Table of contents

We report results of a Monte Carlo study of the kinetics of random sequential deposition of line segments (mostly dimers) on the 1D lattice substrate, already occupied with point-like quenched impurities at low concentration. The area covered by the placed objects grows with time and finally reaches a jamming limit when no more deposition is possible. The jamming coverage values, obtained by numerical simulations, depend on the segment length and on the previous occupation of the substrate by impurities. The rate of late-stage deposition is not disturbed by presence of forbidden sites when the process of deposition starts (t=0). Numerical results, shown in semi-log scale, confirm that area coverage theta (t) approaches the jamming limit theta ( infinity ) exponentially, with the same exponent factor -1 multiplying scaled time, as in the case of random sequential deposition of line segments on the clean 1D lattice (initially non-occupied).

We describe results of Monte Carlo simulations on a model that seems to have the necessary ingredients to describe a disordered type-II superconductor in a magnetic field. We compute the free energy cost to twist the direction of the phase of the condensate and analyse the results by finite-size scaling. The results show convincingly that the model has different behaviour as a function of dimension: in d=4 the model clearly has a finite transition temperature; T_c, while for d=2 only there is only a transition at T=0.

Methods of supersymmetric quantum mechanics (SUSYQM) are used to derive relations between phase functions which occur in the context of the variable-phase approach to potential scattering. It is shown that the phase-equivalent potentials obtained by the application of SUSYQM are not off-shell equivalent. A case study is presented in support of this.

A q-deformed harmonic oscillator (q-HO) in a finite-dimensional Hilbert space (FDHS) is presented. Some basic characteristics of the q-HO in the FDHS is discussed. It is found that the quantum algebra su_q (1,1) can be realized in terms of the q-HO in the FDHS on the root of unit.

Using the R matrix formalism we construct a system of q-superoscillators, bosonic as well as fermionic, covariant under the coaction of the quantum orthosymplectic supergroup OS_p,q(2/2).

The path integral treatment of statistics for identical particles is extended to the case where they possess spin by virtue of a continuous internal coordinate. In this case the restriction to bosons or fermions begins at space dimension two. The role of the 'diagonal' (points of coordinate coincidence) in these arguments is clarified.

The multipoint time-dependent correlation functions of the impenetrable Bose gas are considered. The determinant formula for the correlation functions with an arbitrary order of field operators is proposed.

The q-deformed thermofield dynamics is constructed and in terms of this dynamics some statistical mechanical properties of a system of q-harmonic oscillators are discussed.

The square Ising lattice with nearest- and next-nearest-neighbour interactions is considered. On the basis of the simplest ansatz on the principal eigenvector of the transfer matrix of the model and of the CLE condition, developed in previous papers, the correlation length along a column is calculated for all T>or=T_c, in the ferromagnetic and antiferromagnetic regions. A comparison is made with results obtained through other methods. For competing interactions, it is shown that there is a mechanism which leads to a correlation length made of two analytic pieces.

We consider the su(p) spin chains with long-range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities of the dynamical models and we diagonalise them. In the spin chain case, we establish the connection between the degeneracies of the spectrum and the representation theory of the Yangians. We use a correspondence with the dynamical models to diagonalise the Hamiltonian. Finally, we extend the previous results to the case of a trigonometric R-matrix.

We introduce a new algorithm for domain growth in disordered systems at low temperatures based on the tendency for such a system to freeze into metastable states. We apply the algorithm to the d=1 random-bond Ising model for a variety of bond distributions. The analytical forms we obtain for the two-point correlation function and autocorrelation function agree well with numerical simulations of the model. These forms are not the same as those of the pure Ising model with Glauber dynamics.

We present a Monte Carlo study of a classical lattice-gas Hamiltonian formerly introduced by Parrinello and Tosatti (1982) in order to describe the coexistence of atomic and molecular phases. The nature of the ordering transition undergone by the dense molecular fluid into a molecular-crystal phase is investigated in two dimensions through phenomenological finite-size-scaling techniques.

The critical behaviour of the one-dimensional q-state Potts model with long-range interactions decaying with distance r as r^{-(1+ sigma} has been studied in the wide range of parameters 0< sigma <or=1 and 1/16 <or=q<or=64. A transfer matrix has been constructed for a truncated range of interactions for integer and continuous q, and finite range scaling has been applied. Results for the phase diagram and the correlation length critical exponent are presented.

