Symmetric Equilibrium Configurations Research Articles

Equilibrium of bi-stable flexural-tensegrity segmental beams

One dimensional discrete systems composed of a simple chain of bi-stable springs, with nearest neighbor interaction, have been used to interpret the complex equilibrium states of materials supporting multiple crystallographic phases, foldable macromolecules and biological structures. A discrete system is here proposed for bending within the broad class of flexural-tensegrity beams, which consist of segments in unilateral contact, with tailored-shaped contact surfaces, pre-stressed by an unbonded tendon. Any contact joint shows a bi-stable response to its relative rotation such as a snap-spring hinge, thanks to the internal carving of the segments that increases the mobility of the tendon within them. The constitutive response is nonlocal, because the tendon is free to slide within well lubricated sheaths. The case of pure bending, representing the counterpart of uniaxial tension in the one-dimensional lattice chain, shows that the system can support stable and metastable configurations, possibly containing one snap-spring hinge in the spinodal part of the energy landscape. Remarkably, not only the maximum hysteresis paths, but also the Maxwell paths, are strain-hardening in type. This is due to the nonlocal effect from the unbonded tendon: the rotation of any contact joint stiffens all the other joints, so that the orderly snaps of the spring-hinges occur at an increasing bending moment. An experimental program has been conducted on 3D printed physical models either in a hard or soft device. Symmetric and non-symmetric equilibrium configurations are obtained that are in perfect agreement with the theoretical predictions. Possible applications are envisaged, but they are yet to be fully appreciated.

Oscillation modes of ultralight BEC dark matter cores

Structure formation simulations of ultralight bosonic dark matter, ruled by the Gross-Pitaevskii-Poisson (GPP) system of equations, show that dark matter clumps to form structures with density profiles consisting of a core surrounded by a power-law distribution. The core has a density profile similar to that of spherically symmetric equilibrium configurations of the GPP system. These configurations have been shown to be stable under a variety of perturbations and to have attractor properties. It is interesting to know the dominant frequencies of oscillation of these configurations. One reason is that in galaxies the effects of perturbations can trigger observable effects for specific frequency modes. Another reason is that during the process of structure formation, the oscillation modes can help to characterize a core. Based on these motivations, in this manuscript we present a systematic numerical analysis of the reaction of equilibrium configurations to various axi-symmetric perturbations with modes $l=0,1,2,3,4$. We then calculate the first few oscillation frequencies of equilibrium configurations for each mode.

Equilibrium and stability of relativistic stars in extended theories of gravity

We study static, spherically symmetric equilibrium configurations in extended theories of gravity (ETG) following the notation introduced by Capozziello et {\it al}. We calculate the differential equations for the stellar structure in such theories in a very generic form i.e., the Tolman-Oppenheimer-Volkoff generalization for any ETG is introduced. Stability analysis is also investigated with special focus on the particular example of scalar-tensor gravity.

