Table of contents

A dimerized quantum Heisenberg or XY antiferromagnetic chain has a gap in the spectrum. We show that a weak incommensurate modulation around a dimerized chain produces a zero-temperature quantum critical point. As the incommensuration wavelength is varied, there is a transition to a modulated gapless state. The critical behaviour is in the universality class of the classical commensurate - incommensurate (Pokrovsky - Talapov) transition. An analogous metal - insulator transition can also take place for an incommensurate chain.

Yangian Y(sl(2)) constructed by fermionic field operators is extended to the q-deformed case which can be realized in the massless Thirring model. This leads to local U(1) gauge-invariance of Y(sl(2)) and evolution of the double-time Green function in q-statistics.

Motivated by recent experiments on large quantum dots, we consider the energy spectrum in a system consisting of N particles distributed among K<N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it. On a large scale, the ground-state energy E(N) of such a system grows quadratically with N, but in general there is no simple relation such as . The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) is a fluctuating quantity. Regarding the numbers as values assumed by a certain random variable , we obtain a closed-form expression for its distribution . Its main feature is that the corresponding density has a maximum at the point . As the density is Poissonian, namely, .

The role of the passing mechanism in traffic flows is examined. Specifically, we consider passing rates that are proportional to the difference between the velocities of the passing car and the passed car. From a Boltzmann equation approach, steady-state properties of the flow such as the flux, average cluster size, and velocity distributions are found analytically. We show that a single dimensionless parameter determines the nature of the flow and helps distinguish between dilute and dense flows. For dilute flows, perturbation expressions are obtained, while for dense flows a boundary layer analysis is carried out. In the latter case, extremal properties of the initial velocity distribution underly the leading scaling asymptotic behaviour. For dense flows, the stationary velocity distribution exhibits a rich `triple-deck ' boundary layer structure. Furthermore, in this regime fluctuations in the flux may become extremely large.

The universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems, and proved in some particular cases, as the Hermitian one-matrix model with polynomial potential. Here, we consider the more difficult case of a unidimensional chain of Hermitian matrices with first-neighbour couplings and polynomial potentials.

An asymptotic expression of the orthogonal polynomials and a generalization of the Darboux-Christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain.

Eventually, we consider the limit of the infinite chain of matrices, which can be interpreted as a time-dependent Hermitian one-matrix model, and give the correlation functions of eigenvalues at different times.

Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields and ) of the three-dimensional field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N-vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined.

The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values.

Finally, because an error has been discovered in the last order of the published expansions (order ), we have also re-analysed the determination of exponents from the -expansion.

The conclusion is that the general agreement between -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.

We rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of Young tableaux and integer partitions. In this way we provide a proof of a result, previously only conjectured, for the total number of stars.

We study self-avoiding and neighbour-avoiding walks and lattice trails on two semiregular lattices, the lattice and the lattice. For the lattice we find the exact connective constant for both self-avoiding walks, neighbour-avoiding walks and trails. For the lattice we generate long series which permit the accurate estimation of the connective constant for self-avoiding walks and trails.

Extensive Monte Carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three-dimensional percolation on the simple cubic (s.c.), face-centred cubic (f.c.c.), and body-centred cubic (b.c.c.) lattices. Systems with were studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The excess number of clusters per unit length was confirmed to be a universal quantity with a value . Likewise, the critical crossing probability in the direction, with periodic boundary conditions in the plane, was found to follow a universal exponential decay as a function of for large r. Simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f.c.c. and b.c.c. lattices, yielding , . We also report the value for site percolation.

Recently, Santos obtained a generalized entropy using four assumptions which stated that an entropy must: (i) be a continuous function of the probabilities ; (ii) be a monotonic increasing function of the number of states W, in the case of equiprobability; (iii) satisfy (where A and B are two independent systems) and (iv) satisfy the relation , where ( and ). Santos showed that the only function which satisfies all of these properties is the generalized Tsallis entropy. In this paper we perform a similar analysis and we obtain a family of entropies which are equivalent to the Tsallis entropy. We also discuss the Shannon inequality in the context of the generalized Tsallis entropy.

