Regular exact models for a nonstatic gas sphere in general relativity

Some new exact models for an expanding or a contracting gaseous sphere (i.e., the density is to vanish at the outer boundary together with the pressurep) are given. The physical properties of the models are investigated, and it is found that both the pressure and the density are positive inside the outer boundary of the sphere, and their respective gradients are negative. The density is increasing for contracting spheres, and it is decreasing for expanding spheres. It is also shown that this is the case for the pressure at any moment for the layers close to the boundary of the spheres. For these layers it is further shown that the adiabatic speed of sound is less than the speed of light, and the trace of the energy-momentum tensor is positive. The rate of change of the circumference as measured by an observer riding on the boundary of the sphere is increasing for expanding spheres and it is decreasing for collapsing spheres. We also find that the physical radius is an increasing function of comoving radial coordinate. The mass function is further shown to be positive.     

Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum

We consider a system ofN hard spheres in the Boltzmann-Grad limit (i.e.d0,N,Nd ^2–1>0, whered is the diameter of the spheres). If is sufficiently large, and if the joint distribution densities factorize at time zero, with the one particle distribution decaying sufficiently rapidly in space and velocities, we prove that the time evolved one-particle distribution converges for all times to the solution of the Boltzmann equation with the same initial datum. This result improves and is based on a previous paper [1], valid only in two dimensions.Partially supported by MPI and GNFM (CNR)     

Quantum Even Spheres Σ<Superscript><Emphasis Type="Italic">2n</Emphasis></Superscript><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript> from Poisson Double Suspension

We define even dimensional quantum spheres ²ⁿ _q that generalize to higher dimension the standard quantum two-sphere of Podle and the four-sphere ⁴ _q obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podle two-sphere. The Poisson spheres that we get have the same kind of symplectic foliation consisting of a degenerate point and a symplectic ²ⁿ and, after quantization, have the same C ^* –algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.     

Krichever-Novikov-like bases on punctured Riemann surfaces

Bases of holomorphic -differentials on N-punctured Riemann surfaces of arbitrary genus are constructed. The resulting extension of the Virasoro algebra on N-punctured spheres is displayed explicitly.     

On the energy per particle in three- and two-dimensional Wigner lattices

We come back to the 1979 controversy about the value of the energy per particle, in an infinite Wigner lattice of electrons in a uniform compensating background. For simplicity we restrict ourselves to the simple cubic (and square) lattice. We present an accurate calculation of the energy _el of one electron in the field of the other electrons plus background for the case that the system (system I) is considered as an infinite arrangement of neutral cubes (Wigner-Seitz cells). The value obtained is checked by computer calculations. We confirm the conclusion of de Wette that for this system the relation _i=1/2 _el (often accepted without discussion) does not hold and we calculate the difference, which represents the average potential in the system. On the other hand, if the system is considered as the limit of a set of spheres with increasing radii, such that the spheres are neutral (system II), we obtain a different value of _el and in this case _i=1/2 _el. We show explicitly that the Ewald method of summation, used by Fuchs and others, leads to the same analytical expression as the limit obtained for a set of neutral spheres (system II). We extend the calculations to the two-dimensional square lattice. Here the equality _i=1/2 _el holds also in the case of an infinite arrangement of neutral squares (system I).This paper is dedicated to our friend and colleague Nico van Kampen in honor of his lifelong dedication to science in general and to theoretical physics in particular.     

H-theorem for the (modified) nonlinear Enskog equation

We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that _t H 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=– k_B H becomes the exact equilibrium entropy of the hard-sphere fluid.     