General relativistic polytropes with a repulsive cosmological constant

Spherically symmetric equilibrium configurations of perfect fluid obeying a polytropic equation of state are studied in spacetimes with a repulsive cosmological constant. The configurations are specified in terms of three parameters---the polytropic index $n$, the ratio of central pressure and central energy density of matter $\ensuremath{\sigma}$, and the ratio of energy density of vacuum and central density of matter $\ensuremath{\lambda}$. The static equilibrium configurations are determined by two coupled first-order nonlinear differential equations that are solved by numerical methods with the exception of polytropes with $n=0$ corresponding to the configurations with a uniform distribution of energy density, when the solution is given in terms of elementary functions. The geometry of the polytropes is conveniently represented by embedding diagrams of both the ordinary space geometry and the optical reference geometry reflecting some dynamical properties of the geodesic motion. The polytropes are represented by radial profiles of energy density, pressure, mass, and metric coefficients. For all tested values of $n>0$, the static equilibrium configurations with fixed parameters $n$, $\ensuremath{\sigma}$, are allowed only up to a critical value of the cosmological parameter ${\ensuremath{\lambda}}_{\mathrm{c}}={\ensuremath{\lambda}}_{\mathrm{c}}(n,\ensuremath{\sigma})$. In the case of $n>3$, the critical value ${\ensuremath{\lambda}}_{\mathrm{c}}$ tends to zero for special values of $\ensuremath{\sigma}$. The gravitational potential energy and the binding energy of the polytropes are determined and studied by numerical methods. We discuss in detail the polytropes with an extension comparable to those of the dark matter halos related to galaxies, i.e., with extension $\ensuremath{\ell}>100\text{ }\text{ }\mathrm{kpc}$ and mass $M>1{0}^{12}\text{ }{\mathrm{M}}_{\ensuremath{\bigodot}}$. For such largely extended polytropes, the cosmological parameter relating the vacuum energy to the central density has to be larger than $\ensuremath{\lambda}={\ensuremath{\rho}}_{\text{vac}}/{\ensuremath{\rho}}_{\mathrm{c}}\ensuremath{\sim}{10}^{\ensuremath{-}9}$. We demonstrate that the extension of the static general relativistic polytropic configurations cannot exceed the so-called static radius related to their external spacetime, supporting the idea that the static radius represents a natural limit on the extension of gravitationally bound configurations in an expanding universe dominated by the vacuum energy.

Thermally-induced chemical-order transitions in medium–large alloy nanoparticles predicted using a coarse-grained layer model

A new coarse-grained layer model (CGLM) for efficient computation of axially symmetric elemental equilibrium configurations in alloy nanoparticles (NPs) is introduced and applied to chemical-order transitions in Pt-Ir truncated octahedra (TOs) comprising up to tens of thousands of atoms. The model is based on adaptation of the free energy concentration expansion method (FCEM) using coordination-dependent bond-energy variations (CBEV) as input extracted from DFT-computed elemental bulk and surface energies. Thermally induced quite sharp transitions from low-T asymmetric quasi-Janus and quasi ball-and-cup configurations to symmetric multi-shells furnish unparalleled nanophase composite diagrams for 1289-, 2406- and 4033-atom NPs. At even higher temperatures entropic atomic mixing in the multi-shells gradually intensifies, as reflected in broad heat-capacity Schottky humps, which become sharper for much larger TOs (e.g., ∼10 nm, ∼30,000 atoms), due to transformation to solid-solution-like cores.

Two-vortex equilibrium in the flow past a flat plate at incidence

AbstractThe two-dimensional inviscid incompressible steady flow past an inclined flat plate is considered. A locus of asymmetric equilibrium configurations for vortex pairs is detected. It is shown that the flat geometry has peculiar properties compared to other geometries: (i) in order to satisfy the Kutta condition at both edges, which ensures flow regularity, the total circulation and the force acting on the plate must be zero; and (ii) the Kutta condition and the free vortex equilibrium conditions are not independent of each other. The non-existence of symmetric equilibrium configurations for an orthogonal plate is extended to more general asymmetric flows.

Distinguishing black holes from naked singularities through their accretion disc properties

We show that, in principle, a slowly evolving gravitationally collapsing perfect fluid cloud can asymptotically settle to a static spherically symmetric equilibrium configuration with a naked singularity at the center. We consider one such asymptotic final configuration with a finite outer radius, and construct a toy model in which it is matched to a Schwarzschild exterior geometry. We examine the properties of circular orbits in this model. We then investigate the observational signatures of a thermal accretion disc in this spacetime, comparing them with the signatures expected for a disc around a black hole of the same mass. Several notable differences emerge. A disc around the naked singularity is much more luminous than one around an equivalent black hole. Also, the disc around the naked singularity has a spectrum with a high frequency power law segment that carries a major fraction of the total luminosity. Thus, at least some naked singularities can, in principle, be distinguished observationally from the black holes of the same mass. We discuss the possible implications of these results.

