Table of contents

We study a cyclic Lotka - Volterra model of N interacting species populating a d-dimensional lattice. In the realm of a Kirkwood approximation, a critical number of species above which the system fixates is determined analytically. We find in dimensions d = 1,2,3, in remarkably good agreement with simulation results in two dimensions.

For confining potentials of the form , where p(r) decays rapidly and is smooth for r>0, it is proved that q(r) can be uniquely recovered from the data . Here are energies of bound states and are the values , where are the normalized eigenfunctions, . An algorithm is given for finding q(r) from the knowledge of few first data, corresponding to assuming that the rest of the data are the same as for .

We present a soluble variant of the replicator model well established in theoretical biology and game theory. By using methods of statistical physics we derive an analytical solution to our model which becomes exact in the long time limit. We apply our model to the iterated prisoner's dilemma game and compare our results to numerical simulations.

The Hamiltonian dynamics of the classical model on a two-dimensional square lattice is investigated by means of numerical simulations. The macroscopic observables are computed as time averages. The results clearly reveal the presence of the continuous phase transition at a finite energy density and are consistent both qualitatively and quantitatively with the predictions of equilibrium statistical mechanics. The Hamiltonian microscopic dynamics also exhibits critical slowing down close to the transition. Moreover, the relationship between chaos and the phase transition is considered, and interpreted in the light of a geometrization of dynamics.

A universal model of an n-stage combined Carnot cycle system is established. Several major irreversibilities which often exist in real thermodynamic cycles, such as finite-rate heat transfer in the heat-exchange processes, heat leak losses of the heat source, and internal dissipation of the working fluid, are included in the model so that many models of irreversible and endoreversible Carnot cycles which appear in the literature can be regarded as special cases of the universal cycle model. The efficiency, power output and rate of heat input are optimized. Some characteristic curves of the cycle system are presented. Some important performance bounds are given. The optimal combined conditions between two adjacent cycles in the combined cycle system are determined. The optimal performance of an arbitrary-stage irreversible, endoreversible, and reversible combined Carnot cycle system can be directly derived for specific choices of some parameters. The results obtained here are of general significance for both physics and engineering.

On the basis of a representation in terms of photon-number states we derive an analytically solvable set of ordinary differential equations for the matrix elements of the density operator belonging to the Jaynes-Cummings model. We allow for atomic detuning, spontaneous emission, and cavity damping, but we do not take into account the presence of thermal photons. The exact results are employed to perform a careful investigation of the evolution in time of atomic inversion and von Neumann entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in a coherent state. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. In the undamped case the measure is found to increase during the first few revivals of Rabi oscillations. Hence, the influence of the surroundings on the atom does not decrease monotonically from time zero onwards. A further non-Markovian feature of the dynamics is given by the strong dependence of our measure on the initial atomic state, even for times at which damping brings about irreversible decay. For weak damping and high initial energy density the atomic evolution exhibits a crossover between quasireversible revival dynamics and irreversible Markovian decay. During this stage the state of maximum entropy acts as an attractor for the trajectories in atomic phase space. Subsequently, all trajectories follow a unique route to the atomic ground state, for which the off-diagonals of the atomic density matrix equal zero. From our entropy studies one learns what kind of difficulties must be overcome in establishing formulae for entropy production, the use of which is not limited to semigroup-induced dynamics.

Given a sequence of N positive real numbers , the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized. In the case in which the 's are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite-range, random antiferromagnetic Ising model. We employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large N limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips, and found that it vanishes like .

We locate the critical probability of bond percolation on two-dimensional random lattices as . Because of the symmetry with respect to permutation of the two axes for random lattices, we expect that for an aspect ratio of unity and sufficiently large lattices, the probability of horizontal spanning equals the probability of vertical spanning. This is confirmed by our Monte Carlo simulations. We show that the ideas of universal scaling functions and non-universal metric factors can be extended to random lattices by studying the existence probability and the percolation probability P on finite square, planar triangular, and random lattices with periodic boundary conditions using a histogram Monte Carlo method. Our results also indicate that the metric factors may be the same between random lattices and planar triangular lattices provided that the aspect ratios are 1 and .

A closed analytical expression is derived for the joint distribution function of the real and the imaginary parts of the eigenenergies of the operator for the one-channel case, where is taken from the Poissonian or one of the Gaussian ensembles with universality index , and where the squared moduli of the components of W are assumed to be -distributed with universality index . In the strong coupling limit and for the special case the joint distribution function of the real parts of the eigenvalues of H becomes identical with the joint energy distribution function of the eigenvalues of .

The site percolation problem is studied on d-dimensional generalizations of the Kagomé lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d. The site percolation thresholds are calculated numerically for d=3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: instead of . The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagomé lattice. The return probability of a random walker on these lattices is also shown to scale as . For bond percolation on d-dimensional diamond lattices these results imply .

Formulae for the matrix elements of in the Morse oscillator basis are derived in the p-representation. Explicit expressions are given for k = 1,2,3 and 4.

A shifted-l expansion technique is introduced to calculate the energy eigenvalues for the Klein-Gordon (KG) equation with Lorentz vector and/or Lorentz scalar potentials. Although it applies to any spherically symmetric potential, those that include Coulomb-like terms are only considered. Exact eigenvalues for a Lorentz vector or a Lorentz scalar, and an equally mixed Lorentz vector and Lorentz scalar coulombic potentials are reproduced. Highly accurate and rapidly converging ground-state energies for Lorentz vector Coulomb with a Lorentz vector or a Lorentz scalar linear potential, , and and , respectively, are obtained. Moreover, a simple straightforward closed-form solution for a KG-particle in coulombic Lorentz vector and Lorentz scalar potentials is presented.

