Gauge Theory for Finite-Dimensional Dynamical Systems

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This theory has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems with implications to numerical integration of differential equations. We distinguish between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry is, in essence, re-scaling of the independent variable, while descriptive gauge symmetry is a Yang-Mills-like transformation of the velocity vector field, adapted to finite-dimensional systems. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a remarkable connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse engineering and scientific fields, including quantum mechanics, chemistry, rigid-body dynamics and information theory

Quantifying solute spreading and mixing in reservoir rocks using 3-D PET imaging

We report results of an experimental investigation into the effects of small-scale (mm-cm) heterogeneities on solute spreading and mixing in a Berea sandstone core. Pulse-tracer tests have been carried out in the Péclet number regime Pe = 6-40 and are supplemented by a unique combination of two imaging techniques. X-ray computed tomography (CT) is used to quantify subcore-scale heterogeneities in terms of permeability contrasts at a spatial resolution of approximately 10 mm3, while [11C] positron emission tomography (PET) is applied to image the spatial and temporal evolution of the full tracer plume non-invasively. To account for both advective spreading and local (Fickian) mixing as driving mechanisms for solute transport, a streamtube model is applied that is based on the one-dimensional advection-dispersion equation. We refer to our modelling approach as semideterministic, because the spatial arrangement of the streamtubes and the corresponding solute travel times are known from the measured rock's permeability map, which required only small adjustments to match the measured tracer breakthrough curve. The model reproduces the three-dimensional PET measurements accurately by capturing the larger-scale tracer plume deformation as well as subcore-scale mixing, while confirming negligible transverse dispersion over the scale of the experiment. We suggest that the obtained longitudinal dispersivity (0.10±0.02 cm) is rock rather than sample specific, because of the ability of the model to decouple subcore-scale permeability heterogeneity effects from those of local dispersion. As such, the approach presented here proves to be very valuable, if not necessary, in the context of reservoir core analyses, because rock samples can rarely be regarded as 'uniformly heterogeneous'

Phase transitions in simple and not so simple binary fluids

Compared to pure fluids, binary mixtures display a very diverse phase behavior, which depends sensitively on the parameters of the microscopic potential. Here we investigate the phase diagrams of simple model mixtures by use of a microscopic implementation of the renormalization group technique. First, we consider a symmetric mixture with attractive interactions, possibly relevant for describing fluids of molecules with internal degrees of freedom. Despite the simplicity of the model, slightly tuning the strength of the interactions between unlike species drastically changes the topology of the phase boundary, forcing or inhibiting demixing, and brings about several interesting features such as double critical points, tricritical points, and coexistence domains enclosing `islands' of homogeneous, mixed fluid. Homogeneous phase separation in mixtures can be driven also by purely repulsive interactions. As an example, we consider a model of soft particles which has been adopted to describe binary polymer solutions. This is shown to display demixing (fluid-fluid) transition at sufficiently high density. The nature and the physical properties of the corresponding phase transition are investigated.Comment: 6 pages + 3 figures, presented at the 5th EPS Liquid Matter Conference, Konstanz, 14-18 September 200

A model colloidal fluid with competing interactions: bulk and interfacial properties

Using a simple mean-field density functional theory theory (DFT), we investigate the structure and phase behaviour of a model colloidal fluid composed of particles interacting via a pair potential which has a hard core of diameter $\sigma$, is attractive Yukawa at intermediate separations and repulsive Yukawa at large separations. We analyse the form of the asymptotic decay of the bulk fluid correlation functions, comparing results from our DFT with those from the self consistent Ornstein-Zernike approximation (SCOZA). In both theories we find rich crossover behaviour, whereby the ultimate decay of correlation functions changes from monotonic to long-wavelength damped oscillatory decay on crossing certain lines in the phase diagram, or sometimes from oscillatory to oscillatory with a longer wavelength. For some choices of potential parameters we find, within the DFT, a $\lambda$-line at which the fluid becomes unstable with respect to periodic density fluctuations. SCOZA fails to yield solutions for state points near such a $\lambda$-line. The propensity to clustering of particles, which is reflected by the presence of a long wavelength $\gg \sigma$, slowly decaying oscillatory pair correlation function, and a structure factor that exhibits a very sharp maximum at small but non zero wavenumbers, is enhanced in states near the $\lambda$-line. We present density profiles for the planar liquid-gas interface and for fluids adsorbed at a planar hard wall. The presence of a nearby $\lambda$-transition gives rise to pronounced long-wavelength oscillations in the one-body densities at both types of interface.Comment: 14 pages, 11 figure

Mechanism of temperature dependence of the magnetic anisotropy energy in ultrathin Cobalt and Nickel films