We investigate the energy landscape of two-dimensional network models for covalent glasses by means of the lid algorithm. For three different particle densities and for a range of network sizes, we exhaustively analyse many configuration space regions enclosing deep-lying energy minima. We extract the local densities of states and of minima, and the number of states and minima accessible below a certain energy barrier, the `lid'. These quantities show on average a close to exponential growth as a function of their respective arguments. We calculate the configurational entropy for these pockets of states and find that the excess specific heat exhibits a peak at a critical temperature associated with the exponential growth in the local density of states, a feature of the specific heat also observed in real glasses at the glass transition.

A constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed. The method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high-order ordinary differential equation. Using this method new non-Lie ansätze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities, which describe real processes in physics, chemistry, and biology. The analysis of the found solutions and the relation of the proposed method to some approaches, which have been suggested in several recently published papers, are presented.

We analyse quantal Brownian motion in d dimensions using the unified model for diffusion localization and dissipation, and Feynman-Vernon formalism. At high temperatures the propagator possesses a Markovian property and we can write down an equivalent master equation. Unlike the case of the Zwanzig-Caldeira-Leggett model, genuine quantum mechanical effects manifest themselves due to the disordered nature of the environment. Using Wigner's picture of the dynamics we distinguish between two different mechanisms for destruction of coherence: scattering perturbative mechanism and smearing non-perturbative mechanism. The analysis of dephasing is extended to the low-temperature regime by using a semiclassical strategy. Various results are derived for ballistic, chaotic, diffusive, both ergodic and non-ergodic motion. We also analyse loss of coherence at the limit of zero temperature and clarify the limitations of the semiclassical approach. The condition for having coherent effect due to scattering by low-frequency fluctuations is also pointed out. It is interesting that the dephasing rate can be either larger or smaller than the dissipation rate, depending on the physical circumstances.

The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymptotically free propagation (the only possible regime for the RTC), CTC allows bound-state regimes, various intermediate regimes when one (or several) group(s) of particles form bound state(s), singular and degenerate solutions. These results can be used, for example, in describing the N-soliton train interactions of the nonlinear Schrödinger equation. Explicit expressions for the solutions in terms of minimal sets of scattering data are proposed for all classical series -.

We introduce a geometrical framework for the description of constrained mechanical systems, and we analyse different kinds of symmetries and their relationships. We propose a new definition for non-holonomic Lagrangian mechanical systems, and we give a geometrical characterization for the Helmholtz conditions related to the inverse problem.

Induced representations of from with are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients of with up to a normalization factor are derived by using the linear equation method. Weyl tableaux for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining Clebsch-Gordan coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebras is proposed. Some isoscalar factors of for the resulting irrep with are tabulated.

A set of orthonormalized and complete functions of two real variables in complex representation involving Laguerre polynomials as a substantial part is introduced and is referred to as the set of Laguerre two-dimensional (2D) functions. The properties of the set of Laguerre 2D-functions are discussed and the Fourier and Radon transforms of these functions are calculated. The Laguerre 2D-functions form a basis for a realization of the five-dimensional Lie algebra to the Heisenberg-Weyl group for a two-mode system. Real representation of this set of functions by a sum over products of Hermite functions involving Jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between Laguerre and Hermite polynomials in both directions. The set of Laguerre 2D-functions is the most appropriate set of functions for the Fock-state representation of quasi-probabilities in quantum optics. The Wigner quasi-probability in Fock-state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding Laguerre 2D-function. The properties of orthonormality and completeness of the Laguerre 2D-functions provide the Fock-state matrix elements of the density operator directly from the quasi-probabilities. The Perina-Mista representation of the Glauber-Sudarshan quasi-probability can be represented with advantage by the Laguerre 2D-functions. The Fock-state matrix elements of the displacement operator and the scalar product of displaced Fock states are closely related to Laguerre 2D-functions.