The scalar Yukawa (or Wick-Cutkosky) model, in which complex scalar fields, and , interact via a real scalar field, , is reformulated by using covariant Green functions. It is shown that exact few particle eigenstates of the resulting truncated quantum field theory Hamiltonian can be obtained in the Feshbach-Villars formulation. Analytic solutions for the arbitrary mass two-body case are obtained for massless chion exchange in (3 + 1) dimensions. The binding energy is found to increase more rapidly with strength of coupling than in the case of corresponding results obtained using the regular and light-cone ladder Bethe-Salpeter approximations.

Diffraction of atoms by a particular absorbing `crystal of light', that has been studied experimentally, is described by the complex potential . The diffracted beam intensities can be calculated exactly, as a function of (dimensionless) potential strength , angle of incidence , and crystal thickness . Only the beams have nonzero intensity; this `lop-sidedness' is a dramatic violation of Friedel's law. The nth beam is strong at Bragg angles , and the peaks get sharper with increasing ; they correspond to degeneracies of the (non-Hermitian) governing matrix. For normal incidence , the are periodic in , and as n increases they approach a self-similar function of .

It is demonstrated that several series of conformal field theories, while satisfying braid group statistics, can still be described in the conventional setting of the DHR theory, i.e. their superselection structure can be understood in terms of a compact DHR gauge group. Besides theories with only simple sectors, these include (the untwisted part of) c = 1 orbifold theories and level-two WZW theories. We also analyse the relation between these models and theories of complex free fermions.

Quantum Hall effect wavefunctions corresponding to the filling factors , are shown to form a basis of irreducible cyclic representations of the quantum algebra at . Thus, the wavefunctions possessing filling factors where Q is odd and P,Q are relatively prime integers are classified in terms of

Starting from the non-relativistic field theory of spin- fermions interacting through the Abelian Chern-Simons term, we show that the quantized field theory leads, in the two-particle sector, to a two-particle Aharonov-Bohm-like Schrödinger equation with an antisymmetric (fermionic) wavefunction and without a delta function term. Calculating perturbatively the field-theoretic two-particle scattering amplitude up to one-loop order, we show that, in contrast to the scalar theory, the contribution of all the one-loop diagrams is finite and null, and that of the tree level ones coincides with the exact amplitude. Further, the Pauli matter-magnetic field interaction term is shown not to contribute to the amplitude to this order.

Starting from the q-discrete form of the Painlevé VI equation we obtain its degenerate forms by applying the procedure of coalescence of singularities. The whole cascade of degenerate forms is thus obtained leading to new forms for the discrete Painlevé IV and V equations. The Lax pairs of these discrete Painlevé equations are explicitly constructed, thus confirming their integrability.

q-analogues of Stirling number identities are formulated, and the interconsistency among the q-analogues of the Stirling numbers and of the binomial coefficients is investigated. The close relation with the normal ordering problem for Arik-Coon-type q-bosons plays a central role in the derivations presented.

This paper is the first of two papers devoted to the study of the Clebsch-Gordan (CG) problem for the three-dimensional Lorentz group in an elliptic (or SO(2)) basis. Here we describe the reduction of the tensor product of two unitary irreducible representations (UIRs) of the continuous series, i.e. belonging to either the principal or complementary series. The corresponding CG coefficients are defined as matrix elements of an intertwining operator between the tensor product representation and the irreducible component appearing in the decomposition. We then obtain an expression for CG coefficients in terms of a single function, namely in terms of the bilateral series with unit argument defined in the complex space of the variable . In the general case the functions are expressed in terms of two hypergeometric functions with unit argument; however, it reduces to the single function if at least one of the coupling UIRs belong to a discrete series. We derive a completeness relation for CG coefficients for all the cases under consideration.

An elastica has been recently quantized with the Bernoulli-Euler functional in two-dimensional space using the modified Korteweg-de Vries hierarchy. In this paper a Willmore surface is quantized, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, it is shown that the density of states of the partition function for the quantized Willmore surface is expressed by the volume of a subspace of the moduli of the MNV equation.

Recently there has been a revival of interest in gauge theories with disconnected compact gauge groups. Gauge fields such that this disconnectedness has non-trivial physical effects are called Alice configurations. The question of the existence of such configurations is surprisingly intricate, and no complete theory is known. Here we give some simple techniques for establishing the existence of Alice configurations, with an emphasis on practical, readily verifiable conditions requiring a minimum of topological information on the base manifold and the gauge group.

The irreducible representations of all of the 80 diperiodic groups, being the symmetries of the systems translationally periodical in two directions, are calculated. To this end, each of these groups is factorized as the product of a generalized translational group and an axial point group. The results are presented in the form of the tables, containing the matrices of the irreducible representations of the generators of the groups. General properties and some physical applications (degeneracy and topology of the energy bands, selection rules, etc) are discussed.

In this paper, we apply the symplectic integration method using jolt factorization described in an earlier paper to the symplectic map describing the nonlinear pendulum Hamiltonian. We compare results obtained with this method with those obtained using non-symplectic methods and demonstrate that our results are much better.

The fidelity for two displaced squeezed thermal states is computed using the fact that the corresponding density operators belong to the oscillator semigroup. A novel calculation technique is developed for the computation of the traces of product of Gaussian density operators and of square roots of Gaussian density operators. The method is exemplified for the one-mode case but it is also applicable to the multimode case.

A class of dynamical systems of the 2-torus is considered. These systems have the form of a skew product between the Bernoulli endomorphism , , defined on the 1-torus and a translation on itself. Symbolic dynamics techniques allow one to single out wide classes of observables which show an exponential decay of correlations. For some observables the rate of correlation decay can be explicitly estimated.