Table of contents

We consider a particle diffusing in the y-direction, dy/dt = η(t), subject to a transverse shear flow in the x-direction, dx/dt = f(y), where x ⩾ 0 and x = 0 is an absorbing boundary. We treat the class of models defined by f(y) = ±v_±(±y)^α where the upper (lower) sign refers to y > 0 (y < 0). We show that the particle survives with probability Q(t) ∼ t^−θ with θ = 1/4, independent of α, if v₊ = v₋. If v₊ ≠ v₋, however, we show that θ depends on both α and the ratio v₊/v₋, and we determine this dependence.

We establish a family of point-like impurities which preserve the quantum integrability of the nonlinear Schrödinger model in 1+1 spacetime dimensions. We briefly describe the construction of the exact second quantized solution of this model in terms of an appropriate reflection–transmission algebra. The basic physical properties of the solution, including the spacetime symmetry of the bulk scattering matrix, are also discussed.

The stability of photon trajectories in models of the universe that have constant spatial curvature is determined by the sign of the curvature: they are exponentially unstable if the curvature is negative and stable if it is positive or zero. We demonstrate that random fluctuations in the curvature provide an additional stabilizing mechanism. This mechanism is analogous to the one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean curvature is negative it is capable of stabilizing the photon trajectories; when the mean curvature is zero or positive it determines the characteristic frequency with which neighbouring trajectories oscillate about each other. In constant negative curvature models of the universe that have compact topology, exponential instability implies chaos (e.g. mixing) in the photon dynamics. We discuss some consequences of stochastic stabilization in this context.

We define an integrable lattice model which, in the notation of Yang, in addition to the conventional two-particle R-matrices also contains non-reducible three-particle R-matrices. The corresponding modified Yang--Baxter equations are solved, and an expression for the transfer matrix is found as a normal-ordered exponential of a (non-local) Hamiltonian.

We introduce a general class of statistical-mechanics models, taking values in an Abelian group, which includes examples of both spin and gauge models, both ordered and disordered. The model is described by a set of 'variables' and a set of 'interactions'. Each interaction is associated with a linear combination of variables; these are summarized in a matrix J. A Gibbs factor is associated with each variable (one-body term) and with each interaction. Then we introduce a duality transformation for systems in this class. The duality exchanges the Abelian group with its dual, the Gibbs factors with their Fourier transforms and the interactions with the variables. High (low) couplings in the interaction terms are mapped into low (high) couplings in the one-body terms. If the matrix J is interpreted as a vector representation of a matroid, duality exchanges the matroid with its dual. We discuss some physical examples. The main idea is to generalize the known models up to eventually include randomness into the pattern of interaction. We introduce and study a random Gaussian model, a random Potts-like model and a random variant of discrete scalar QED. Although the classical procedure as given by Kramers and Wannier does not extend in a natural way to such a wider class of systems, our weaker procedure applies to these models, too. We shortly describe the consequence of duality for each example.

We study a finite-range spin-glass model in arbitrary dimension, where the intensity of the coupling between spins decays to zero over some distance γ⁻¹. We prove that, under a positivity condition for the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington–Kirkpatrick model, in the Kac limit γ → 0. We study the implication of this convergence for the local order parameter, i.e., the local overlap distribution function and a family of susceptibilities associated with it, and we show that locally the system behaves like its mean field analogue. Similar results are obtained for models with p-spin interactions. Finally, we discuss a possible approach to the problem of the existence of long-range order for finite γ, based on a large deviation functional for overlap profiles. This will be developed in future work.

A property of dynamical correlation functions for nonequilibrium states is discussed. We consider arbitrary-dimensional quantum spin systems with local interaction and translationally invariant states with nonvanishing current over them. A correlation function between local charge and local Hamiltonian at different spacetime points is shown to exhibit slow decay.

