Boltzmann equation extended to reacting gases. The main objective is a detailed analysis of the non-equilibrium effects arising in the reactive system A₁ + A₂ ⇌ A₃ + A₄, in a flow regime which is considered not far away from thermal, mechanical and chemical equilibrium. A first-order perturbation solution technique is applied to the macroscopic field equations for the spatially homogeneous gas system, and the trend to equilibrium is studied in detail. Adopting elastic hard-spheres and reactive line-of-centres cross sections and an appropriate choice of the input distribution functions—which allows us to distinguish the two cases where the constituents are either at same or different temperatures—explicit computations of the linearized production terms for mass, momentum and total energy are performed for each gas species. The departures from the equilibrium states of densities, temperatures and diffusion fluxes are characterized by small perturbations of their corresponding equilibrium values. For the hydrogen–chlorine system, the perturbations are plotted as functions of time for both cases where the species are either at the same or different temperatures. Moreover, the trend to equilibrium of the reaction rates is represented for the forward and backward reaction H₂ + Cl ⇌ HCl + H.

In our previous papers we proposed a continuum model for the dynamics of the systems of self-propelling particles with conservative kinematic constraints on the velocities. We have determined a class of stationary solutions of this hydrodynamic model and have shown that two types of stationary flow, linear and axially symmetric (vortical) flow, are possible. In this paper we consider the stability properties of these stationary flows. We show, using a linear stability analysis, that the linear solutions are neutrally stable with respect to the imposed velocity and density perturbations. A similar analysis of the stability of the vortical solution is found to be not conclusive.