The phase transition of the three-dimensional random field Ising model with a discrete (+or-h) field distribution is investigated by extensive Monte Carlo simulations. Values of the critical exponents for the correlation length, specific heat, susceptibility, disconnected susceptibility and magnetization are determined simultaneously via finite size scaling. While the magnetization appears to be discontinuous, the specific heat appears to saturate, indicating no latent heat. Sample-to-sample fluctuations of the susceptibility are consistent with the droplet picture for the transition.

The phase diagram of a three-state microemulsion model at low temperatures is discussed. It is shown how, taking into account low-energy excitations, the ground-state phase diagram is modified and the degeneracy of a coexistence line bordering the region of lamellar phase is removed.

The non-equilibrium critical relaxation of systems in which the order parameter couples reversibly to conserved densities is investigated. The initial state which may be macroscopically prepared by a quench from a temperature T >> T_c to the critical temperature T_c is characterized by short-range correlations. In the case where the order parameter itself is conserved the relaxation shows universal scaling. If, on the other hand, the order parameter is not conserved the scaling is governed by a new exponent which depends on the width of the initial distribution of the conserved fields. However, the response function of the conserved fields still shows universal scaling.

The main aim of this work is to present an interpretation of Ising-type models as a kind of field theory in the framework of non-commutative geometry. We present the method and construct sample models of field theory on discrete spaces using the tools of discrete geometry introduced. We write the action for a few models, then we compare them with various models of statistical physics. We also construct the gauge theory with a discrete gauge group.

The purpose of the paper is to illustrate how the cluster method, introduced originally in classical statistical mechanics and later on generalized to describe correlation effects in quantum systems, can be extended for treating collective behaviour in diffusion limited recombination A+B to 0: This involves exploring a many-body state vector in second quantization form with pairing correlations between particles only. An algebraic approach for deriving kinetics equations for pair correlation functions g_{lambda lambda '}, (r, t) is proposed, where lambda , lambda '=a, b. solution of those equations is presented. In two-body approximation, g_ab(r,t) and the corresponding time-dependent rate constant reproduce the Smoluchowski results. Many-body effects are analysed both for a monomolecular regime (c_b>>c_a(t)) (MR) and a bimolecular regime (c_b(t)=c_a(t)) (BR). For MR at C_b to 0, g_ab(r, t) is shown to have exponential screening on the characteristic length inversely proportional to square root c_b, and the rate constant acquires additional correction, proportional to square root c_b, to the Smoluchowski value. For BR at the later stage of recombination, the pair correlation function is predominantly determined by the initial condition, i.e. g_ab(r, 0), and the average concentration c_a is found to exhibit time dependence approximately=t^-3/4 at the pre-asymptotic stage. However, at the true asymptotic, t to infinity , the model gives c_a approximately=t^-1. We confront these results with the previous analytical results and find a qualitative consistency in the MR case and some discrepancy for the BR case. The proposed description is rather simple and can be applied for exploring collective and correlation effects in other non-equilibrium classical systems.

Two-velocity kinetic models are used to derive, in the appropriate limit, the equations which govern the macroscopic density of fluid systems. Such equations are obtained from an asymptotic expansion in powers of a small parameter related to the microscopic mean free path. It is shown that the density of a fluid interacting with a non-equilibrium background satisfies a linear diffusion equation, and that the hierarchy of equations arising from the asymptotic expansion can be completely solved by a recursive scheme. For a system of interacting particles, a nonlinear diffusion equation is obtained and some of its solutions are analysed. Finally, the density of a system of particles undergoing chemical reactions is shown to satisfy a nonlinear equation which formally coincides with the reaction-diffusion equation proposed ad hoc at the macroscopic level.

Based on a commonly used model for the dynamical evolution of extensive variables, a novel formulation of nonequilibrium thermodynamics is proposed which emphasizes the fundamental role played by the Boltzmann constant k in fluctuations. The equivalence of this and the stochastic formulation is demonstrated. k to 0 for this theory yields the classical description of nonequilibrium thermodynamics. The new formulation possesses unique features which bear two important results, namely the thermodynamic uncertainty principle and the quantization of the entropy production rate for stationary states.

We study numerically the eigenfunctions and their Wigner phase space distributions of the two-dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by Robnik (1983). This system is a generic KAM system and displays a transition from integrability to almost ergodicity as the billiard shape changes. We clearly identify two classes of states: the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions, whilst the mixed type states were not found, in support of Percival's conjecture (1973). We confirm the existence of(extremely) intense scars in the classically chaotic regions, and demonstrate their association with classical periodic orbits. Three classes of scars are revealed: one-orbit scars, many-orbit-one-family scars (of statistically similar orbits in the homoclinic neighbourhood), and many-orbit-many-family scars. We argue that it is impossible to find an a priori semiclassical theory of individual eigenstates, but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates.