Temperature dependent FMR-measurements of Ni and Co films are analysed using a microscopic theory for ultrathin metallic systems. The mechanism governing the temperature dependence of the magnetic anisotropy energy is identified and discussed. It is reduced with increasing temperature. This behavior is found to be solely caused by magnon excitations.Comment: 3 pages, 4 figures III Joint European Magnetic Symposia, San Sebastian, Spai

Smooth cutoff formulation of hierarchical reference theory for a scalar phi4 field theory

The phi4 scalar field theory in three dimensions, prototype for the study of phase transitions, is investigated by means of the hierarchical reference theory (HRT) in its smooth cutoff formulation. The critical behavior is described by scaling laws and critical exponents which compare favorably with the known values of the Ising universality class. The inverse susceptibility vanishes identically inside the coexistence curve, providing a first principle implementation of the Maxwell construction, and shows the expected discontinuity across the phase boundary, at variance with the usual sharp cutoff implementation of HRT. The correct description of first and second order phase transitions within a microscopic, nonperturbative approach is thus achieved in the smooth cutoff HRT.Comment: 8 pages, 4 figure

Effect of antiferromagnetic exchange interactions on the Glauber dynamics of one-dimensional Ising models

We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional (1D) Ising models with spin $\pm 1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors (nnn) is assumed to compete with a nearest-neighbor (nn) exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent z, defined as ${1/\tau} \approx ({1/ {\xi}})^z$ (where \tau is the longest relaxation time and \xi is the correlation length of the chain), suggests that the T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.Comment: 5 figures. Phys. Rev. B (accepted July 12, 2007

Measuring and modelling supercritical adsorption of CO2 and CH4 on montmorillonite source clay

The porosity of clay minerals is dominated by nanoscale pores that provide a large surface area for physical and chemical interactions with the surrounding fluids, including gas adsorption. Measuring gas adsorption at subsurface conditions is difficult, because elevated pressures are required and the interactions between the supercritical gas and the clay are relatively weak. Here, we report on the measurement of adsorption isotherms of CO2 and CH4 on the source clay Na-montmorillonite (SWy-2) at different temperatures (25–115°C) over a wide range of pressures (0.02–25 MPa). The experimental observations are thoroughly analysed by considering both net and excess adsorbed amounts, and by extracting adsorption metrics, such as the Henry's constants and enthalpy of adsorption. The results consistently indicate that SWy-2 favours adsorption of CO2 over CH4 with selectivity, . The experimental data are successfully described using a Lattice Density Functional Theory (LDFT) model. The adsorption energetics estimated by the model compare well with the experimentally obtained enthalpy of adsorption. It is further shown that even at the highest pressure the pore space of the clay is only partially filled and that the degree of saturation increases upon approaching the critical temperature of the gas. The ability of the LDFT model to reveal pore-dependent adsorption behaviours demonstrates its potential against empirical models, such as the Langmuir equation, which fail at capturing the complexities of supercritical gas adsorption at subsurface conditions

Phase behavior of a fluid with competing attractive and repulsive interactions

Fluids in which the interparticle potential has a hard core, is attractive at moderate separations, and repulsive at greater separations are known to exhibit novel phase behavior, including stable inhomogeneous phases. Here we report a joint simulation and theoretical study of such a fluid, focusing on the relationship between the liquid-vapor transition line and any new phases. The phase diagram is studied as a function of the amplitude of the attraction for a certain fixed amplitude of the long ranged repulsion. We find that the effect of the repulsion is to substitute the liquid-vapor critical point and a portion of the associated liquid-vapor transition line, by two first order transitions. One of these transitions separates the vapor from a fluid of spherical liquidlike clusters; the other separates the liquid from a fluid of spherical voids. At low temperature, the two transition lines intersect one another and a vapor-liquid transition line at a triple point. While most integral equation theories are unable to describe the new phase transitions, the Percus Yevick approximation does succeed in capturing the vapor-cluster transition, as well as aspects of the structure of the cluster fluid, in reasonable agreement with the simulation results.Comment: 15 pages, 20 figure

Recent developments of the Hierarchical Reference Theory of Fluids and its relation to the Renormalization Group

The Hierarchical Reference Theory (HRT) of fluids is a general framework for the description of phase transitions in microscopic models of classical and quantum statistical physics. The foundations of HRT are briefly reviewed in a self-consistent formulation which includes both the original sharp cut-off procedure and the smooth cut-off implementation, which has been recently investigated. The critical properties of HRT are summarized, together with the behavior of the theory at first order phase transitions. However, the emphasis of this presentation is on the close relationship between HRT and non perturbative renormalization group methods, as well as on recent generalizations of HRT to microscopic models of interest in soft matter and quantum many body physics.Comment: 17 pages, 5 figures. Review paper to appear in Molecular Physic