Table of contents

Shafranov's virial theorem implies that nontrivial magnetohydrodynamical equilibrium configurations must be supported by externally supplied currents. Here we extend the virial theorem to field theory, where it relates to Derrick's scaling argument on soliton stability. We then employ virial arguments to investigate a realistic field theory model of a two-component plasma, and conclude that stable localized solitons can exist in the bulk of a finite-density plasma. These solitons entail a nontrivial electric field, which implies that purely magnetohydrodynamical arguments are insufficient for describing stable, nontrivial structures within the bulk of a plasma.

We show that non-locality in the conservation of both the order parameter and a non-critical density (model-D dynamics) leads to new fixed points for critical dynamics. Depending upon the parameters characterizing the non-locality in the two fields, we find four regions: (i) model-A-like, where both conservations are irrelevant; (ii) model-B-like, with the conservation in the order parameter field relevant and the conservation in the coupling field irrelevant; (iii) model-C-like, where the conservation in the order parameter field is irrelevant but the conservation in the coupling field is relevant; and (iv) model-D-like, where both conservations are relevant. While the first three behaviours are already known in dynamical critical phenomena, the last one is a novel phenomenon due entirely to the non-locality in the two fields.

We study spatially homogeneous inelastic gases using the Boltzmann equation. We consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient . In the unforced case, we find that the velocity distribution decays algebraically, P(v,t)~v^-σ, for sufficiently large velocities. The exponent σ(d,) exhibits nontrivial dependence on the spatial dimension and the dissipation coefficient.

We consider the problem of on-lattice cluster anisotropy in the diffusion-limited aggregation (DLA) model. On the basis of a recent paper (Bogoyavlenskiy V A 2001 Phys. Rev. E 64 066303), we derive an isotropic quadratic ratio for a set of aggregation probabilities, in order to grow on-lattice DLA clusters without anisotropy.

Time-recurrent networks are considered. Synaptic plasticity is defined by a simple Hebb rule. It is well known that this Hebbian mechanism can support learning and memory.

We show that this plasticity is a computational instrument with large possibilities. In particular, the synaptic matrix can store different information, both dynamic and static. For example, the network can perform the Fourier and wavelet transformations and calculate probability distributions of unknown parameters. These networks can analyse and identify dynamics, calculate likelihood, study autoregression etc. They can resolve even more sophisticated problems, for example decoding fractal images.

We present a detailed classification of random Dirac Hamiltonians in two spatial dimensions based on the implementation of discrete symmetries. Our classification is slightly finer than that of random matrices and contains 13 classes. We also extend this classification to non-Hermitian Hamiltonians with and without a Dirac structure.

A kinetic one-dimensional Ising model is coupled to two heat baths, such that spins at even (odd) lattice sites experience a temperature T_e (T_o). Spin-flips occur with Glauber-type rates generalized to the case of two temperatures. Driven by the temperature differential, the spin chain settles into a non-equilibrium steady state which corresponds to the stationary solution of a master equation. We construct a perturbation expansion of this master equation in terms of the temperature difference and compute explicitly the first two corrections to the equilibrium Boltzmann distribution. The key result is the emergence of additional spin operators in the steady state, increasing in spatial range and order of spin products. We comment on the violation of detailed balance and entropy production in the steady state.

We study a Gaussian Potts-Hopfield model. Whereas for Ising spins and two disorder variables per site the chaotic pair scenario is realized, we find that for q-state Potts spins q (q-1)-tuples occur. Beyond the breaking of a continuous stochastic symmetry, we study the fluctuations and obtain the Newman-Stein metastate description for our model.

The q-deformed supersymmetric t-J model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the quantum affine superalgebra

We give the bosonization of the boundary states. We give an integral expression for the correlation functions of the boundary model, and derive the difference equations which they satisfy.

We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we rediscuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centred on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves.

We study the evolution of geometric invariants for equations such as the Davey–Stewartson and Novikov–Veselov equations.

The bc-system of higher rank introduced recently is examined further. It is shown that the correlation functions are connected with certain non-Abelian θ-functions and it is discussed how quasi-determinants arise.

The star product and Moyal bracket are introduced using the coherent states corresponding to quantum systems with non-linear spectra. Two kinds of coherent state are considered. The first kind is the set of Gazeau-Klauder coherent states and the second kind are constructed following the Perelomov-Klauder approach. The particular case of the harmonic oscillator is also discussed.

For a discrete excited state coupled to a continuum, we prove that the positions of the resonances are, in general, multivalued functions of the coupling constant and the width of the continuum, even if these parameters are real. In such a model, taken from quantum electrodynamics, if the spatial extension of the discrete state is above a certain value, we prove that the bound state which appears has little to do with the excited state. For more general coupling constants, we bring some new information about the behaviour of the resonance corresponding to the excited state when the coupling constant is increased.

We discuss the case of a Markovian master equation for an open system, as it is frequently found from environmental decoherence. We prove two theorems for the evolution of the quantum state. The first one states that for a generic initial state the corresponding Wigner function becomes strictly positive after a finite time has elapsed. The second one states that also the P-function becomes exactly positive after a decoherence time of the same order. Therefore, the density matrix becomes exactly decomposable into a mixture of Gaussian pointer states.

Previous work extending the Kohn–Sham theory to excited states was based on replacing the study of the ground-state energy as a functional of the ground-state density by a study of an ensemble average of the Hamiltonian as a functional of the corresponding average density. We suggest and develop an alternative to this description of excited states that utilizes the matrix of the density operator taken between any two states of the included space. Such an approach provides more detailed information about the states included, for example transition probabilities between discrete states of local one-body operators. The new theory is also based on a variational principle for the trace of the Hamiltonian over the space of states that we wish to describe, viewed, however, as a functional of the associated array of matrix elements of the density. This finds expression in a matrix version of the Kohn–Sham theory. To illustrate the formalism, we study a suitably defined weak-coupling limit, which is our equivalent of the linear response approximation. On this basis, we derive an eigenvalue equation that has the same form as an equation derived directly from the time-dependent Kohn–Sham equation and applied recently with considerable success to molecular excitations. We provide an independent proof, within the defined approximations, that the eigenvalues can be interpreted as true excitation energies.