Table of contents

We use an approximation procedure to find the partition function of a gas composed by anharmonic oscillators in a region where perturbation theory breaks down. The results are still quite accurate for small values of the coupling constant.

The operator algebra C(S − ) = CD − DC = (S + )C, D + D = DS − SD is shown to represent the stochastic dynamics of symmetric hopping of hard-core particles in one dimension and to describe the Heisenberg quantum chain. The particle or spin state is specified by strings of the operators C and D, and S is related to a current. Recursive reductions and matrix representations are used to obtain stationary and time-dependent properties, including the evolving profile for a system driven by a density gradient between open boundaries. Generalizations to other models are outlined.

We consider a simple model for the dielectric response of a water-rich surfactant sponge phase (L₃), in which an insulating bilayer separates two continuous brine domains ("in" and "out"). Experiments of Vinches, Coulon and Roux (J. Phys. II, 4 (1994) 1165) show for high salt concentrations a bimodal relaxation. We associate their higher-frequency mode with behaviour on the scale of a single pore of the sponge, and the lower-frequency one with fluctuations in the "in/out" order parameter. For a perfect bilayer, we predict a divergence of the real part of the relative permittivity at the in/out symmetry breaking transition. This is consistent with the extremely high values (ε ≃ 10⁷) observed experimentally in some high salt samples.

We consider a model of degenerate optical parametric oscillator which includes the possibility of an arbitrary level of breaking of the translational symmetry. On the basis of the example of the stripe pattern, we examine the impact of quantum noise on nonlinear optical patterns. We define and focus on the phenomenon of quantum image and show that, in the present example, it is related to the wave and particle aspects of the e.m. radiation.

We show that the Kolmogorov-1941 picture of fully developed hydrodynamic turbulence (with the scaling of the structure functions S_n(R) ∝ R^n/3) necessarily leads to an anomalous scaling for correlation functions which include the rate of energy dissipation ε(t,r), these correlation functions being described by an independent index. The mechanism for anomalous scaling, suggested on the basis of the Navier-Stokes equation, is the multi-step interaction of eddies from the inertial interval with eddies at the viscous scale via a set of eddies of intermediate scales.

High-resolution numerical simulations (up to 256³ modes) are performed for the Navier-Stokes equations with hyperviscosity (−1)^h+1 Δ^h (h = 8). With random white-in-time forcing, the inertial-range energy spectrum has the scaling law 1/k^α with α ≈ 1.85 ± 0.05 at small wave numbers k with a flatter region near the ultraviolet cut-off. The transitional region is characterized by a sharp change of the energy transfer mechanism. Spectra of fluctuations of higher-order correlation functions are measured. At inertial-range scales the system seems insensitive to the power h of the hyperviscosity.

It is shown that an exact solution of the non-linear Boltzmann equation does not verify the variational principle for non-equilibrium steady states proposed by Evans and Baranyai (Phys. Rev. Lett., 67 (1991) 2597). Therefore, the principle does not provide a basis for the understanding of far-from-equilibrium steady states.

X-ray diffraction measurements within the region of total external reflection from a thin wetting film of liquid CCl₄ on a Si/SiO₂ substrate are presented. The in-plane structure of the liquid surface yields a peak in the diffuse scattering at the specular position. Since there is no way to separate specular and off-specular contributions, the data were analysed simultaneously, which leads to a consistent set of interface parameters. Although the obtained power spectral density function of the liquid surface is very similar to that used in the case of free capillary waves, a modified cut-off has to be introduced. The difference results from the constrained thin-film geometry and is therefore caused by the interaction with the underlying substrate.

Polymer chains with hard-core interaction of a two-dimensional lattice are modeled by directed random walks. Two models, one with intersecting walks (IW) and another with non-intersecting walks (NIW) are presented, solved and compared. The exact solution of the two models, based on a representation using Grassmann variables, leads, surprisingly, to the same analytic expression for the polymer density and identical phase diagrams. There are three different phases as a function of hopping probability and single site monomer occupancy, with a transition from the dense polymer system to a polymer liquid (A) and a transition from the liquid to an empty lattice (B). Within the liquid phase there exists a self-dual line with peculiar properties. The derivative of polymer density with respect to the single site monomer occupancy diverges at transitions A and B, but is smooth across and along the self-dual line. The density-density correlation function along the direction x, perpendicular to the axis of directedness has a power law decay 1/x² in the entire liquid phase, in both models. The difference between the two models shows up only in the behavior of the correlation function along the self-dual line: it decays exponentially in the IW model and as 1/x⁴ in the NIW model.

We consider tunnelling across a p-n junction (a tunnelling diode) where the motion on both sides is constrained by quantum wells (QW) in the direction perpendicular to the interface to form 2D n-type and p-type electron gases. Coulomb interactions are neglected. The Hamiltonian near an intersection of the dispersion curves, E(k), of the n side and the p side resembles that of a relativistic Dirac electron. An electron, prepared in one of the QWs, will oscillate periodically between the two QWs with a frequency proportional to the tunnelling amplitude. In the tunnelling process, the velocity parallel to the interface is reversed resulting in periodic velocity oscillations similar to the Zitterbewegung where in the present case the opposite directions of the velocity are separated in space. We also find that the eigenstates have finite "persistent" circular currents which create magnetic moments of several Bohr magnetons, typically. The noise spectrum, and the linear response of the system are also obtained.