Intersecting disks (and spheres) and statistical mechanics. I. Mathematical basis

The calculations of a previous paper on intersecting disks are completed. Further quantities of interest in connection with intersecting disks and spheres are defined. The above considerations are extended to spherical boundary conditions. Then two applications are stated: The penetrable-sphere model of Widom and Rowlinson, and the hard-sphere system. Finally, the generalization toD-dimensional spheres is outlined.     

Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral b ₂(V) related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.     

Chord length distributions in binary stochastic media in two and three dimensions

Simple models for transport through stochastic media usually assume that the chord lengths of materials are distributed exponentially. Theory predicts that, in a medium consisting of disks/spheres that can interpenetrate, chord lengths in the background material (between the disks/spheres) should exactly follow an exponential. In a medium with impenetrable (non-overlapping) disks/spheres, the distribution is only approximately exponential. This paper demonstrates, through direct numerical simulations, that for randomly distributed disks in 2D and spheres in 3D, with distributions of radii, chord lengths in the background material (between the disks/spheres) are accurately described by exponentials over five orders of magnitude when the material is dilute. The chord lengths inside the disks and spheres are not exponentially distributed, but those distributions can be calculated. A scaling relationship between the mean chord lengths in the two materials is presented for an infinite medium. By knowing the mean properties of the disks/spheres in a medium, this relationship allows one to accurately describe the statistical properties of the background material. The stochastic simulations are validated by this infinite medium relationship. When the fraction of space occupied by the disks or spheres becomes large, the distributions are no longer accurately described by an exponential.     

Transport coefficients and mechanical response in hard-disk colloidal suspensions

We investigate the transport properties and mechanical response of glassy hard disks using nonlinear Langevin equation theory.We derive expressions for the elastic shear modulus and viscosity in two dimensions on the basis of thermalactivated barrier-hopping dynamics and mechanically accelerated motion.Dense hard disks exhibit phenomena such as softening elasticity,shear-thinning of viscosity,and yielding upon deformation,which are qualitatively similar to dense hard-sphere colloidal suspensions in three dimensions.These phenomena can be ascribed to stress-induced "landscape tilting".Quantitative comparisons of these phenomena between hard disks and hard spheres are presented.Interestingly,we find that the density dependence of yield stress in hard disks is much more significant than in hard spheres.Our work provides a foundation for further generalizing the nonlinear Langevin equation theory to address slow dynamics and rheological behavior in binary or polydisperse mixtures of hard or soft disks.     

A simple GUI for modeling the optical properties of single metal nanoparticles

We present a graphical user interface, based on the modified long-wavelength approximation, to compute the optical properties of single metal nanoparticles of ellipsoidal geometry (spheres, rods, and disks). The user-friendly interface allows one to readily gauge the accuracy of results obtained using the modified long-wavelength approximation. For spherical particles, up to 10-nm diameter, we confirm that our approach yields an exact correspondence with Mie theory, and gives an approximation error of less than 15% for gold (silver) particles with diameters approaching 40 nm (26 nm). Results are shown to be sensitive to the source data for the optical constants for a given material. The modular nature of the simulation platform provides a quick and intuitive guide for optical characterization experiments.     

Dual Focus Optical Head for 0.6 mm and 1.2 mm Disks

We propose a dual focus optical head with a holographic optical element (HOE) which corrects spherical aberrations in order to read both 0.6 mm disks and 1.2 mm disks. The thin disks are read using transmitted light and the thick disks are read using +1st order diffracted light of the blazed HOE. The good quality of the focused spots and the focusing error signals was experimentally confirmed. No influence of unnecessary diffracted light was observed.     

Modified cell theory: Free volume distributions and the equation of state for D-dimensional hard sphere systems

A generalized cell model, using cells of different sizes, is applied to hard rods, disks and spheres. Structures is discussed in terms of free volumes. The derived equation of state is exact for rods. For disks and spheres it provides a good approximation in the dense fluid and solid state.     