The quantization of the free relativistic spinning particle is revised on the basis of a group approach to quantization. In momentum space, the wavefunctions provide the minimal canonical representation of mass m and spin j of the Poincare group P₊^{up arrow} . The quantization in configuration space requires, as in many other physical systems, polarizations of higher-order type. This higher-order polarization technique turns out to be a natural framework for studying localizability and to provide a position operator.

In this paper the connection between field models and infinite-dimensional Lie groups is widely analysed on the bases of a new group quantization approach. We also relate the Poincare-Cartan form of variational calculus to the symplectic current/structure of the covariant phase-space formulation of (higher-derivative) field theory. The Virasoro and Kac-Moody groups are considered. In the first case the action functional of the 2D-induced gravity in the light-cone formulation is derived. The hidden SL(2, R) simply appears as generated by the kernel of the Lie algebra two-cocycle and plays the role of a gauge-type symmetry. Nevertheless, it is shown that a proper space-like formulation is out of reach of the Virasoro group. The corresponding symplectic structure of the (nonlocal)action functional is determined showing that it is related to the symplectic structure associated with the SL(2, R)-Kac-Moody group. This unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint-orbit approach. The relation between the Kac-Moody groups and the Chern-Simons gauge theory on a disc in the presence of a source is considered using the new approach.

We analyse the present status of Poincare group in considering it as a fundamental object independent of the Minkowski space. We examine the observables associated with it. After having introduced the notion of kinematical observable, we derive the class of 'deformed' Poincare groups which are compatible with physics.

We have analysed a generalized Heisenberg spin chain invariant under the quantum group SU_p,q(2). Instead of the usual periodic boundary condition, non-trivial boundary conditions are imposed a la Sklyanin(1988). R-matrix and Bethe eigenstates are constructed explicitly. The R-matrix describes a vertex type model which can be shown to be connected to the xxz type six vertex model under the Akutsu-Deguchi-Wadati-type symmetry-breaking transformation.

The q-deformed Krichever-Novikov algebra on higher genus Riemann surfaces is obtained by means of the operator product expansion method. The Ward q-identities for correlation functions of primary fields are derived. It is found that the Ward g-identities cannot determine the two-point correlation function.

We show how to construct a multi-mode operator which satisfies the quantum-Heisenberg-Weyl algebra (H-W_q algebra); we use this operator to create new two-mode realizations of the SU_q(2) and SU_q(1,1) quantum algebra. We also investigate the corresponding coherent states.

In this paper we construct a new quantum double by endowing the l-state boson algebra with a non-trivial Hopf algebra structure, which is not a q-deformation of the Lie algebra or superalgebra. The universal R-matrix for the Yang-Baxter equation associated with this new quantum group structure is obtained explicitly. By building the representations of this quantum double, we obtain some R-matrices which can result in new representations of the braid group.

The Green function for a relativistic particle interacting with a gravitational point source and a flux confined at the origin of the ( rho , phi )-space is evaluated using Feynman's summation-over-histories. The bound state energy spectrum is calculated when a uniform magnetic field is applied perpendicular to the ( rho , phi )-space.

The topology of the non-adiabatic parameter space bundle is discussed for evolution of exact cyclic state vectors in Berry's original example of split angular momentum eigenstates. It turns out that the change in topology occurs at a critical frequency. The first Chern number that classifies these bundles is proportional to angular momentum. The non-adiabatic principal bundle over the parameter space is not well defined at the critical frequency.

It is shown that for a Hamiltonian action of a compact Lie group on a symplectic manifold, a reduction scheme can be constructed based on a suitably normalized momentum map. Under the appropriate regularity assumptions it turns out that the reduced spaces are Poisson manifolds or, more precisely, canonical manifolds. As an application, the reduction of a particular class of non-Hamiltonian systems with symmetry is considered.

We discuss a family of variable-coefficient linear wave equations that are exactly solved by a generalization of progressing waves that spread out as they propagate. This can be viewed as a generalization to some variable-coefficient equations of the dispersion theory method that is standard for constant coefficient equations. The method is illustrated by solving some physically interesting equations.

