Lattice gauge theories: Introduction

Lattice gauge theories are based on the notion of parallel transport. They can be considered as non-perturbative regularizations of the continuum gauge theories in the sense of a low-temperature expansion. The chapter is mainly devoted on a study of matterless lattice gauge theories from the point of view of phase transitions. This means many properties of a realistic theory like quantum chromodynamics (QC) cannot be investigated, but the important question of confinement can still be studied: does the theory generate a force between charged particles increasing at large distances, so that heavy quarks in the fundamental representation cannot be separated? More generally, can one find charged asymptotic states like massless vector particles in the theory? Lattice gauge theories have properties quite different from the ferromagnetic systems. In particular the absence of a local order parameter requires a study of the behaviour of a non-local quantity, a functional of loops generally called Wilson's loop, to distinguish between the confined and deconfined phases, characterized by an area or perimeter law, respectively.

On the Definition of Energy Flux in One-Dimensional Chains of Particles

We review two well-known definitions present in the literature, which are used to define the heat or energy flux in one dimensional chains. One definition equates the energy variation per particle to a discretized flux difference, which we here show it also corresponds to the flux of energy in the zero wavenumber limit in Fourier space, concurrently providing a general formula valid for all wavelengths. The other relies somewhat elaborately on a definition of the flux, which is a function of every coordinate in the line. We try to shed further light on their significance by introducing a novel integral operator, acting over movable boundaries represented by the neighboring particles’ positions, or some combinations thereof. By specializing to the case of chains with the particles’ order conserved, we show that the first definition corresponds to applying the differential continuity-equation operator after the application of the integral operator. Conversely, the second definition corresponds to applying the introduced integral operator to the energy flux. It is, therefore, an integral quantity and not a local quantity. More worryingly, it does not satisfy in any obvious way an equation of continuity. We show that in stationary states, the first definition is resilient to several formally legitimate modifications of the (models of) energy density distribution, while the second is not. On the other hand, it seems peculiar that this integral definition appears to capture a transport contribution, which may be called of convective nature, which is altogether missed by the former definition. In an attempt to connect the dots, we propose that the locally integrated flux divided by the inter-particle distance is a good measure of the energy flux. We show that the proposition can be explicitly constructed analytically by an ad hoc modification of the chosen model for the energy density.

Height variation of the cutoff frequency in a sunspot umbra

Context. In the solar atmosphere, the acoustic cutoff frequency is a local quantity that depends on atmospheric height. It separates low-frequency evanescent waves from high-frequency propagating waves. Aims. We measure the cutoff frequency of slow magnetoacoustic waves at various heights of a sunspot umbra and compare the results with the estimations from several analytical formulae. Methods. We analyzed the oscillations in the umbra of a sunspot belonging to active region NOAA 12662 observed in the 10 830 Å spectral region with the GREGOR Infrared Spectrograph and in the Fe I 5435 Å line with the GREGOR Fabry-Pérot Interferometer. Both instruments are attached to the GREGOR telescope at the Observatorio del Teide, Tenerife, Spain. We computed the phase and amplification spectra between the velocity measured from various pairs of lines that sample various heights of the solar atmosphere. The cutoff frequency and its height variation were estimated from the inspection of the spectra. Results. At the deep umbral photosphere the cutoff frequency is around 5 mHz and it increases to 6 mHz at higher photospheric layers. At the chromosphere the cutoff is ~3.1 mHz. A comparison of the observationally determined cutoff with the theoretically predicted values reveals an agreement in the general trend and a reasonable match at the chromosphere, but also significant quantitative differences at the photosphere. Conclusions. Our analyses show strong evidence of the variation of the cutoff frequency with height in a sunspot umbra, which is not fully accounted for by current analytical estimations. This result has implications for our understanding of wave propagation, the seismology of active regions, and the evaluation of heating mechanisms based on compressible waves.

Local Crater Wear Prediction Using Physics-Based Models

A physics-based, pointwise model is developed to predict crater profiles of multilayer coated carbides after a series of turning experiments. Dissolution and abrasion mechanisms, which are identified to be the dominant wear mechanisms at the crater, are reformulated into a pointwise or local quantity to predict the crater profiles based on the temperature and pressure profiles from finite element (FE) simulations. The crater profiles predicted by the proposed model have to be adjusted, however, due to the creep deformation of the carbide substrate occurring under the machining conditions employed in our experiment. The crater predictions correlate pretty well with the crater profiles experimentally observed in the multilayer (TiN–Al2O3–TiCN) coated carbides until the wear front reached the middle of the Al2O3 layer. At this point, the Al2O3 coating undergoes the κ-to-α-phase transformation, which makes the wear prediction difficult due to substantial changes in the thermomechanical properties of the Al2O3 coating.

Supercritical transition to turbulence in an inertially driven von Kármán closed flow

We study the transition from laminar flow to fully developed turbulence for an inertially driven von Kármán flow between two counter-rotating large impellers fitted with curved blades over a wide range of Reynolds number (102–106). The transition is driven by the destabilization of the azimuthal shear layer, i.e. Kelvin–Helmholtz instability, which exhibits travelling/drifting waves, modulated travelling waves and chaos before the emergence of a turbulent spectrum. A local quantity – the energy of the velocity fluctuations at a given point – and a global quantity – the applied torque – are used to monitor the dynamics. The local quantity defines a critical Reynolds number Rec for the onset of time-dependence in the flow, and an upper threshold/crossover Ret for the saturation of the energy cascade. The dimensionless drag coefficient, i.e. the turbulent dissipation, reaches a plateau above this finite Ret, as expected for ‘Kolmogorov’-like turbulence for Re→∞. Our observations suggest that the transition to turbulence in this closed flow is globally supercritical: the energy of the velocity fluctuations can be considered as an order parameter characterizing the dynamics from the first laminar time-dependence to the fully developed turbulence. Spectral analysis in the temporal domain, moreover, reveals that almost all of the fluctuation energy is stored in time scales one or two orders of magnitude slower than the time scale based on impeller frequency.

A theory of correlation dimension for stationary time series

We develop a formal theory of correlation dimension for a class of stationary time series that includes both deterministic outputs and gaussian processes with continuous paths. This theory enables us to completely analyse correlation dimension in gaussian processes via spectral methods. Our approach plus recent results on the convergence behaviour of the sample correlation integral are then used to re-examine the behaviour of gaussian power-law coloured noise. We show that the finite correlation dimension observed by Osborne & Provenzale (1989) is a local quantity entirely due to the non-ergodicity of the simulation model. We also show that non-ergodic and weakly ergodic finite-dimensional dynamical systems (such as the simple circle map) exhibit the same phenomenon.