Table of contents

Using a technique based on self-similar random processes, we construct unitary representations of the group of diffeomorphisms of R describing continuum quantum systems with infinitely many degrees of freedom. These can describe systems undergoing a phase change, from rarefied to condensed, at the critical value of a correlation parameter. The method offers promise for generalization to higher spatial dimensions, and for application to theories of extended quantum objects.

The steady-state behaviour of the irreversible reaction A+B to C with separated reactants in one dimension is studied, when currents J of A and B particles are directed towards each other by means of an analytical approach. The interparticle distance distribution in the reaction front is derived. Fixing the reaction constant, and for large J, the process is limited by reaction and the average interparticle distance at the reaction front (l_AB) behaves as l_AB approximately J^-1. In the limit J to 0 the process is limited by diffusion and the behaviour l_AB approximately J^-1/2 holds. The crossover between both regimes is well described analytically and the crossover current is found to be J_c=k²/(pi D). These results are in excellent agreement with Monte Carlo simulation results.

For all but the simplest open quantum systems, quantum trajectory Monte Carlo methods, including quantum jump and quantum state diffusion (QSD) methods, have besides their intuitive insight into the measurement process a numerical advantage over direct solutions for the density matrix, especially where many degrees of freedom are involved. For QSD the trajectories are continuous, and often localize to small near-minimum uncertainty wave packets which follow approximately classical paths in phase space. The mixed representation discussed here takes advantage of this localization to reduce computing space and time by a further significant factor, using a quantum oscillator representation that follows a classical path. The classical part of this representation describes the time evolution of the expectation values of position and momentum in classical phase space, while the quantal part determines the degree of localization of the quantum mechanical state around this phase space point. The method can be applied whether or not the localization is produced by a measuring apparatus.

The quadrupolar glass model with p-quadrupole interactions, exactly solvable in the limit p to infinity is presented. It has been shown that for p to infinity the energy levels of the system are statistically independent, which gives a quadrupolar counterpart of the spin random energy model. Using the Parisi scheme of replica symmetry breaking it has been calculated that the quadrupolar glass parameter function is a step function for p to infinity as well as for large p.

The three-state Potts antiferromagnetic model on the triangular lattice (3PAFT) has a weak first-order transition in the pure case. The first-order nature follows from analytic arguments and while it has been observed in some simulations, the precision of (and internal disagreement between) these observations has left much to be desired. The 3PAFT is the only two-dimensional nearest-neighbour Potts model with an ordered phase that does not have an infinite degeneracy, that remains without an exact value for its critical point. This model is expected to exhibit interesting glassy behaviour and a possible crossover to a second-order transition upon dilution. A careful characterization of the nature of the pure transition is needed in order to explore the physics of the dilute model. We find clear indication of a first-order transition at T_t=0.62731+or-0.00006, using a cluster method that can be extended to the dilute case and histogram reweighting analysis. Our estimate is two orders of magnitude more precise than previous simulations and falls within the error bounds of a reanalysis of existing series expansion (T_t=0.628+or-0.004) reported on herein.

We determine the eigenvalues of the transfer matrices for integrable open quantum spin chains which are associated with the affine Lie algebras A_2n-1⁽²⁾, B_n⁽¹⁾, C_n⁽¹⁾, D_n⁽¹⁾, and which have the quantum-algebra invariance U_q(C_n), U_q(B_n), U_q(C_n), U_q(D_n), respectively.

We consider oriented self-avoiding walks on the square lattice with different energies between steps that are oriented parallel or antiparallel across a face of the lattice. Rigorous bounds on the free energy and exact enumeration data are used to study the statistical mechanics of this model. We conjecture a phase diagram in the parallel-antiparallel interaction plane, and discuss the order of the associated phase transitions. The question, raised by previous field theoretical considerations, of the existence of an exponent that varies continuously with the energy of interaction is discussed at length. In connection with this we have also studied two oriented walks fixed at a common origin; this being the simplest model of branched oriented polymers in two dimensions. The evidence, although not conclusive, tends to support the field theoretic prediction.

The pair annihilation of identical particles initially distributed at random, and interacting by a tunnelling law is studied. Monte Carlo simulations are performed in one and two dimensions to measure the density and the two-particle correlations which show a tendency to local ordering of the surviving particles. This model is shown to be equivalent to a simpler model that we can solve on a one-dimensional lattice. In this case, the self-ordering property can be proved and characterized. This analysis is then naturally extended to higher dimensionalities.