Chameleon dark matter stars

We consider static, spherically symmetric equilibrium configurations consisting of fermionic dark matter nonminimally coupled to dark energy in the form of a quintessence scalar field. With the scalar field coupling function, the form of which is taken to meet cosmological observations, we estimate the effect of such a nonminimal coupling on the properties of dark matter compact configurations. We show that the masses and sizes of the resulting chameleon dark matter stars are smaller than those of systems with no field present.

A quantum chemical study of silsesquioxanes: H8Si8O12, Me8Si8O12, H@Me8Si8O12, He@Me8Si8O 12 + , and He@Me8Si8O12

Quantum chemical PBE0 and B3LYP/cc-pVTZ, PBE0, B3LYP, RHF and MP2/6-31G(d,p) methods are employed to calculate the structural parameters of octa(silsesquioxane) H8Si8O12 and octa(methylsilsesquioxane) Me8Si8O12. These molecules and complexes H@Me8Si8O12, He@Me8Si8O 12 + , and He@Me8Si8O12 have highly symmetric (O h ) equilibrium configurations. With the use of the PBE0 method and a cc-pVTZ multicenter basis set common for the complex and its components coincidence is achieved between the calculated polarizability of a free He atom and the experimental value of 0.21 A3 and the polarizability depression of 0.17 A3 was found for He@Me8Si8O12. In order to avoid the false conclusion about molecular symmetry the calculations of the structure of silsesquioxanes must be performed with sufficiently high accuracy (Int = ultrafine and Opt = tight in the use of the GAUSSIAN program).

Mixed neutron-star-plus-wormhole systems: Equilibrium configurations

We study gravitationally bound, spherically symmetric equilibrium configurations consisting of ordinary (neutron-star) matter and of a phantom/ghost scalar field which provides the nontrivial topology in the system. For such mixed configurations, we show the existence of static, regular, asymptotically flat general relativistic solutions. Based on the energy approach, we discuss the stability as a function of the core density of the neutron matter for various sizes of the wormhole throat.

Hamiltonian magnetohydrodynamics: Helically symmetric formulation, Casimir invariants, and equilibrium variational principles

The noncanonical Hamiltonian formulation of magnetohydrodynamics (MHD) is used to construct variational principles for continuously symmetric equilibrium configurations of magnetized plasma, including flow. In particular, helical symmetry is considered, and results on axial and translational symmetries are retrieved as special cases of the helical configurations. The symmetry condition, which allows the description in terms of a magnetic flux function, is exploited to deduce a symmetric form of the noncanonical Poisson bracket of MHD. Casimir invariants are then obtained directly from the Poisson bracket. Equilibria are obtained from an energy-Casimir principle and reduced forms of this variational principle are obtained by the elimination of algebraic constraints.

Relativistic polytropic spheres embedded in a chameleon scalar field

In this paper we investigate gravitationally bound, spherically symmetric equilibrium configurations consisting of ordinary (polytropic) matter nonminimally coupled to an external chameleon scalar field. We show that this system has static, regular, asymptotically flat general relativistic solutions. The properties of these spherical configurations, such as total mass, distribution of matter, and size, depend strongly on the surrounding scalar field. The mass is found in terms of the parameter $\sigma$, which is proportional to the central mass density of the ordinary matter. We perform a stability analysis of these spherical solutions and find an upper bound for $\sigma$ where dynamical instability occurs.

