Table of contents

Recent work has demonstrated a new structural transition occurring at an internal defect in a two-dimensional Ising model. The new behaviour is induced by boundary conditions that constrain the interface to lie at an angle across the defect line. This gives rise to the energy–entropy competition familiar from other examples of pinning–depinning transitions. We demonstrate how a horizontal solid-on-solid (SOS) model can be used to obtain comparable results to this exact calculation. This simpler model can then be easily extended to encompass a situation where the interface has a differing stiffness on either side of the grain boundary.

How do charge and density fluctuations compete in ionic fluids near gas–liquid criticality when quantum mechanical effects play a role? To gain some insight, long-range interactions (with σ > 0), which encompass van der Waals forces (when σ = d = 3), have been incorporated in exactly soluble, d-dimensional 1:1 ionic spherical models with charges ±q₀ and hard-core repulsions. In accord with previous work, when d > min{σ, 2} (and q₀ is not too large), the Coulomb interactions do not alter the (q₀ = 0) critical universality class that is characterized by density correlations at criticality decaying as 1/r^d−2+η with η = max{0, 2 − σ}. But screening is now algebraic, the charge–charge correlations decaying, in general, only as 1/r^d+σ+4; thus σ = 3 faithfully mimics known noncritical d = 3 quantal effects. But in the absence of full (+, −) ion symmetry, density and charge fluctuations mix via a transparent mechanism: then the screening at criticality is weaker by a factor r^4−2η. Furthermore, the otherwise valid Stillinger–Lovett sum rule fails at criticality whenever η = 0 (as, e.g., when σ > 2) although it remains valid if η > 0 (as for σ < 2 or in real d ⩽ 3 Ising-type systems).

Notwithstanding radical conceptual differences between classical and quantum mechanics, it is usually assumed that physical measurements concern observables common to both theories. Not so with the eigenvalues (±1) of the parity operator. The effect of such a measurement on a mixture of even and odd states of the harmonic oscillator is akin to separating at a single stroke a pair of shuffled card decks: the result is a set of definite parity, though otherwise mixed. Here we derive the general form of a parity collapsed state, whether pure or mixed. The signature of positive or negative parity is a corresponding spike in the Wigner function which is sharpened by decoherence. We conjecture that states with pure parity always have negative values in their Wigner functions.

A computational procedure that allows the detection of a new type of high-dimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows us to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence of a chaotic saddle. It also allows us to detect heteroclinic connections between different NHIMs. NHIMs control the phase space transport across an equilibrium point of saddle-centre-⋅⋅ ⋅ -centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, 'transformation' in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well-known model in celestial mechanics and which gained much interest, e.g. in the study of the formation of binaries in the Kuiper belt.

We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder-averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs, our macroscopic laws close already in terms of the single-spin path probabilities at zero external field. For the general case of arbitrary graph symmetry, we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.

We study properties of a non-Markovian random walk X⁽ⁿ⁾_l, l = 0, 1, 2, ..., n, evolving in discrete time l on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the rise-and-descent sequences characterizing random permutations π of [n + 1] = {1, 2, 3, ..., n + 1}. We determine exactly the probability of finding the end-point X_n = X⁽ⁿ⁾_n of the trajectory of such a permutation-generated random walk (PGRW) at site X, and show that in the limit n → ∞ it converges to a normal distribution with a smaller, compared to the conventional Pólya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identical to the distribution of the intermediate points X⁽ⁿ⁾_l, l < n, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of 'turns' of the PGRW trajectories.

Poincaré recurrence for a class of circle maps is used to study the properties of the corresponding invariant measures. In the subcritical case, when the map is a diffeomorphism, the return time measure is smooth, and in the critical case, when the map is only a homeomorphism, the measure is only continuous. Furthermore, in the considered class of critical maps the behaviour of the return time entropy depends only on the tail in the continued fraction expansion of the rotation number.

We investigate chaotic behaviour in a 2D Hamiltonian system—oscillators with anharmonic coupling. We compare the classical system with the quantum system. Via the quantum action, we construct Poincaré sections and compute Lyapunov exponents for the quantum system. We find that the quantum system is globally less chaotic than the classical system. We also observe with increasing energy the distribution of Lyapunov exponents approaching a Gaussian with a strong correlation between its mean value and energy.

A special family of solutions explicit in the space and time variables is found for the Fisher equation with density-dependent diffusion. The connection with the known travelling wave solution and the initial conditions from which that evolves is also shown.

We use the smaller alignment index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behaviour of the SALI for the two cases: the index fluctuates around non-zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behaviour for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents σ₁, σ₂ i.e. . Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of two and three degrees of freedom.

We use the Dunkl operator approach to construct one-dimensional integrable models describing N particles with internal degrees of freedom. These models are described by a general Hamiltonian belonging to the centre of the Yangian or the reflection algebra, which ensures that they admit the corresponding symmetry. In particular, the open problem of the symmetry is answered for the B_N-type Sutherland model with spin and for a generalized B_N-type nonlinear Schrödinger Hamiltonian.

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the nth and (n − 1)th minors, whose solution is a representation of the nth minor as an n × n determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order n with respect to the kernel. Our formula is a linear combination of the nth and the (n ± 1)th minors.

We obtain the Hamiltonian theory of the Landau–Lifschitz equation with an easy axis by using a suitable gauge transformation and the standard procedure. Action–angle variables are found and the canonical equation is given.

Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson–Nijenhuis structure from a given type (1, 1) tensor field J on Q. It is argued that the complete lift J^c of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to T*Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case when Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.

We consider a perturbation of an 'integrable' Hamiltonian and give an expression for the canonical or unitary transformation which 'simplifies' this perturbed system. The problem is to invert a functional defined on the Lie-algebra of observables. We give a bound for the perturbation in order to solve this inversion, and apply this result to a particular case of the control theory, as a first example, and to the 'quantum adiabatic transformation', as another example.

In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in ℏ by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows us to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac–Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity.

The symmetry structure of the non-Abelian affine Toda model based on the coset SL(3)/SL(2) ⊗ U(1) is studied. It is shown that the model possess non-Abelian Noether symmetry closing into a q-deformed SL(2) ⊗ U(1) algebra. Specific two-vertex soliton solutions are constructed.

The Casimir energies and pressures for a massless scalar field associated with δ-function potentials in 1 + 1 and 3 + 1 dimensions are calculated. For parallel plane surfaces, the results are finite, coincide with the pressures associated with Dirichlet planes in the limit of strong coupling, and for weak coupling do not possess a power-series expansion in 1 + 1 dimension. The relation between Casimir energies and Casimir pressures is clarified, and the former are shown to involve surface terms, interpreted as the quantum vacuum energies of the surfaces. The Casimir energy for a δ-function spherical shell in 3 + 1 dimensions has an expression that reduces to the familiar result for a Dirichlet shell in the strong-coupling limit. However, the Casimir energy for finite coupling possesses a logarithmic divergence first appearing in third order in the weak-coupling expansion, which seems unremovable. The corresponding energies and pressures for a derivative of a δ-function potential for the same spherical geometry generalizes the TM contributions of electrodynamics. Cancellation of divergences can occur between the TE (δ-function) and TM (derivative of δ-function) Casimir energies. These results clarify recent discussions in the literature.

We point out an error and several inconsistencies in the analysis of Khachatourian and Wistrom. In their force computation, an essential contribution was neglected, leading to an erroneous nonzero torque prediction. Furthermore, the analysis includes several internal contradictions.