Adhesion of small spheres

Bradley 1 Bradley, RS. 1931. Phil. Mag., 11: 846[Taylor & Francis Online] , [Google Scholar] calculated the adhesive force between rigid spheres to be 2πRΔγ, where Δγ is the surface energy of the spheres. Johnson et al. (JKR) [2] calculated the adhesive force between elastic spheres to be (3/2) πRΔγ and independent of the elastic modulus. Derjaguin et al. [3] published an alternative theory for elastic spheres (DMT theory), and concluded that Bradley's value for the pull-off force was the correct one. Tabor [4] explained the discrepancy in terms of the range of action of the surface forces, z ₀, and introduced a parameter μ≡(RΔγ²/E²z ₀³)^1/3, determining which result is applicable. Subsequently, detailed calculations by Derjaguin and his colleagues [5] and others, assuming a surface force law based on the Lennard-Jones 6–12 potential law, covered the full range of the Tabor parameter. Greenwood and Johnson [6] presented a map delineating the regions of applicability of the different theories. Yao et al. [7] repeated the numerical calculations but using an exact sphere shape instead of the usual paraboloidal approximation. They found that the pull-off force could be less than one-tenth of the JKR value, depending on the value of a ‘strength limit’ σ₀/E, and modified the Johnson and Greenwood map correspondingly. Yao et al.'s numerical calculations for contact between an exact sphere and an elastic half-space are repeated and their values confirmed: but it is shown that the drastic reductions found occur only for spheres that are smaller than atomic dimensions. The limitations imposed by large strain elasticity and by the ‘Derjaguin approximation’ are discussed.     

Disks vs. spheres: Contrasting properties of random packings

Collections of random packings of rigid disks and spheres have been generated by computer using a previously described concurrent algorithm. Particles begin as infinitesimal moving points, grow in size at a uniform rate, undergo energy-onconserving collisions, and eventually jam up. Periodic boundary conditions apply, and various numbers of particles have been considered (N2000 for disks,N8000 for spheres). The irregular disk packings thus formed are clearly polycrystalline with mean grain size dependent upon particle growth rate. By contrast, the sphere packings show a homogeneously amorphous texture substantially devoid of crystalline grains. This distinction strongly influences the respective results for packing pair correlation functions and for the distributions of particles by contact number. Rapidly grown disk packings display occasional vacancies within the crystalline grains; no comparable voids of such distinctive size have been found in the random sphere packings. Rattler particles free to move locally but imprisoned by jammed neighbors occur in both the disk and sphere packings.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

Maximum number of collisions among identical hard spheres

It is proved that the maximum number of collisions among three identical hard spheres in more than one dimension is four. It is conjectured that the maximum number of collisions amongn hard spheres ind dimensions is independent ofd, provideddn-1.     

Comments on the mean spherical approximation for hard-core and soft potentials and an application to the one-component plasma

The thermodynamic properties of the mean spherical (MSA), Percus-Yevick (PY), and hypernetted-chain (HNC) approximations are derived by a simple and unified approach by considering the RPA free-energy functionalF and employing an Ewald-type identity. It is demonstrated that with decreasing relative contribution of the hard-core insertion to the thermodynamic functions, the MSA changes its nature from PY-like to HNC-like, withF changing its role from excess pressure to excess free energy, respectively. It is found that the condition of continuity of the MSA pair functions is equivalent to a stationarity condition forF and leads to thermodynamic consistency between the virial and energy equations of state for the (thus defined) soft-MSA (SMSA), withF playing the role of the excess free energy. It is shown that the PY-compressibility and virial equations of state forD-dimensional hard spheres may be simply obtained one from the other without knowing any details of the solution of the model. Using this relation we find an indication that the PY approximation for hard spheres becomes less accurate with increasing dimensionality. A general variational formulation is presented for the application of the MSA for soft potentials, and results for the one-component plasma are discussed and extended.On sabbatical leave from the Nuclear Research Center-Negev, P.O. Box 9001, Beer Sheva, Israel.     

Ionization equilibrium in the electron-proton gas

Using Fefferman's analysis of the quantum electron-proton gas, we give a rigorous proof of ionization equilibrium in this system. Ionization equilibrium phases are obtained as low-density and low-temperature limits, letting the chemical potential(T) approach the ground-state energy of the hydrogen atom as the temperatureT tends to zero. The rate of ionization is determined by the slope of(T) atT=0 and is correctly given by the Saha formula. We also discuss a simpler model where a single quantum particle interacts with a classical gas of hard spheres.     

Spectral Analysis and Zeta Determinant on the Deformed Spheres

We consider a class of singular Riemannian manifolds, the deformed spheres , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian , we study the associated zeta functions . We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in . An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular and . We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2,3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k. Partially supported by FAPESP: 2005/04363-4     

Differential calculus on quantum spheres

Three-dimensional differential calculus on quantum spheres S _infc ^sup2 ,]–1, 1[{0}, c[0, ], is introduced and investigated. Spectra of generalized Laplacians are found. These operators are expressed by generalized directional derivatives. Classical limits of these objects are obtained and a simple approach to quantum mechanics on a quantum sphere is presented.     

