Table of contents

We define a transverse correlation length suitable to discuss the finite-size scaling behaviour of an out-of-equilibrium lattice gas, whose correlation functions decay algebraically with the distance. By numerical simulations we verify that this definition has a good infinite-volume limit independent of the lattice geometry. We study the transverse fluctuations as they can select the correct field-theoretical description. By means of a careful finite-size scaling analysis, without tunable parameters, we show that they are Gaussian, in agreement with the predictions of the field theory proposed by Janssen, Schmittmann, Leung and Cardy.

In an alternative interpretation, the Seiberg–Witten map is shown to be induced by a field-dependent coordinate transformation connecting non-commutative and ordinary spacetimes. Furthermore, following our previous ideas, it has been demonstrated here that the above (field-dependent coordinate) transformation can occur naturally in the Batalin–Tyutin extended space version of the relativistic spinning particle model (in a particular gauge). There is no need to postulate the spacetime non-commutativity in an ad hoc way: it emerges from the spin degrees of freedom.

This review reports on the research done during past years on violations of the fluctuation–dissipation theorem (FDT) in glassy systems. It is focused on the existence of a quasi-fluctuation–dissipation theorem (QFDT) in glassy systems and the current supporting knowledge gained from numerical simulation studies. It covers a broad range of non-stationary aging and stationary driven systems such as structural glasses, spin glasses, coarsening systems, ferromagnetic models at criticality, trap models, models with entropy barriers, kinetically constrained models, sheared systems and granular media. The review is divided into four main parts: (1) an introductory section explaining basic notions related to the existence of the FDT in equilibrium and its possible extension to the glassy regime (QFDT), (2) a description of the basic analytical tools and results derived in the framework of some exactly solvable models, (3) a detailed report of the current evidence in favour of the QFDT and (4) a brief digression on the experimental evidence in its favour. This review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the QFDT as well as for the more expert readers who may be interested in more specific results.

Finite-size scaling (FSS) is a standard technique for measuring scaling exponents in spin glasses. Here we present a critique of this approach, emphasizing the need for all length scales to be large compared to microscopic scales. In particular we show that the replacement, in FSS analyses, of the correlation length by its asymptotic scaling form can lead to apparently good scaling collapses with the wrong values of the scaling exponents.

Nanotubes as well as polymers and quasi-1D subsystems of 3D crystals have line group symmetry. This allows two types of quantum numbers: roto-translational and helical. The roto-translational quantum numbers are linear and total angular (not conserved) momenta, while the helical quantum numbers are helical and complementary angular momenta. Their mutual relations determine some topological properties of energy bands, such as systematic band sticking or van Hove singularities related to parities. The importance of these conclusions is illustrated by the optical absorption in carbon nanotubes: parity may prevent absorption peaks at van Hove singularities.

Contact matrices provide a coarse grained description of the configuration ω of a linear chain (polymer or random walk) on ⁿ: _ij(ω) = 1 when the distance between the positions of the ith and jth steps are less than or equal to some distance a and _ij(ω) = 0 otherwise. We consider models in which polymers of length N have weights corresponding to simple and self-avoiding random walks, SRW and SAW, with a the minimal permissible distance. We prove that to leading order in N, the number of matrices equals the number of walks for SRW, but not for SAW. The coarse grained Shannon entropies for SRW agree with the fine grained ones for n ⩽ 2, but differs for n ⩾ 3.

We have developed a parallel algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 110. We have also extended the series for the first 10 area-weighted moments and the radius of gyration to 100. Analysis of the resulting series yields very accurate estimates of the connective constant μ = 2.638 158 530 31(3) (biased) and the critical exponent α = 0.500 000 1(2) (unbiased). In addition, we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations.

If a point particle moves chaotically through a periodic array of scatterers the associated transport coefficients are typically irregular functions under variation of control parameters. For a piecewise linear two-parameter map we analyse the structure of the associated irregular diffusion coefficient and current by numerically computing dimensions from box-counting and from the autocorrelation function of these graphs. We find that both dimensions are fractal for large parameter intervals and that both quantities are themselves fractal functions if computed locally on a uniform grid of small but finite subintervals. We furthermore show that there is a simple functional relationship between the structure of fractal fractal dimensions and the difference quotient defined on these subintervals.

The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and, therefore, more easily accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity.

We construct generalized coherent states for the one-dimensional double-well potential and calculate their Mandel parameter, uncertainty relation and Wigner functions. The singularities of their autocorrelation function undergo bifurcations as the mean energy of the system is varied, and we analyse their structure. In the high-energy regime these states consist of a coherent superposition of two minimum-uncertainty Gaussians (they are Schrödinger catlike states).

We consider the general problem of a single two-level atom interacting with a multimode radiation field (without the rotating-wave approximation), and additionally take the field to be coupled to a thermal reservoir. Using the method of bosonization of the spin operators in the Hamiltonian, and working in the Bargmann representation for all the boson operators, we obtain the propagator for the composite system using the techniques of functional integration, under a reasonable approximation scheme. The propagator is explicitly evaluated for a simplified version of the system with one spin and a dynamically coupled single-mode field. The results are also checked on the known problem of quantum Brownian motion.

The nonadiabatic transition probabilities in the two-level systems are calculated analytically by using the monodromy matrix determining the global feature of the underlying differential equation. We study the time-dependent 2 × 2 Hamiltonian with the tanh-type plus sech-type energy difference and with constant off-diagonal elements as an example to show the efficiency of the monodromy approach. We also discuss the application of this method to multi-level systems.

The stability of flows of Newtonian fluids induced by a surface acoustic wave (SAW) along the deformable walls in a confined parallel-plane microchannel or slab in the laminar flow regime is investigated. The governing equation which was derived by considering the weakly nonlinear coupling between the deformable wall and viscous flow is linearized and then the eigenvalue problem is solved by a numerical code together with the associated interface and boundary conditions. The value of the critical Reynolds number was found to be near 613.26 which is much smaller than the static-wall case: 5772 for conventional pressure-driven flows.

The computable cross norm (CCN) criterion for separability is neither weaker nor stronger than the positive partial transpose (PPT) criterion.