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We simulate freely jointed chains to investigate how knotting affects the overall shapes of freely fluctuating circular polymeric chains. To characterize the shapes of knotted polygons, we construct enveloping ellipsoids that minimize volume while containing the entire polygon. The lengths of the three principal axes of the enveloping ellipsoids are used to define universal size and shape descriptors analogous to the squared radius of gyration and the inertial asphericity and prolateness. We observe that polymeric chains forming more complex knots are more spherical and also more prolate than chains forming less complex knots with the same number of edges. We compare the shape measures, determined by the enveloping ellipsoids, with those based on constructing inertial ellipsoids and explain the differences between these two measures of polymer shape. [ABSTRACT FROM AUTHOR]

The depletion interactions of ellipsoidal colloidal particles in a solution of long polymer chains are analyzed. Of primary concern are the limiting cases in which the ellipsoid reduces to a cylinder of infinite length and finite radius and a “needle” of finite length and vanishing radius. Relations are obtained between the polymer effects induced by a needle that is much shorter than the polymer size and by a cylinder with radius much smaller than the polymer size. These imply exact quantitative results for the orientation-dependent depletion interaction between a short needle and a wall. Qualitative differences between the needle and thin disk are discussed. [ABSTRACT FROM AUTHOR]

We have carried out NPT molecular dynamics simulations of isolated ellipsoids in a glass forming binary mixture to gain insight into the nature of orientational relaxation (OR) in a viscous liquid. At high pressures when the liquid is highly viscous, the OR is found to occur mainly via correlated hopping, sometimes involving participation of several neighboring atoms, placed in a ring like tunnel. In the glassy state, hopping is found to be accompanied by larger fluctuations in the total energy and the volume of the system. Both orientational and translational hopping are found to be gated, restricted primarily by the entropic bottlenecks, with the orientational motion becoming increasingly slower than the translation as the pressure is increased. Orientational relaxation is found to occur with a wide distribution of decay times. [ABSTRACT FROM AUTHOR]

q-deformations of functions and distributions have been used in the literature to explain several experimental observations. In this work, we study the dynamics of the Tinkerbell map under q -deformations. The system exhibits a rich variety of dynamical behavior as q varies, including occurrences of interior crises, paired cascades, simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcations, and co-existence of attractors and multistability. Numerical analysis reveals the existence of 3 limit cycles occurring simultaneously in a certain parameter regime. An appropriate choice of initial conditions enables one to choose a desired attractor for the system among other co-existing ones, thus switching the system between different dynamical states. We demonstrate the possibility of secure encryption and decryption of messages with the q -deformed Tinkerbell map. The system's sensitivity to the initial conditions and to the deformation parameter makes the cryptic message secure, and decrypting the original message difficult. We propose the use of the q -deformed map as a novel method for transmission of messages securely. [ABSTRACT FROM AUTHOR]

The transition from the isotropic to nematic phase is investigated for a fluid comprised of bendable ellipsoids. When the bending amplitude is chosen such that the bodies remain convex, density functional-Onsager methods can be employed to estimate the phase transition properties. The convexity requirement places a restriction on the size of the effects; however, a trend is discernible. As the density of the fluid increases, in both the isotropic and nematic phases, the mean bending amplitude is suppressed. Furthermore, the density of the isotropic to nematic transition increases with the bending amplitude in accord with the predictions of the Khokhlov–Semenov theory. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper we describe a general approach which allows the hypernetted chain (HNC) and Percus–Yevick (PY) integral equation theories to be solved numerically for fluids of hard nonspherical particles. Explicit results are given for fluids of hard ellipsoids of revolution and comparisons are made with recent Monte Carlo calculations. It is found that for dense systems of highly anisotropic ellipsoids the HNC and PY closures give significantly different results. The HNC theory is superior predicting the existance of a nematic phase in qualitative agreement with computer simulations. The PY approximation strongly and erroneously suggests that the isotropic phase is stable throughout the liquid regime. [ABSTRACT FROM AUTHOR]

