Geometry of three convex bodies applicable to three-molecule clusters in polyatomic gases

An expression for the configuration integral for three overlapping convex bodies, which is a generalization of Hadwiger-Isihara's formula for two convex bodies, has been found. As an application of this expression, two- and three-molecule cluster integrals (or second and third virial coefficients) for polyatomic molecules in gases are discussed on the basis of a squarewell potential with convex cores.     

Two examples of non-radiative systems—An application of homogeneous functions to Synge's approximation method

A method of finding approximations for the gravitational field of two non-radiative systems is given. The first system consists of a shrinking body with convex boundary, having certain symmetries. The second system consists of two shrinking bodies which, in the first approximation, approach each other along thex ₁-axis with a certain constant relative velocity. The two bodies are assumed to have rotational symmetry around thex ₁-axis.Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.Supported by N.R.C. Grant No. A-5205.     

The excluded volume of hard sphero-zonotopes

Excluded volume effects can account for most ordering transitions in simple liquids and liquid crystals. Starting with the work of Onsager, this has been demonstrated in the case of liquid crystals for a number of simple convex bodies, e.g. sphero-cylinders, for which the orientation-dependent pair-excluded volume could be written down analytically. However, in recent years, experiments and simulations have been reported on ordering transitions in suspensions of more complex convex colloids. For these systems, theoretical understanding is hampered by the fact that no analytical expressions for the pair-excluded volume were available. Here we show that it is possible to obtain explicit expressions for the pair-excluded volume of a much larger class of convex bodies: the so-called sphero-zonotopes. These bodies are obtained by ‘padding’ a special class of convex polytopes with a blanket of uniform thickness. The resultant family of particles encompasses a wide range of shapes that have been considered as models for fluid and liquid crystalline behaviour e.g. spheres, cubes, sphero-cylinders, sphero-platelets. We discuss two explicit examples: sphero-cuboids, the 3D core generalization of the sphero-cylinder and the sphero-platelet, and hexagonal prisms that are models for the recently synthesized colloidal gibbsite platelets. Employing the fact that a cylinder is a zonoid, i.e. the limit of a sequence of right regular prisms, we are able to compute the excluded volume of the ‘true’ sphero-cylinder, a uniformly padded cylinder, of which the oblate-spherocylinder is a known example. Our approach en passant provides a relatively elementary rederivation of Onsager's classical result on cylinders.     

Dense regular packings of irregular nonconvex particles

We present a new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner. We employ this method to analyze regular packings of odd-shaped bodies, both from a nanoparticle and from a computational geometry perspective. Besides determining close-packed structures for 17 irregular shapes, we confirm several conjectures for the packings of a large set of 142 convex polyhedra and extend upon these. We also prove that we have obtained the densest packing for both rhombicuboctahedra and rhombic enneacontrahedra and we have improved upon the packing of enneagons and truncated tetrahedra.     

航天器表面环境散射返回流TPMC模拟

介绍试验粒子Monte Carlo(test particle Monte Carlo,TPMC)方法,并采用该方法对4种航天器表面出气分子形成的环境散射返回流进行数值模拟.其中,圆球出气表面的计算结果与已有的DSMC(direct simulation Monte Carlo)结果一致,验证了方法的正确性.此外,对不同出气和来流条件下圆形平板、凸半球和凹半球3种航天器简化表面出气分子形成的环境散射返回流进行计算,结果表明:出气表面外形是影响返回通量比的一个重要因素;圆形平板和凹半球出气表面的返回通量比远大于凸半球表面的;凹半球表面的出气分子会直接和出气表面碰撞形成直接流污染,且其量级远大于返回流污染.因此,在航天器设计中尽可能使用凸形表面作为敏感的出气表面可以有效降低出气分子污染.     

A unified treatment of the equation of state of hard linear bodies

An unified treatment of the equation of state of convex (spherocylinders) and non-convex (dumbells) hard bodies is presented. Comparison of our results with simulation shows very good agreement for diatomics and excellent for triatomics.     

Monte-Carlo integration for virial coefficients re-visited: hard convex bodies,spheres with a square-well potential and mixtures of hard spheres

Techniques to adapt the hit-and-miss Monte-Carlo numerical integration are proposed with the aim to determine virial coefficients up to eighth order in fluids of hard convex bodies, hard spheres with an attractive square-well potential and a two-component mixture of hard spheres. These algorithms make use of look-up tables of all the blocks contributing to the coefficients. Each type of block is represented in the tables by several entries. These correspond to all possible topologically equivalent graphs that can be generated by the Monte-Carlo process. This rendered the Monte-Carlo method statistically more efficient. In the case of a two-component system the look-up tables had to have representations of blocks having two sorts of vertices. The reported data are: improved values of the seventh and eighth virial coefficients for hard spheres, the sixth, seventh and eighth coefficients of spheroids, spherocylinders and cutspheres, fifth virial coefficient of spheres with a square-well potential of relative range 1.25; 1.5; 1.75 and 2.0 and the partial contributions of the sixth virial coefficient for a mixture of hard spheres with the size ratio 0.1.     

