Table of contents

The error threshold transition in a stochastic (i.e. finite population) version of the quasispecies model of molecular evolution is studied using finite-size scaling. For the single-sharp-peak replication landscape, the deterministic model exhibits a first-order transition at , where Q is the probability of exact replication of a molecule of length , and a is the selective advantage of the master string. For sufficiently large population size, N, we show that in the critical region the characteristic time for the vanishing of the master strings from the population is described very well by the scaling assumption , where is an a-dependent scaling function.

We study the synchronization phenomena in systems of globally coupled oscillators, each possessing finite inertia, with particular attention to the noise effects. The self-consistency equation for the order parameter as well as the probability distribution is obtained from the Smoluchowski equation, and analyzed in the presence of thermal noise. It is found that the hysteresis present in the system without noise disappears as the thermal noise comes into the system. Numerical simulations are also performed to give results generally consistent with the analytical ones.

Recently, a phase transition to synchronized congested traffic has been observed in empirical highway data (Kerner B S and Rehborn H 1997 Phys. Rev. Lett.79 4030). This hysteretic transition has been described by a non-local, gas-kinetic-based traffic model (Helbing D and Treiber M 1998 Phys. Rev. Lett. 81 3042) that did not, however, display a wide scattering of synchronized states. Here, it is shown that the latter can be reproduced by a mixture of different vehicle types like cars and trucks. The simulation results are in good agreement with Dutch highway data.

Using finite-size scaling techniques we obtain accurate results for critical quantities of the Ising model and the site percolation, in three dimensions. We pay special attention to parametrizing the corrections-to-scaling, which is necessary to bring the systematic errors below the statistical ones.

This paper deals with dynamical systems depending on a slowly varying parameter. We present several physical examples illustrating memory effects, such as metastability and hysteresis, which frequently occur in these systems. The examples include the delayed appearance of convection rolls in Rayleigh-Bénard convection with slowly varying temperature gradient, scaling of hysteresis area for ferromagnets in a low-frequency magnetic field, and a pendulum on a rotating table displaying chaotic hysteresis. A mathematical theory is outlined, which allows us to prove the existence of hysteresis cycles, and determine related scaling laws.

We investigate the anisotropic integrable chain consisting of spins and s = 1. Depending on the anisotropy and the ratio of the coupling constants it has different (antiferromagnetic) ground states, manifesting themselves in a different string content in the Bethe ansatz framework. In this paper we continue the study of those regions with different string content, below called sectors. First we compare the dispersion relations for the sectors with infinite Fermi zones. Further we calculate the speeds of sound for regions close to sector borders, where the Fermi radii either vanish or diverge, and compare the results.

Energy eigenvalues and order parameters are calculated by exact diagonalization for the isotropic spin- model on square lattices of up to sites. The finite-size scaling behaviour is in excellent agreement with the effective Lagrangian predictions of Hasenfratz and Niedermayer (Hasenfratz P and Niedermayer F 1993 Z. Phys. B 92 91). Estimates are obtained for the bulk ground-state energy per site, the spontaneous magnetization, the spin-wave velocity and the spin-wave stiffness.

We consider the behaviour of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast-switching, random external field. After introducing a general formalism for describing such systems, we consider here the mean-field theory. A novel type of first-order phase transition related to spontaneous symmetry breaking and dynamic freezing is found. The nonequilibrium stationary state has a complex structure, which changes as a function of parameters from a singular-continuous distribution with Euclidean or fractal support to an absolutely continuous one. These transitions are reflected in both finite size effects and sample-to-sample fluctuations.

We consider the behaviour of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast-switching, random external field. In paper I, we introduced a general formalism for describing such systems and presented the mean-field theory. In this article we derive results for the one-dimensional case, which can be only partially solved. Monte Carlo simulations performed on a square lattice indicate that the main features of the mean-field theory survive the presence of strong fluctuations.

In this paper, an ultra-discrete version of Burger's equation, which includes the rule-184 CA model, is extended to treat a higher velocity. The extended model has multiple states at the transition region of car density from free to congested phase in the fundamental diagram. The state of free phase at high density is unstable under perturbation, and its stability is discussed in detail.

We explain the results recently obtained for the damage-spreading behaviour in the Bak-Sneppen (BS) model (Tamarit et al 1998 Eur. Phys. J. B 1 545). We do this by relating the BS model to a much simpler one, which includes many features of the BS model and provides a clear explanation for the occurrence of power-law growth of the distance.

Squeezed spin states are defined through canonical transformations analogous to the squeezed states for boson systems and variances of spin components are obtained. The defined concept of squeezed spin states has been applied to Heisenberg interaction for the spin system with S = 1.

For a system with one degree of freedom, coherent states that are parametrized by classical canonical action-angle variables are introduced. These states also possess continuity of labelling, a resolution of unity, and temporal stability. The insistence on canonical action-angle variables strongly restricts any remaining arbitrariness in the coherent state definition. Such states are introduced for semibounded Hamiltonian operators having either a discrete or a continuous spectrum. Hamiltonians that have both discrete and continuous parts in their spectrum are also discussed.

The eigenvalue bounds obtained earlier (Hall R L and Saad N 1998 J. Phys. A: Math. Gen. 31 963) for smooth transformations of the form are extended to N dimensions. In particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential , , , and is valid for all discrete eigenvalues, arbitrary angular momentum l and spatial dimension N.

Within the framework of the phase-space representation of quantum mechanics recently developed by Torres-Vega and Frederick we have solved for the exact solutions of the Schrödinger equation of a Morse oscillator whose structures reveal the special complexity.

A t-J model for correlated electrons with impurities is proposed. The impurities are introduced in such a way that integrability of the model in one dimension is not violated. The algebraic Bethe ansatz solution of the model is also given and it is shown that the Bethe states are highest weight states with respect to the supersymmetry algebra .

The left module structure of finite-dimensional quantum algebras is analysed using the theory of primitive idempotents. In particular, a complete structural result in terms of principal (projective) indecomposable modules (p.i.m.) is given in the case of (q a root of 1) by finding a complete set of primitive idempotents. The structure of p.i.m. is analysed in detail. The Jacobson radical of the algebra is investigated and its significance in the study of nonsemisimple symmetries in physical systems is discussed.

The tensor product of two modules, q a root of 1, is decomposed into indecomposable summands for both irreducible and indecomposable modules. Clebsch-Gordan coefficients in the general case are computed. An apparently new identity is derived and some possible applications are conjectured.

We propose a theory of deterministic chaos for discrete systems, based on their representations in binary state spaces , homeomorphic to the space of symbolic dynamics. This formalism is applied to neural networks and cellular automata; it is found that such systems cannot be viewed as chaotic when one uses the Hamming distance as the metric for the space. On the other hand, neural networks with memory can in principle provide examples of discrete chaos; numerical simulations show that the orbits on the attractor present topological transitivity and a dimensional phase space reduction. We compute this by extending the methodology of Grassberger and Procaccia to . As an example, we consider an asymmetric neural network model with memory which has an attractor of dimension for N = 49.