Experimental Measurement of Asymmetric Fluctuations of Poloidal Magnetic Field in Damavand Tokomak at Different Plasma Currents

Toroidal and Poloidal magnetic fields have an important effect on the tokomak topology. Damavand Tokomak is a small size tokomak characterized with k = 1.2, B t = 1T, R 0 = 36 cm, maximum plasma current is about 35 KA with a discharge time of 21 ms. In this experimental work, the variation of poloidal magnetic field on the torodial cross section is measured and analyzed. In order to measure the polodial magnetic field, 18 probes were installed on the edge of tokomak plasma with ∆θ = 18°, while a limiter was installed inside the torus. Plasma current, I p, induces a polodial magnetic field, B p, smaller than the torodial magnetic field B t. Magnetic lines B produced as a combination of B t and B p, are localized on the nested toroidal magnetic surfaces. The presence of polodial magnetic field is necessary for particles confinement. Mirnov oscillations are the fluctuations of polodial magnetic field, detected by magnetic probes. Disrupted instability in Tokomak typically starts with mirnov oscillations which appear as fluctuations of polodial magnetic field and is detected by magnetic probes. Minor disruptions inside the plasma can contain principal magnetic islands and their satellites can cause the annihilation of plasma confinement. Production of thin layer of turbulent magnetic field lines cause minor disruption. Magnetic limiter may cause the deformation of symmetric equilibrium configuration and chaotic magnetic islands reveal in plasma occurring in thin region of chaotic field lines close to their separatrix. The width of this chaotic layer in the right side of poloidal profile of Damavand Tokomak is smaller than the width in the left side profile because of Shafranov displacement. Ergodic region in the left side of profile develops a perturbation on the magnetic polodial field lines, B p, that are greater in magnitude than that in the right side, although the values of B p on the left side are smaller than that on the right side of the profile. The Left side of profile is close to the principal magnetic axis and the right side is away from Z axis of Tokamak.

Analysis of a Class of Symmetric Equilibrium Configurations for a Territorial Model

Motivated by an animal territoriality model, we consider a centroidal Voronoi tessellation algorithm from a dynamical systems perspective. In doing so, we discuss the stability of an aligned equilibrium configuration for a rectangular domain that exhibits interesting symmetry properties. We also demonstrate the procedure for performing a center manifold reduction on the system to extract a set of coordinates which capture the long term dynamics when the system is close to a bifurcation. Bifurcations of the system restricted to the center manifold are then classified and compared to numerical results. Although we analyze a specific set-up, these methods can in principle be applied to any bifurcation point of any equilibrium for any domain. AMS subject classifications: 37N25, 37G10

Chemo-dynamical evolution of dwarf galaxies: from flat to cuspy dark matter density profiles

A model of an isolated dwarf spherical galaxy (dSph) is considered in its chemo-dynamical evolution with time. The system is composed by 3 γ-model density profiles: gas, stellar and dark matter, and it is realized in a spherical symmetric equilibrium configuration. The total masses used in our simulations are covering the dwarf galaxies mass range. The stability of this configuration is first tested for the system evolving under the gravity effect alone and then evolved taking into account for the most relevant stellar astrophysical processes implemented with a Smooth Particle Hydrodynamic approach. The two different kinds of evolution are compared. The dark matter evolves naturally from a centrally cuspy density profile into a flatter one within a timescale of several Gyr. The effect manifests itself naturally, without any tuned initial conditions, as soon as few standard criteria on star formation are assumedand the SN feedback on the ISM has been adopted the prescription in (Cioffi & Shull 1991) and (Bradamante et al. 1998). This result is expected to be a possible natural explanation for the discrepancy between observations that want flatter dark matter profiles (e.g. de Block 2005), and N-body simulations that predict cuspy dark matter profiles (Navarro et al. 1997). Chemical considerations are presented as a tool to follow with observational parameters the theory predictions.

Axially symmetric membranes with polar tethers

Axially symmetric equilibrium configurations of the conformally invariant Willmore energy are shown to satisfy an equation that is two orders lower in derivatives of the embedding functions than the equilibrium shape equation, not one as would be expected on the basis of axial symmetry. Modulo a translation along the axis, this equation involves a single free parameter c. If c ≠ 0, a geometry with spherical topology will possess curvature singularities at its poles. The physical origin of the singularity is identified by examining the Noether charge associated with the translational invariance of the energy; it is consistent with an external axial force acting at the poles. A one-parameter family of exact solutions displaying a discocyte to stomatocyte transition is described.