Molecular Fluids at High Dimensionality

We extend the analysis of orientational first-order phase transitions in anisotropic molecular fluids at high spatial dimensionality to hard-disk fluids, and then to mixture of hard disks and hard spheres. The effect of hard-sphere admixture depends sensitively on the relative sizes of the two geometrical objects, and large spheres completely quench the disk transition. An introductory study is made of spatially ordered states.     

Segregation in binary mixtures under gravity

We employ kinetic theory for a binary mixture to study segregation by size and/or mass in a gravitational field. Simple segregation criteria are obtained for spheres and disks that are supported by numerical simulations.     

Photothermal radiometry with spherical solids

We extend the applications of photothermal diagnostics to solid spheres, in which both theoretical and experimental PTR studies on spherical geometries and thermal diffusivity of the sample are discussed. Based on the Green function method, a full thermal-wave field distribution of a spherical solid is obtained. Experimental results with steel spheres of different diameters exhibit good agreement between the theory and the experiments.     

The acoustic basis for target discrimination by FM echolocating bats

Past experiments show that echolocating bats of the species Myotis lucifugus and Eptesicus fuscus can discriminate among airborne sonar targets presented in the context of pursuit maneuvers for the interception of prey. These bats distinguish between edible mealworms and inedible spheres of various sizes. Myotis can distinguish between disks and mealworms similar enough in size that the bat's performance requires the ability to perceive the acoustic equivalent of target shape. Previously observed small differences in the spectrum of echoes from mealworms and disks appear insufficient to distinguish these targets at the performance levels achieved by bats. We measured the acoustic properties of the targets by broadcasting ultrasonic impulses at mealworms, spheres, and disks and recording their echoes, displaying the results in terms of impulse echo waveforms and the frequency response of targets derived from the target transfer function. The echoes from disks and mealworms at various orientations convey the range-axis profile of the target (number and spacing of reflecting points or glints distributed at different ranges) in terms of the impulse structure of their waveforms and in terms of the locations and spacing of notches or nulls in their spectra. For targets that bats can discriminate and that reflect echoes which do not clearly differ in overall amplitude, the targets appear distinguishable from the acoustic representation of their range profile, which is a feature of targets that bats can perceive with great acuity.     

Some comments on the electrostatic forces between circular electrodes

We study the force between two circular electrodes in different configurations. A formula analogous to Kelvin's formula for the spheres is given in the case of equal disks held at the same potential and when one plate is earthed. An expression for the force at short distance between two arbitrarily charged disks is found: the generic case shows a logarithmic repulsive force, also for disks carrying charges of opposite sign. Some numerical computations support the results. A classification for the possible behaviors of the force is proposed on the basis of a decomposition of the capacitance matrix. It is shown that the forces depend strongly on the dimensionality of the contact zone between the conductors. The analysis is supported by a numerical computation carried for the case of two disks of different radii.     

On spherical harmonics for fuzzy spheres in diverse dimensions

We construct spherical harmonics for fuzzy spheres of even and odd dimensions, generalizing the correspondence between finite Matrix algebras and fuzzy two-spheres. The finite Matrix algebras associated with the various fuzzy spheres have a natural basis which falls in correspondence with tensor constructions of irreducible representations of orthogonal groups SO(n). This basis is useful in describing fluctuations around various D-brane constructions of fuzzy spherical objects. The higher fuzzy spheres are non-associative algebras that appear as projections of associative algebras related to matrices. The non-associativity (as well as the non-commutativity) disappears in the leading large N limit, ensuring the correct classical limit. Some simple aspects of the combinatorics of the fuzzy four-sphere can be accounted by a heuristic picture of giant fractional instantons.     

Extended cohomological field theories and noncommutative Frobenius manifolds

We construct some extension of cohomological field theories (stable field theory). The stable field theory is a system of homomorphisms to some vector spaces generated by spheres and disks with punctures. It is described by a formal tensor series, satisfying to some system of “differential equations”. In points of convergence the tensor series generate special noncommutative analogues of Frobenius algebras, describing ‘open-closed’ topological field theories.     

