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

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

Static density response of one-dimensional soft bosons across the clustering transition

One-dimensional bosons interacting via a soft-shoulder potential are investigated at zero temperature. The flatness of the potential at short distances introduces a typical length, such that, at relatively high densities and sufficiently strong interactions, clusters are formed, even in the presence of a completely repulsive potential. We evaluate the static density response function of this system across the transition from the liquid to the cluster liquid phases. Such quantity reveals the density modulations induced by a weak periodic external potential, and is maximal at the clustering wavevector. It is known that this response function is proportional to the static structure factor in the classical regime at high temperature, while for this zero-temperature quantum systems, we extract it from the dynamical structure factor evaluated with quantum Monte Carlo methods.Comment: 7 pages, 3 figures. Proceedings of the "Recent Progress in Many-Body Theories" international conference, Pohang, South Korea, 201

A thermodynamically self-consistent theory for the Blume-Capel model

We use a self-consistent Ornstein-Zernike approximation to study the Blume-Capel ferromagnet on three-dimensional lattices. The correlation functions and the thermodynamics are obtained from the solution of two coupled partial differential equations. The theory provides a comprehensive and accurate description of the phase diagram in all regions, including the wing boundaries in non-zero magnetic field. In particular, the coordinates of the tricritical point are in very good agreement with the best estimates from simulation or series expansion. Numerical and analytical analysis strongly suggest that the theory predicts a universal Ising-like critical behavior along the $\lambda$-line and the wing critical lines, and a tricritical behavior governed by mean-field exponents.Comment: 11 figures. to appear in Physical Review

Surface spin-flop transition in a uniaxial antiferromagnetic Fe/Cr superlattice induced by a magnetic field of arbitrary direction

We studied the transition between the antiferromagnetic and the surface spin-flop phases of a uniaxial antiferromagnetic [Fe(14 \AA)/Cr(11 \AA]$_{\rm x20}$ superlattice. For external fields applied parallel to the in-plane easy axis, the layer-by-layer configuration, calculated in the framework of a mean-field one-dimensional model, was benchmarked against published polarized neutron reflectivity data. For an in-plane field $H$ applied at an angle $\psi \ne 0$ with the easy axis, magnetometry shows that the magnetization $M$ vanishes at H=0, then increases slowly with increasing $H$. At a critical value of $H$, a finite jump in $M(H)$ is observed for $\psi<5^{\rm o}$, while a smooth increase of $M$ $vs$ $H$ is found for $\psi>5^{\rm o}$. A dramatic increase in the full width at half maximum of the magnetic susceptibility is observed for $\psi \ge 5^{\rm o}$. The phase diagram obtained from micromagnetic calculations displays a first-order transition to a surface spin-flop phase for low $\psi$ values, while the transition becomes continuous for $\psi$ greater than a critical angle, $\psi_{\rm max} \approx 4.75^{\rm o}$. This is in fair agreement with the experimentally observed results.Comment: 24 pages, 7 figure

Theory and simulation of short-range models of globular protein solutions

We report theoretical and simulation studies of phase coexistence in model globular protein solutions, based on short-range, central, pair potential representations of the interaction among macro-particles. After reviewing our previous investigations of hard-core Yukawa and generalised Lennard-Jones potentials, we report more recent results obtained within a DLVO-like description of lysozyme solutions in water and added salt. We show that a one-parameter fit of this model based on Static Light Scattering and Self-Interaction Chromatography data in the dilute protein regime, yields demixing and crystallization curves in good agreement with experimental protein-rich/protein-poor and solubility envelopes. The dependence of cloud and solubility points temperature of the model on the ionic strength is also investigated. Our findings highlight the minimal assumptions on the properties of the microscopic interaction sufficient for a satisfactory reproduction of the phase diagram topology of globular protein solutions.Comment: 17 pages, 8 figures, Proc. of Conference "Structural Arrest Transitions in Colloidal Systems with Short-Range Attractions", Messina (ITALY) 17-20 December 200

Does the longevity of one or both parents influence the health status of their offspring?