Hard-sphere heat conductivity via nonequilibrium molecular dynamics

We use an Evans-Gillan driving forceF ^d, together with isokinetic and isoenergetic constraint forcesF ^c, to drive steady heat currents in periodic systems of 4 and 32 hard spheres. The additional driving and constraint forces produce curved trajectories as well as additional streaming and collisional contributions to the momentum and energy fluxes. Here we develop an analytic treatment of the collisions so that the simulation becomes approximately ten times faster than our previous numerical treatment. At low field strengths, for less than 0.4, where is the hard-sphere diameter, the 32-sphere conductivity is consistent with Alder, Gass, and Wainwright's 108-sphere value. At higher field strengths the conductivity varies roughly as ^1/2, in parallel with the logarithmic dependence found previously for three hard disks.     

Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

We evaluate the virial coefficients B_k for for hard spheres in dimensions Virial coefficients with k even are found to be negative when This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when . Further analysis provides evidence that negative virial coefficients will be seen for some k > 10 for D = 4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D = 3.     

Physical properties of some McVittie metrics

A class of McVittie's new nonquadratic solutions is investigated in some detail with regard to its physical properties. It is found that decreasing pressure and density are not compatible with center regularity for these perfect fluid spheres. It is further seen that for gaseous spheres (i.e., the density drops to zero at the outer boundary of the sphere together with the pressurep) oscillatory motions arenot possible. For these gaseous models the pressure and the density are both positive inside the outer surface, and their respective gradients are negative. For the outer gaseous shells models are constructed where for a certain time interval the pressure is increasing for contracting models. Without any restriction with respect to time, for these shell models it found that the density is increasing for contracting models, and the adiabatic speed of sound is less than the speed of light. It is also found that the trace of the energy-momentum tensor is positive, the total mass is negative, and for collapsing shells the rate of change of circumference as measured by an observer riding on the shell is an increasing function of time. However, all these models have the strange geometric feature that the physical radius is a decreasing function of comoving radial coordinate.     

The incoherent scattering function and related correlation functions in hard sphere fluids at short times

For a classical fluid of hard spheres and hard disks exact expressions for all densities and wave vectors are derived for the coefficients oft ⁿ in the short-time expansion of the incoherent intermediate scattering function (n = 0, 1,..., 4) and the velocity correlation function (n=0,1,2). Similarly, we obtain the coefficient of the leading term in the short-time behavior of the cumulants of the displacements. Furthermore,S(k, ) has a high-frequency tail ^–4, characteristic for the hard-sphere fluid, which leads to a modification of the standard sum rules. We present estimates for the frequency range, in which this tail may be observed in neutron scattering off noble gases. The results are also compared with Enskog's theory and molecular dynamics calculations.     

Noncommutative Riemann Surfaces by Embeddings in $${\mathbb{R}^{3}}$$

We introduce C-Algebras of compact Riemann surfaces as non-commutative analogues of the Poisson algebra of smooth functions on . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.     

Effect of neutral particles on high-frequency conductivity of plasma

Diagram technique of the temperature-dependent Green's functions is used to study the effect of neutral particles on the high-frequency conductivity of high-temperature plasma in the Born approximation. Assuming that the frequency of the electric field is much larger than the collision frequency of the electrons with neutral particles, expressions are derived for Re ^en () in two cases: 1. the neutral particles and electrons are regarded as hard, elastic spheres, 2. the neutral particles are hydrogen atoms in the ground state.In conclusion the author thanks J. Teichmann, C.Sc., for suggesting this work and for valuable discussions.     

The temperature dependence of the refractive index of an aqueous suspension of polystyrene microspheres

We report an experimental measurement, based on an interferometric technique, for the determination of the temperature dependence of the refractive index n _L of a liquid suspension of dielectric spheres in the thermal range 20–52° C. In this range we have measured values of n _L /T of order of magnitude 10^–4° C^–1. The observed values are compared with those obtained for water.