The Effect of Sphere Size on the Phase Behaviors in the Rod and Sphere Mixture System

The addition of polysaccharides, such as chondroitin sulfate (Chs), to the aqueous suspension of tobacco mosaic virus (TMV) results in the TMV aggregation at very dilute TMV concentration compared with the addition of polyethylene oxide (PEO). The Chs chain has a semirigid nature, whereas the PEO has a flexible nature. In this study, we investigated the effect of the size of spherical polymer coils on the phase behaviors in the mixtures of rods and spheres using Monte Carlo simulations. As a model for TMV, we used the spherocylinder particle. The Chs and PEO chains were simplified to spherical particles having different sizes. With the addition of large spherical particles, the system changed from miscible phase to isotropic–isotropic phase separation and then isotropic–nematic phase separation states, whereas with the addition of small spherical particles, the system transformed from miscible isotropic phase to miscible nematic phase. We found that the sphere size had an important influence on the phase behaviors in the mixtures of rods and spheres.     

Confinement of spherical colloid particles in a soft fluid membrane tube

We investigate the structure of multiple spherical particles confined in a soft membrane tube that originally has a cylindrical shape. Assuming an attraction energy between the surface of the spherical particle and the inner wall of the membrane tube, we show that a variety of conformational structures can be stabilized on the basis of analyzing a Helfrich energy for the soft tube. Using a numerical approach, we calculate the phase diagram in terms of basic parameters in the system. Structures that prefer close contact of spheres and structures that contain well separated spheres are found in this calculation.     

A criterion for percolation threshold in a random array of plates

We give a dimensionless criterion for percolation of a random assembly of 3-dimensional flat disks of radius r with a density N per unit volume as (Nr³)_c ～0.15 to 0.3. The invariant at threshold is obtained by analogy with excluded volume results for arrays of needles or with the active filling factor approach used for spheres. As the result can be applied to the permeability of fractured rocks, we discuss the results for polydisperse assemblies and we show how the invariant can be measured from a random cut of the array of disks.     

T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates

There has been recently a growing interest in the development of what is usually known as the T-matrix method (better to be named: T-matrix formulation), in connection with studies concerning light scattering by nonspherical particles. Another line of research has been devoted to the development of generalized Lorenz-Mie theories dealing with the interaction between arbitrary electromagnetic shaped beams and some regular particles, allowing one to solve Maxwell’s equations by using a method of separation of variables. Both lines of research are conjointly considered in this paper. Results of generalized Lorenz-Mie theories in spherical coordinates (for homogeneous spheres, multilayered spheres, spheres with an eccentrically located spherical inclusion, assemblies of spheres and aggregates) are modified from scalar results in the framework of the Bromwich method to vectorial expressions using vector spherical wave functions (VSWFs) in order to match the T-matrix formulation, and to express the T-matrix. The results obtained are used as a basis to clarify statements, some of them erroneous, concerning the T-matrix formulation and to provide recommendations for better terminologies.     

Experimental observation and characterization of the magnetorotational instability

Differential rotation occurs in conducting flows in accretion disks and planetary cores. In such systems, the magnetorotational instability can arise from coupling Lorentz and centrifugal forces to cause large radial angular momentum fluxes. We present the first experimental observation of the magnetorotational instability. Our system consists of liquid sodium between differentially rotating spheres, with an imposed coaxial magnetic field. We characterize the observed patterns, dynamics, and torque increases, and establish that this instability can occur from a hydrodynamic turbulent background.     

Casimir Effect for Concentric Spheres in de Sitter Spacetime with Signature Change

We revisit the Casimir effect for two concentric spherical shells in de Sitter background with a new geometric configuration, namely Euclidean signature between and Lorentzian signature outside the spheres with different cosmological constants, for a massless scalar field satisfying Dirichlet boundary conditions on the spheres. It is shown that an extra constant pressure emerges due to this signature changing configuration. Some interesting aspects of this extra term are then discussed.