According to the findings of some recent studies, the centenarians' offspring appear to represent a promising model for research on longevity and healthy aging. This study compares the health status and the functional status of three groups of subjects: 1. individuals with two long-lived parents (one of whom centenarian), 2. individuals with only one long-lived (centenarian) parent, and 3. individuals with no long-lived parents. The goal is to verify whether the centenarians' offspring display any advantage over the offspring of both non-long-lived parents and to evaluate whether the longevity of the non-centenarian parent provides a further advantage. A total of 374 subjects (mean age approximately 70 years) was examined. A threshold for longevity was established for non-centenarian parents through demographic data available for Italy (males surviving to at least 81 years of age and females to 87 years). The participants were assessed for their health and functional status by means of a standardized questionnaire and tests of physical performance. Data were analyzed using multivariate regression models adjusted for socio-demographic characteristics and risk factors for age-related pathologies. The results of the study show that centenarians' offspring have a better functional status, a reduced risk for several age-related pathologies and reduced drug consumption than the offspring of non-long-lived parents. In addition, the health status of centenarians' offspring does not appear to be influenced by the longevity of the second parent. It therefore seems possible to conclude that at ages around 70 years the genetic contribution to health status deriving from having one centenarian parent is not substantially improved if the other parent is also long-lived

Critical behavior in colloid-polymer mixtures: theory and simulation

We extensively investigated the critical behavior of mixtures of colloids and polymers via the two-component Asakura-Oosawa model and its reduction to a one-component colloidal fluid using accurate theoretical and simulation techniques. In particular the theoretical approach, hierarchical reference theory [Adv. Phys. 44, 211 (1995)], incorporates realistically the effects of long-range fluctuations on phase separation giving exponents which differ strongly from their mean-field values, and are in good agreement with those of the three-dimensional Ising model. Computer simulations combined with finite-size scaling analysis confirm the Ising universality and the accuracy of the theory, although some discrepancy in the location of the critical point between one-component and full-mixture description remains. To assess the limit of the pair-interaction description, we compare one-component and two-component results.Comment: 15 pages, 10 figures. Submitted to Phys. Rev.

Dipolar interaction between two-dimensional magnetic particles

We determine the effective dipolar interaction between single domain two-dimensional ferromagnetic particles (islands or dots), taking into account their finite size. The first correction term decays as 1/D^5, where D is the distance between particles. If the particles are arranged in a regular two-dimensional array and are magnetized in plane, we show that the correction term reinforces the antiferromagnetic character of the ground state in a square lattice, and the ferromagnetic one in a triangular lattice. We also determine the dipolar spin-wave spectrum and evaluate how the Curie temperature of an ensemble of magnetic particles scales with the parameters defining the particle array: height and size of each particle, and interparticle distance. Our results show that dipolar coupling between particles might induce ferromagnetic long range order at experimentally relevant temperatures. However, depending on the size of the particles, such a collective phenomenon may be disguised by superparamagnetism.Comment: 11 pages, 5 figure

Phase equilibria and glass transition in colloidal systems with short-ranged attractive interactions. Application to protein crystallization

We have studied a model of a complex fluid consisting of particles interacting through a hard core and a short range attractive potential of both Yukawa and square-well form. Using a hybrid method, including a self-consistent and quite accurate approximation for the liquid integral equation in the case of the Yukawa fluid, perturbation theory to evaluate the crystal free energies, and mode-coupling theory of the glass transition, we determine both the equilibrium phase diagram of the system and the lines of equilibrium between the supercooled fluid and the glass phases. For these potentials, we study the phase diagrams for different values of the potential range, the ratio of the range of the interaction to the diameter of the repulsive core being the main control parameter. Our arguments are relevant to a variety of systems, from dense colloidal systems with depletion forces, through particle gels, nano-particle aggregation, and globular protein crystallization.Comment: 20 pages, 10 figure