Table of contents

The formation of local denaturation zones (bubbles) in double-stranded DNA is an important example of conformational changes of biological macromolecules. We study the dynamics of bubble formation in terms of a Fokker–Planck equation for the probability density to find a bubble of size n base pairs at time t, on the basis of the free energy in the Poland–Scheraga model. Characteristic bubble closing and opening times can be determined from the corresponding first passage time problem, and are sensitive to the specific parameters entering the model. A multistate unzipping model with constant rates recently applied to DNA breathing dynamics (Altan-Bonnet et al 2003 Phys. Rev. Lett.90 138101) emerges as a limiting case.

It is commonly assumed that in statistical closures describing fluid or magnetohydrodynamic turbulence, the only ingredient needed for an understanding of the turbulent evolution is the energy spectrum. This assumption is shown to be valid only in very special cases. The key feature invalidating the unique significance of the energy spectrum is the existence of more than one non-vanishing quadratic invariant for a non-dissipative system, such as the presence of non-zero energy and helicity invariants. The latter plays a key role in the evolution of a turbulent MHD dynamo. These results have serious implications for the development of practical closures for inhomogeneous turbulence.

Recent work on stochastic interacting particle systems with two particle species (or single-species systems with kinematic constraints) has demonstrated the existence of spontaneous symmetry breaking, long-range order and phase coexistence in nonequilibrium steady states, even if translational invariance is not broken by defects or open boundaries. If both particle species are conserved, the temporal behaviour is largely unexplored, but first results of current work on the transition from the microscopic to the macroscopic scale yield exact coupled nonlinear hydrodynamic equations and indicate the emergence of novel types of shock waves which are collective excitations stabilized by the flow of microscopic fluctuations. We review the basic stationary and dynamic properties of these systems, highlighting the role of conservation laws and kinetic constraints for the hydrodynamic behaviour, the microscopic origin of domain wall (shock) stability and the coarsening dynamics of domains during phase separation.

The quantum Hall effects in all even dimensions are uniformly constructed. Contrary to some recent accounts in the literature, the existence of quantum Hall effects (QHE) does not crucially depend on the existence of division algebras. For QHE on flat space of even dimensions, both the Hamiltonians and the ground-state wavefunctions for a single particle are explicitly described. This explicit description immediately tells us that QHE on a higher even-dimensional flat space shares common features such as incompressibility with QHE on a plane.

In this work we solve the 19-vertex models with the use of algebraic Bethe ansatz for diagonal reflection matrices (Sklyanin K-matrices). The eigenvectors, eigenvalues and Bethe equations are given in a general form. Quantum spin chains of spin one derived from the 19-vertex models were also discussed.

After a short review of various ways to calculate the Maslov index appearing in semiclassical Gutzwiller type trace formulae, we discuss a coordinate-independent and canonically invariant formulation recently proposed by Sugita (2000 Phys. Lett. A 266 321, 2001 Ann. Phys., NY288 227). We give explicit formulae for its ingredients and test them numerically for periodic orbits in several Hamiltonian systems with mixed dynamics. We demonstrate how the Maslov indices and their ingredients can be useful in the classification of periodic orbits in complicated bifurcation scenarios, for instance in a novel sequence of seven orbits born out of a tangent bifurcation in the Hénon–Heiles system.

We provide a new proof, inspired by quantum field theory, of the Lagrange--Good multivariable inversion formula.

The kinematical and dynamical symmetries of equations describing the time evolution of quantum systems such as the supersymmetric harmonic oscillator in one space dimension and the interaction of a non-relativistic spin one-half particle in a constant magnetic field are reviewed from the point of view of the vector field prolongation method. Generators of supersymmetries are then introduced so that we get Lie superalgebras of symmetries and supersymmetries. This approach does not require the introduction of Grassmann-valued differential equations but a specific matrix realization and the concept of dynamical symmetry. The Jaynes–Cummings model and supersymmetric generalizations are then studied. We show how it is closely related to the preceding models. Lie algebras of symmetries and supersymmetries are also obtained.

In this paper we are concerned with rational solutions, algebraic solutions and associated special polynomials with these solutions for the third Painlevé equation (P_III). These rational and algebraic solutions of P_III are expressible in terms of special polynomials defined by second-order, bilinear differential-difference equations which are equivalent to Toda equations. The structure of the roots of these special polynomials is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Using the Hamiltonian theory for P_III, it is shown that these special polynomials satisfy pure difference equations, fourth-order, bilinear differential equations as well as differential-difference equations. Further, representations of the associated rational solutions in the form of determinants through Schur functions are given.

The coupled Kadomtsev–Petviashvili (cKP) equation possesses N-soliton solutions with more parametric freedom than the solitons of the usual KP equation. Its solutions can therefore be expected to model far more complex interactions than their KP counterparts. The existence of 'web'-like structures (on a finite scale) for cKP solutions (Isojima S, Willox R and Satsuma J 2002 J. Phys. A: Math. Gen.35 6893–6909) is a manifestation of this greater freedom. In this paper, we propose a new method to analyse the behaviour of solitons which we demonstrate in some examples. In addition we discuss 'essentially three-body collisions', described by a two-soliton solution of the cKP equation.

We investigate entanglement production in a class of quantum baker's maps. The dynamics of these maps is constructed using strings of qubits, providing a natural tensor-product structure for application of various entanglement measures. We find that, in general, the quantum baker's maps are good at generating entanglement, producing multipartite entanglement amongst the qubits close to that expected in random states. We investigate the evolution of several entanglement measures: the subsystem linear entropy, the concurrence to characterize entanglement between pairs of qubits and two proposals for a measure of multipartite entanglement. Also derived are some new analytical formulae describing the levels of entanglement expected in random pure states.

The operatorial theory of optical polarization devices can provide some of the clearest examples in which von Neumann's model of orthogonal measurement is violated. This paper analyses the global and spectral properties of some two- and three-layer optical polarizers of non-orthogonal kind by calculating the dyadic expressions of their operators in a Dirac-algebraic form. These constitute some of the most expressive examples of non-Hermitian operators corresponding to generalized observables and it is expected that the theory of generalized quantum measurement will take advantage of the simplicity of their physical realization.

For chaotic systems there is a theory for the decay of the survival probability, and for the parametric dependence of the local density of states. This theory leads to the distinction between 'perturbative' and 'non-perturbative' regimes, and to the observation that semiclassical tools are useful in the latter case. We discuss what is 'left' from this theory in the case of one-dimensional systems. We demonstrate that the remarkably accurate uniform semiclassical approximation captures the physics of all the different regimes, though it cannot take into account the effect of strong localization.