Stability of Cyclically Symmetric Strained Pumpkin Balloons and the Formation of Undesired Equilibria

NASA's effort to develop a large payload, high altitude, long-duration balloon, the ultralong duration balloon, focuses on a pumpkin shape superpressure design. It has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Hoop stress considerations in the pumpkin design lead to choosing the lowest possible bulge radius, whereas robust deployment is favored by a large bulge radius. Mechanical locking may be a contributing factor in the formation of undesired equilibria. Long term success of the pumpkin balloon for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria. This paper uses the notion of stability to classify balloon designs. When we applied our finite element model to a balloon based on the NASA Phase IV-A pumpkin design, we found the fully inflated/fully deployed strained equilibrium float configuration was unstable. To demonstrate our approach for exploring the stability of constant bulge radius designs and their sensitivity to parameter changes we carry out a number of parametric studies. We focus on analytical studies, but we also compare our results with flight data whenever possible.

Cleft formation in pumpkin balloons

NASA’s development of a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, centers on a pumpkin shape super-pressure design. Under certain circumstances, it has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Success of the pumpkin balloon for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria and developing of means for the quantitative assessment of design measures that prevent the occurrence of undesired equilibrium. In this paper, we will use the concept of stability to classify cyclically symmetric equilibrium states at full inflation and pressurization. Our mathematical model for a strained equilibrium balloon, when applied to a shape that mimics the Phase IV-A balloon of Flight 517, predicts instability at float. Launched in Spring 2003, this pumpkin balloon failed to deploy properly. Observations on pumpkin shape type super-pressure balloons that date back to the 1980s suggest that within a narrowly defined design class of pumpkin shape super-pressure balloons where individual designs are fully described by the number of gores n g and by a single measure of the bulging gore shape, the designs tend to become more vulnerable with the growing number of gores and with the diminishing size of the bulge radius r B Weight efficiency considerations favor a small bulge radius, while robust deployment into the desired cyclically symmetrical configuration becomes more likely with an increased bulge radius. In an effort to quantify this dependency, we will explore the stability of a family of balloon shapes parametrized by ( n g, r B) which includes a design that is very similar, but not identical, to the balloon of Flight 517. In addition, we carry out a number of simulations that demonstrate other aspects related to multiple equilibria of pumpkin balloons.

THE EAST END, THE WEST END, AND KING'S CROSS: ON CLUSTERING IN THE FOUR‐PLAYER HOTELLING GAME

We study experimentally a standard four‐player Hotelling location game with a uniform density of consumers and inelastic demand. The pure strategy Nash equilibrium configuration consists of two firms located at one quarter of the “linear city,” and the other two at three quarters. We do not observe convergence to such an equilibrium. In our experimental data we find three clusters. Besides the direct proximity of the two equilibrium locations, this concerns the focal midpoint. Moreover, we observe that whereas this midpoint appears to become more notable over time, other focal points fade away. We explain how these observations are related to best‐response dynamics, and to the fact that the players rely on best‐responses in particular when they are close to the equilibrium configuration.

Axisymmetric Self‐similar Equilibria of Self‐gravitating Isothermal Systems

We show that there are two families of axisymmetric self-similar equilibria of self-gravitating, rotating, isothermal systems: (1) cylindrically symmetric solutions in which the density varies with cylindrical radius as R-2(n+1), with -1 ≤ n ≤ 0 and (2) axially symmetric solutions in which the density varies as f(θ)/r2, where r is the spherical radius and θ is the colatitude. The singular isothermal sphere is a special case of the latter class with f(θ) = constant. The axially symmetric equilibrium configurations form a two-parameter family of solutions and include equilibria which are asymmetric with respect to the equatorial plane. The asymmetric equilibria are, however, not force-free at the singular points r = 0,∞, and their relevance to real systems is unclear. For each hydrodynamic equilibrium, we determine the phase-space distribution of the collisionless analog.