First numerical investigation of a conjecture by N. N. Nekhoroshev about stability in quasi-integrable systems

We investigate numerically a conjecture by N. N. Nekhoroshev about the influence of a geometric property, called steepness, on the long term stability of quasi-integrable systems. In a Nekhoroshev's 1977 paper, it is conjectured that, among the steep systems with the same number ν of frequencies, the convex ones are the most stable, and it is suggested to investigate numerically the problem. Following this suggestion, we numerically study and compare the diffusion of the actions in quasi-integrable systems with different steepness properties in a large range of variation of the perturbation parameter ? and different dimensions of phase space corresponding to ν?=?3 and ν?=?4 (ν ≤ 2 is not significant for the conjecture). For six dimensional maps (ν?=?4), our numerical experiments perfectly agree with the Nekhoroshev conjecture: for both convex and non convex cases, the numerically computed diffusion coefficient D of the actions is compatible with an exponential fit, and the convex case is definitely more stable than the steep one. For four dimensional maps (ν?=?3), since we find that in the steep case D(?) has large oscillations around an exponential behaviour, the agreement of our numerical experiments with the conjecture is not sharp, and it is found by considering a sup over different initial conditions.     

Morphological thermodynamics for hard bodies from a controlled expansion

ABSTRACT

The morphometric approach is a powerful ansatz for decomposing the chemical potential for a complex solute into purely geometrical terms. This method has proven accuracy in hard spheres, presenting an alternative to comparatively expensive (classical) density functional theory approaches. Despite this, fundamental questions remain over why it is accurate and how one might include higher-order terms to improve accuracy. We derive the morphometric approach as the exact resummation of terms in the virial series, providing further justification of the approach. The resulting theory is less accurate than previous morphometric theories, but provides fundamental insights into the inclusion of higher-order terms and to extensions to mixtures of convex bodies of arbitrary shape.     

Analytical Model of Doppler Spectra of Light Backscattered from Rotating Convex Bodies of Revolution in the Global Cartesian Coordinate System

We present an analytical model of Doppler spectra in backscattering from arbitrary rough convex bodies of revolution rotating around their axes in the global Cartesian coordinate system. This analytical model is applied to analyse Doppler spectra in backseatter from two cones and two cylinders, as well as two ellipsoids of revolution. We numerically analyse the influences of attitude and geometry size of objects on Doppler spectra. The analytical model can give contribution of the surface roughness, attitude and geometry size of convex bodies of revolution to Doppler spectra and may contribute to laser Doppler velocimetry as well as ladar applications.     

Master-slave synchronization of Lur'e systems with sector and slope restricted nonlinearities

This Letter presents a synchronization method for Lur'e systems with sector and slope restricted nonlinearities. A static error feedback controller based on the Lyapunov stability theory is proposed for asymptotic synchronization. The Lyapunov function candidate is chosen as a quadratic form of the error states and nonlinear functions of the systems. The nonlinearities are expressed as convex combinations of sector and slope bounds by using convex properties of the nonlinear function so that equality constraints are converted into inequality constraints. Then, the feedback gain matrix is derived through a linear matrix inequality (LMI) formulation. Finally, a numerical example shows the effectiveness of the proposed method.     

Entanglement and Distillability in Qutrit-Qutrit Systems by Convex Linear Combination

We discuss entanglement and distillability in qutrit-qutrit systems by convex linear combination. Inspired by the seminal Horodecki quantum states, we explicitly construct three pairs of qutrit-qutrit systems. We find that convex linear combination states can evolve from entangled into non-entangled states and from distillable into non-distillable states, and vice versa. The new and different angle from convex linear combination states can be helpful for us to understand entanglement and distillability.     

Remarks on Radial Centres of Convex Bodies

The paper concerns a family of selectors for convex bodies in Rⁿ, radial centre maps, defined in the article of M. Moszyńska, Looking for selectors of star bodies, Geom. Dedicata 81 (2000), 131–147. A radial centre of a convex body A is the maximizer of a suitable generalized dual volume of A. We give physical interpretations of the notion of radial centre and study its geometric properties. We prove that these selectors are continuous with respect to the Hausdorff metric and solve the problem of direct additivity for radial centre of order α, which corresponds to the dual volume of order α. Mathematics Subject Classifications (2005) 52A20, 52A40, 51P05, 85A25, 86A20.     

Comparison of two non-probabilistic approaches for the energy flow uncertainty in structural vibrating system

Two non-probabilistic, set-theoretical methods for determining the energy flow between two structural multimodal systems coupled by a joint with uncertain parameters are presented. They are based on the theories of interval mathematics and convex models. The uncertain parameters of the joint are assumed to be a convex set, hyper-rectangle or ellipsoid. For both non-probabilistic methods, less prior information about the uncertain nature is required than that which is required concerning the probabilistic model. The properties of the standard interval arithmetic, the interval Taylor expansion, the convex models and the Monte Carlo Method solutions are investigated and compared.     

Existence and vanishing of the breathing mode in strongly correlated finite systems

One of the fundamental eigenmodes of finite interacting systems is the mode of uniform radial expansion and contraction-the breathing mode (BM). Here we show in a general way that this mode exists only under special conditions: (i) for harmonically trapped systems with interaction potentials of the form 1/rgamma (gamma in R not equal 0) or log(r), or (ii) for some systems with special symmetry such as single-shell systems forming platonic bodies. Deviations from the BM are demonstrated for two examples: clusters interacting with a Lennard-Jones potential and parabolically trapped systems with Yukawa repulsion. We also show that vanishing of the BM leads to the occurrence of multiple monopole oscillations which is of importance for experiments.