Recently it has been found (Pruessner G and Jensen H J 2003 Phys. Rev. Lett.91 244303) that a totally asymmetric variant of the Oslo model (Christensen K et al 1996 Phys. Rev. Lett.77 107) represents the entire universality class of the Oslo model with anisotropy. The totally asymmetric model can be solved without scaling assumptions by finding recursively the eigenvectors of the Markov matrix, which can be suitably modified to produce the moment generating function of the relevant observable. This method should be applicable to many other stochastic processes.

We investigate stochastic resonance (SR) in a periodically driven Langevin system on a circle. The bifurcation parameter of the system and the amplitude of the periodic signal are controlled. The response amplitude that characterizes the degree of the amplification of the input periodic signal and the quality factor that is a coherence measure of the spontaneous periodic motion sustained by noise are computed and plotted as functions of the noise intensity. According to different parameter values, four types of behaviour are observed, as the results of interplay between two types of stochastic resonances, the spontaneous SR and conventional SR, respectively.

An irreversible cycle model of the quantum refrigeration cycle using an ideal Fermi gas as the working substance is established. The cycle consists of two adiabatic and two isobaric processes and consequently may be simply referred to as the Fermi Brayton refrigeration cycle. The performance of the cycle is investigated, based on the equation of state of an ideal Fermi gas. Expressions for several important performance parameters, such as the coefficient of performance, work input and refrigeration load, are derived. The influence of the quantum degeneracy of the Fermi gas and the irreversibility in the cycle on the performance of the Fermi Brayton refrigeration cycle is analysed. The minimum pressure ratio of the cycle is determined. The optimally operating problems of the cycle and several special cases are discussed in detail. The results obtained here are general and may reveal the general performance characteristics of the Fermi Brayton refrigeration cycle.

Introducing the associated Laguerre functions in terms of two non-negative integers, we obtain simultaneously and separately realization of the laddering equations with respect to each of the integers by means of two pairs of ladder operators. Besides, two different types of shape-invariance symmetries are realized. This approach leads to a derivation of shape-invariance equations of third type which are realized by two simultaneous raising and lowering operators of two parameters.

In this paper we present a general theory, based on a Lyapunov–Schmidt reduction, for the linearized stability of a perturbed Hamiltonian system with a number of symmetries. The reduction leads to an eigenvalue problem in a small subspace of the original system, whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system. The Krein signature of the eigenvalues can also be obtained in this framework. The method is applied to the case example of the coupled nonlinear Schrödinger equations giving excellent agreement with the full numerical linear stability results.

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient (1 + 1)-dimensional nonlinear diffusion–convection equations of the general form f(x)u_t = (D(u)u_x)_x + K(u)u_x. We obtain new interesting cases of such equations with the density f localized in space, which have non-trivial invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.

We consider a quantum system of two particles in dimension three interacting via a smooth potential. We characterize the asymptotic dynamics in the limit of small mass ratio for an initial state given in product form, with an explicit control of the error. An application to the decoherence effect produced on the heavy particle is also discussed.

We introduce a multi-coin discrete quantum walk where the amplitude for a coin flip depends upon previous tosses. Although the corresponding classical random walk is unbiased, a bias can be introduced into the quantum walk by varying the history dependence. By mixing the biased walk with an unbiased one, the direction of the bias can be reversed leading to a new quantum version of Parrondo's paradox.

An infinite-dimensional Lie group of symmetries of the anisotropic Chew–Goldberger–Low (CGL) plasma equilibrium equations is introduced. The symmetries are used to construct families of new anisotropic plasma equilibria. An infinite-dimensional family of transformations between solutions to the isotropic magnetohydrodynamic (MHD) equilibrium equations and solutions to the anisotropic CGL plasma equilibrium equations is presented. The transformations depend on the topology of the original solutions and produce a wide class of anisotropic plasma equilibrium solutions, including 3D solutions with no geometrical symmetries.

The energy of an nth-gradient fluid depends on its Eulerian velocity gradients of order n. A variational principle is introduced for the dynamics of nth-gradient fluids and their properties are reviewed in the context of Noether's theorem. The stability properties of Craik–Criminale solutions for first and second gradient fluids are examined.