We investigate the problem of minimizing the moment of inertia among convex surfaces in R3 having a specified surface area. First, we prove that a minimizing surface exists, and derive a necessary condition holding at points of positive curvature. Then we show that an equilateral triangular prism is the optimal triangular prism, that the cube is the optimal rectangular prism, and that the sphere is (locally) optimal among ellipsoids. Many examples of convex surfaces are examined, among which the lowest moment of inertia is achieved by a truncated tetrahedron. The problem of finding the global minimizing surface remains open. The analogous problem in two dimensions has been solved by Sachs and later by Hall, who showed that the equilateral triangle minimizes the moment of inertia, among all convex curves with given length. [ABSTRACT FROM AUTHOR]

A finite line of uniform charge is either ignored or handled incompletely in most textbooks. New simple and practical expressions are presented for the electric potential and electric field for this charge distribution. The equipotentials are shown to be prolate ellipsoids and the electric field lines follow hyperboloids confocal to the ellipsoids. The common foci are shown to be the endpoints of the charged line segment. These geometries are demonstrated with simple VPython programs. [ABSTRACT FROM AUTHOR]

The kinetic relaxation of a monatomic gas in a state of gross rest is studied by use of Maxwell’s second-order moment equation. The molecular models considered range from the hard sphere to the Maxwell molecule. The development is exact for an ellipsoidal distribution and becomes a good approximation for more general distributions. The results show that the relaxation of second-order moments is nearly exponential for power-law molecules and that the characteristic time μ/p appears as the principal controlling parameter while the power-law constant plays a secondary role. Comparisons between results of numerical simulations using Bird’s direct simulation Monte Carlo method and theory show excellent agreement for both hard-sphere and Maxwell molecules. [ABSTRACT FROM AUTHOR]

An analytical method to study the magnetization behavior of a system composed of two magnetic particles is presented, when the magnetic field is applied in-plane. The study is based on a single-domain model, assuming that the particles are two ellipsoids with both exchange and magnetostatic coupling. The model is used to study the synthetic antiferromagnetic structure and a pair of particles with only dipolar interaction for bond angles β=0 and π/2. When the magnetic field is applied along the easy axis a complete description for the system’s minima, activation energy, and possible switches is performed. [ABSTRACT FROM AUTHOR]

We present a strategy for control of chaos in open flows and provide its experimental validation in the near field of a transitional jet flow system. The low-dimensional chaotic dynamics studied here results from vortex ring formation and their pairings over a spatially extended region of the flow that was excited by low level periodic forcing of the primary instability. The control method utilizes unstable periodic orbits (UPO) embedded within the chaotic attractor. Since hydrodynamic instabilities in the open flow system are convective, both monitoring and control can be implemented at a few locations, resulting in a simple and effective control algorithm. Experiments were performed in an incompressible, initially laminar, 4 cm diameter circular air jet, at a Reynolds number of 23 000, housed in a low-noise, large anechoic chamber. Distinct trajectory bundles surrounding the dominant UPOs were found from experimentally derived, time-delayed embedding of the chaotic attractor. Velocity traces from a pair of probes placed at the jet flow exit and farther downstream were used to empirically model the UPOs and compute control perturbations to be applied at the jet nozzle lip. Open loop control was used to sustain several nearly periodic states. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamics of self-gravitating ellipsoids in n dimensions, from the point of view of the Poisson structure of the dual of a suitable Lie algebra. When n=3 this was done by Rosenteel. In this setting we describe explicitly the ring of invariant functions. In the two-dimensional case we apply a technique due to Pedersen to find globally defined Darboux coordinates on the coadjoint orbits. Finally we derive a characterization of tensors coming from potentials, and use it to exhibit the dynamical equations of the ellipsoid in the Hamiltonian form. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]