We have introduced a cellular automaton to investigate self-organized criticality in the activity of neural populations. The model is composed of pulse-coupled integrate-and-fire neurons and stimulated by continuous driving. Under an appropriate condition, the system is found to exhibit a robust self-organized critical behaviour accompanied with the large-scale synchronized activities among the units. It indicates the close relationship between self-organized criticality and synchronization.

We study the critical properties of the weakly disordered p-component ferromagnet in terms of the renormalization group (RG) theory generalized to take into account the replica symmetry breaking (RSB) effects coming from the multiple local minima solutions of the mean-field equations. Recently it has been shown that for p<4 the traditional RG flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to the RSB potentials as found in spin glasses, and a new type of stable one-step RSB fixed point has been discovered. Here it is demonstrated that for a general type of the Parisi RSB structures there exist no stable fixed points, and the RG flows lead to the strong-coupling regime at the finite scale R_* approximately exp(1/u), where u is the small parameter describing the disorder. The physical consequences of the obtained RG solutions are discussed. In particular, we argue that discovered RSB strong-coupling phenomena indicate the onset of a new spin-glass-type critical behaviour in the temperature interval tau < tau _* approximately exp(-1/u) near Tr. The possible relevance of the considered RSB effects for the Griffith phase is also discussed.

We have recently interpreted 2D quasi-periodic patterns in terms of substitutions. In this paper we extend this interpretation to 3D. In particular we describe Danzer tilings in terms of word sequences of L systems.

We derive a generating function for the second moment of the distinct number of sites visited by an n-step lattice random walk. The formalism allows us to find asymptotic forms for the second moment as previously given in the mathematical literature using much more complicated techniques. The general technique can, in principle, be utilized in a derivation of higher moments.

We study the coagulation (A+A to A) and annihilation (A+A to 0) reactions with input probability epsilon and reaction probability p in a one-dimensional lattice. In the steady state we find two different behaviours for the density of nearest-neighbour occupied sites Gamma against the density of particles rho . These behaviours correspond to the diffusion-limited regime ( rho to 0) and to the reaction-limited regime ( rho approximately 1). Using a scaling ansatz for Gamma against rho we derive an approximation for rho as a function of epsilon and p that agrees well with Monte Carlo numerical results.

Using exact results, we determine the complex-temperature phase diagrams of the 2D Ising model on three regular heteropolygonal lattices, (3.6.3.6) (kagome), (3.12²) and (4.8²) (bathroom tile), where the notation denotes the regular n-sided polygons adjacent to each vertex. We also work out the exact complex-temperature singularities of the spontaneous magnetization. A comparison with the properties on the square, triangular, and hexagonal lattices is given. In particular, we find the first case where, even for isotropic spin-spin exchange couplings, the non-trivial non-analyticities of the free energy of the Ising model lie in a two-dimensional, rather than one-dimensional, algebraic variety in the z=e^-2K plane.

We study the complex spatiotemporal behaviour of a coupled map lattice with a one-humped chaotic map and an unstable Laplacian coupling. Bifurcations are numerically investigated and interpreted using low-dimensional approximations corresponding to the relevant degrees of freedom of the infinite-dimensional system. Varying the control parameter we find different phases in the chaotic domain such as localized chaos or propagating chaos, spatiotemporal intermittency and transient chaos. According to our results, unstable coupling leads to a number of new features, including the effects that (i) the first bifurcation becomes discontinuous and (ii) the chaotic regime sets in sooner than for a single map.

We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e. the fully-frustrated model on an infinite-dimensional hypercube and the so-called sine model. We use the mean-field (or TAP) equations which we derive by resumming the high-temperature expansion of the Gibbs free energy. In some special non-random cases, we can find the absolute minimum of the free energy. For the random case we compute the average number of solutions to the TAP equations. We find that the configurational entropy (or complexity) is extensive in the range T_RSB<T<T_M. We also present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of TAP solutions.

The non-stationary quadratic quantum system which can be considered as a quantum model of a damped oscillator is investigated in the framework of the Wigner representation. The explicit expressions of the ordinary and smoothed Wigner functions for this system are obtained.

In this article we make a new connection between the linear wave equation and the linear heat equation. In this way we are able to construct new solutions of the linear wave equation, using symmetries and conditional symmetries of the heat equation.

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e. the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.