The linearized relativistic Vlasov-Maxwell system for a hot inhomogeneous relativistic magnetized electron plasma is studied through particle orbit theory using the techniques of Fourier transform. An analytical integral expression in the (k, omega )-space is obtained for the current density for waves propagating across an externally applied uniform static magnetic field (k is the wavenumber and omega is the wave frequency). After applying inverse Fourier transform, differential equations for the electric field are obtained from the expression for the current density combined with Maxwell's equations. These fully relativistic equations are correct up to second order in r_c/L. where r_c is the electron gyroradius and L is the gradient length of the plasma inhomogeneity.

Classical and quantum dynamics of wave packets for spin- 1/2 particles are analysed in the presence of an electromagnetic field. An almost exact agreement is found between these two approaches, both in the strong and the weak coupling limits. The question of negative energy states is discussed. and it is shown that they are not produced in such an interaction. The divergence problem of the perturbation expansion of the particles wavefunction is discussed, the source of which is explained.

We give a direct and general proof of the equality of the energy levels of a pair of different quantum problems: the double well and the unstable anharmonic oscillator defined by complex translation. The 'resonances' of the two problems are also equal and are the analytic continuations of the anharmonic oscillator energy levels.

An experimental arrangement is proposed which combines the idea of two-photon interferometry with that of geometric phases. A pair of photons generated by parametric down-conversion is fed into a pair of Mach-Zehnder interferometers, in each of which there is phase shift produced by the helically wound optical fibre in the manner of Tomita and Chiao(1986). If the winding angle is a multiple of 2 pi there is a closed path in parameter space, in accordance with the pioneering analysis of Berry(1984). More general windings are also considered in which the path in parameter space is not closed, and the geometric phase for such open paths is calculated in accordance with the gauge-invariant prescription of Aitchison and Wanelik. If the proposed experiment is realized, it will provide a test for the noncyclic geometric phase for photons and for the set of prescriptions equivalent to that of Aitchison and Wanelik(1992). Furthermore, the geometric phase in this arrangement will be unequivocally quantum mechanical because of the non-classical character of two-photon interferometry.

Schrodinger energy-momentum tensor for a matter field interacting with an external electromagnetic field is represented in phase space by means of Terletsky distribution function. The explicit form of this distribution function in the case of a hydrogen-type atom is found. The structure of the Schrodinger energy-momentum tensor in phase space is compared with the structure of a real fluid and its local pressure and local energy are presented in the fundamental state of a hydrogen-type atom. The negative values of the Terletsky distribution function are not rejected in view of their participation in the phase space energy-momentum tensor. The negative part of the quantum distribution function could be interpreted as phase space density for anti-constituents in a hypothetical sub-quantum level of the matter.

Expressions for moments of the spectral density distribution of a system described by a non-symmetric N-particle Hamiltonian are derived. In particular, this kind of Hamiltonian may represent N coupled oscillators; then its spectrum models the vibrational spectrum of a molecule. It is assumed that the Hamiltonian is defined in a finite-dimensional space formed by products of orthogonal eigenstates of some one-body Hamiltonians. Possible applications of the formalism are briefly discussed.

We investigate the accidental degeneracy of resonances mixed by a Hermitian interaction. We give general expressions for the codimension in parameter space of any degeneracy of resonances. In the case of a degeneracy of two resonances which produces only one simple pole in the S matrix, the codimension is four for time reversal invariant interactions and six when the quantum system is not time reversal invariant. Close to this type of degeneracy the corresponding energy surfaces are two double cones lying in orthogonal subspaces with a common vertex at a double diabolic point. When the degeneracy of two resonances leads to one simple plus one double pole of S the codimension is two irrespective of the time reversal invariant character of the quantum system. Close to this type of degeneracy, the energy surfaces are a hyperbolic cone and a sphere which lie in orthogonal subspaces and touch one another at all points on a diabolic circle.

We show that many second-class constrained Hamiltonian systems can be reformulated as first-class systems within the same phase space.

We describe graded contractions of Casimir operators of Lie algebras. The formalism applies to any Casimir operator given as a symmetric form. We deal with the quadratic Casimir operator in exhaustive detail and indicate the modifications for operators of higher degree.

The above comment by Munoz (ibid., vol.26, p.4475, 1993) claims that a recent paper by Rattray and McKane (ibid., vol.24, p.4375, 1991) is incorrect. This is not so. To refute these claims we first expose the errors in Munoz's comment and then restate the basic elements of the procedure we used previously to calculate stationary probability distributions. There are neither errors nor 'puzzling features' in this approach when it is properly understood.

We discuss the supersymmetric structure of the Aharonov-Casher effect.