We review the existing evidence on the (non)integrability of the mixmaster universe model. We show how a local Painleve analysis can be used to study the possible existence of essential singularities. In agreement with recent studies, we find that the mixmaster model possesses critical essential singularities, the multivalued character of which is incompatible with integrability. Our analysis is complemented by a numerical study of the spectrum of the stretching numbers of the system. We show that the zero-energy case is characterized by a vanishing maximal Lyapunov characteristic number.

Curved quantum waveguides are known to bind particles. We show that a number of charged fermions in such a trap can be tuned by an external electrostatic field; if the latter is slowly increased, the bent duct can serve as a single particle ejector up to the spin degeneracy.

Some aspects of the bound state and scattering properties of a quantum mechanical particle in an arbitrary scalar N-prong potential are considered. Such a study is relevant in applications to mesoscopic devices. Multi-prong potentials are in some way intermediate between one- and two-dimensional systems. In contrast to the one-dimensional situation, where there is no degeneracy, the energy levels for the case of N identical prongs exhibit an alternating pattern of non-degeneracy and (N-1)-fold degeneracy. We generalize the techniques of supersymmetric quantum mechanics to multi-prong systems and generate solutions to new N-prong systems. Solutions for prongs of arbitrary lengths are developed. Scattering on piecewise constant potentials and tunnelling in N-well potentials are discussed in detail. Since our treatment is for general values of N, the results can be studied in the large N-limit. A somewhat surprising result is that a free particle incident on an N-prong vertex undergoes continuously increased backscattering as the number of prongs is increased.

The importance of non-local symmetries of differential equations lies in their manifestation as Lie point symmetries of the equations resulting from reduction of order. The reason for the determination of these symmetries in second-order equations with only one Lie point symmetry is self-evident. However, the disadvantage of non-local symmetries is that no systematic approach to their determination exists. We present such an approach (applicable to differential equations of any order) and apply it to some second-order ordinary differential equations and show that they have a rich occurrence. We also look at possible generalizations of the concept of non-local symmetries.

It is shown that all the information yielded by the reduction methods of Bluman and Cole (1974) and Clarkson and Kruskal (1989) can be obtained using the singular manifold expansion. The Burgers' equation, modified Korteweg-de Vries equation, Caudrey-Dodd-Gibbon equation and the Fitzhugh-Nagumo equation are used as illustrative examples. Several new exact solutions are presented.

Classical scattering in the Liouville field theory (LFT) is essentially finite-dimensional, and in some cases the classical S-matrix can be represented as a transformation of the Poisson group SL(2,R). Motivated by this and using conjectural quantum analogues of some ingredients of the classical model, we find an exact quantum S-matrix without constructing the quantum LFT in full. The quantum S-matrix is explicitly represented as a transformation of the quantum group SL_q(2, R), and for a particular implementation, the S-matrix is shown to be unitary (unitarily generated).

We investigate a class of localized stationary numerical solutions to the Maxwell-Dirac system of classical nonlinear field equations in 3+l dimensions. The solutions are discrete energy eigenstates bound predominantly by the self-produced electric field.

Solutions of the inhomogeneous wave equation propagating into space-like wedge regions are used to construct solutions of the Poisson equation for arbitrary non-localized locally integrable inhomogeneities. This, for instance, allows for a general proof of existence of standard gauges for the classical electromagnetic field in four-dimensional spacetime.

Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wavepackets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle (1995), we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second-harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.

We prove that quantum fluctuations can suppress structural phase transitions. We give a rigorous proof for a one-component (R¹) quantum crystal with local double-well anharmonism under the condition that the masses of the atoms in the lattice sites of Z^d (d>or=3) are light enough.

Three neural network training algorithms are presented which are robust to nonlearnable problems. The first algorithm converges to the Gardner stability limit if the learning problem is linearly separable, and otherwise finds a locally maximally stable solution. The second algorithm is a robust version of Rosenblatt's perceptron learning algorithm which will converge to a solution of the learning problem if one exists, and otherwise will converge locally to a solution with a certain fraction of wrongly mapped patterns. The third algorithm is suited most favourably to unlearnable problems: it will always find a solution if the problem is learnable and otherwise it locally maximizes the number of patterns which are stored correctly. The error rate of this algorithm and other known algorithms for unlearnable problems are compared for two benchmark problems. Proofs of the existence of solutions are given. Convergence is proven as well to be global in the case of learnable and local in unlearnable cases.