Colour conductivity of hard spheres

We present an analytic solution for the d-dimensional (d > 1) hard-sphere free flight trajectories in a thermostatted colour field. The solution shows that particles can only reach a finite distance in the direction perpendicular to the field in the absence of collisions. Using a numerical algorithm we designed to simulate many-body hard-sphere systems with curved trajectories, we study the onset of the instability leading to phase separation in the two-dimensional case for a range of field strengths and three densities. For the two fluid densities we find that phase separation occurs for sufficiently strong fields regardless of the initial configuration, and that the phase-separated state eventually becomes a collisionless, non-ergodic steady state. For solid densities the phase-separated configuration is stable and conducting, but is not an attractor for other charge distributions because of the impossibility of particle rearrangement.     

Comparative Studies of the Exchange-Correlation Potential Formulas for Coulomb Systems

The exchange-correlation part (xc) to the free energy is numerical evaluated in the RPA at arbitrary degree of degeneracy. The results are compared with numerical data of easy-to-use analytic fit-formulas or Padé approximants of the xc-term. All together results show very high accuracy at extremly high densities (r_s ≈ 1). The agreements disappear between the several formulas for increasing Brueckner parameter r_s. Numerical results for the xc-potentials (pressure and chemical potential) at finite temperatures for an electron-ion system are given. The xc-part of the ground state energy of our electron-ion model is compared with the ground state energy for metallic hydrogen and with Monte-Carlo calculations.     

The many vacua of gauged extended supergravities

A novel method is presented which employs advanced numerical techniques used in the engineering sciences to find and study the properties of nontrivial vacua of gauged extended supergravity models. While this method only produces approximate numerical data rather than analytic results, it overcomes the previous limitation of only being able to find vacua with large residual unbroken gauge symmetry groups. The effectiveness of this method is demonstrated by applying it to the technically most challenging D ≥ 3 scalar potential—that of SO(8) × SO(8) gauged Chern–Simons Supergravity in D = 3. Extensive data on the properties of 99 different vacua (92 of them new) of this model are given. Furthermore, techniques are briefly discussed which should allow using this numerical information as an input to the construction of semi-automatic stringent analytic proofs on the locations and properties of vacua. It hence is argued that these combined techniques presumably are powerful enough to systematically map all the nontrivial vacua of every supergravity model.     

SU(1, 1) Random Polynomials

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N ^–1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N. Our results are supported through various numerical simulations. We then extend results of Hannay⁽¹⁾ and Bleher et al. ⁽²⁾ to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |z _j |>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.     

q-Gaussians in the porous-medium equation: stability and time evolution

The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, , the porous-medium equation, is investigated through both numerical and analytical approaches. An analysis of the kurtosis of the distributions strongly suggests that an initial q-Gaussian, characterized by an index q_i, approaches asymptotically the final, analytic solution of the porous-medium equation, characterized by an index q, in such a way that the relaxation rule for the kurtosis evolves in time according to a q-exponential, with a relaxation index q_rel ≡q_rel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (q_i ≥ 5/3) into a finite-variance one (q < 5/3), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.     

Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals

We study some typical defect problems in one-dimensional (1D) hexagonal and two-dimensional (2D) octagonal quasicrystals. The first part of this investigation addresses in detail a uniformly moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6mm. A general solution is derived in terms of two functions φ ₁, φ ₂, which satisfy wave equations, and another harmonic function φ ₃. Elementary expressions for the phonon and phason displacements, strains, stresses, electric potential, electric fields and electric displacements induced by the moving screw dislocation are then arrived at by employing the obtained general solution. The derived solution is verified by comparison with existing solutions. Also obtained in this part of the investigation is the total energy of the moving screw dislocation. The second part of this investigation is devoted to the study of the interaction of a straight dislocation with a semi-infinite crack in an octagonal quasicrystal. Here the crack penetrates through the solid along the period direction and the dislocation line is parallel to the period direction. We first derive a general solution in terms of four analytic functions for plane strain problem in octagonal quasicrystals by means of differential operator theory and the complex variable method. All the phonon and phason displacements and stresses can be expressed in terms of the four analytic functions. Then we derive the exact solution for a straight dislocation near a semi-infinite crack in an octagonal quasicrystal, and also present the phonon and phason stress intensity factors induced by the straight dislocation and remote loads.      

Approximate analytic chiral SU2×SU2 skyrmion solution

An approximate analytic chiral SU₂×SU₂ skyrmion solution that incorporates all the boundary conditions as well as the asymptotic behaviour of the exact solution is suggested. The form of the analytic solution as well as various integrals calculated from it are in excellent agreement with those obtained from the numerical solution.     

The next-to-leading order (NLO) gluon distribution from DGLAP equation and the logarithmic derivatives of the proton structure functionF 2(x, Q 2) at lowx

At lowx, an analytic solution of the DGLAP equation for gluon in the next-to-leading order (NLO) is obtained by applying the method of characteristics. Its compatibility with double leading logarithmic approximation (DLLA) asymptotics is discussed and comparison with the exact ones like GRV98NLO is made. The solution is then utilized to calculate the derivatives∂F ₂ (x,Q ²)/∂ lnQ ² and ∂ lnF ₂(x,Q ²)/∂ ln (1/x) and compared with the recent HERA data. Our solution is found to reproduce most of the essential features of the data on the derivatives.     

Dirac Equation with Coupling to 1/<Emphasis Type="Italic">r</Emphasis> Singular Vector Potentials for all Angular Momenta

We consider the Dirac equation in 3+1 dimensions with spherical symmetry and coupling to 1/r singular vector potential. An approximate analytic solution for all angular momenta is obtained. The approximation is made for the 1/r orbital term in the Dirac equation itself not for the traditional and more singular 1/r ² term in the resulting second order differential equation. Consequently, the validity of the solution is for a wider energy spectrum. As examples, we consider the Hulthén and Eckart potentials.     

The permittivity tensor and bounce resonance effects on wave dissipation in toroidal plasmas

The asymptotic solution of Vlasov's equation in the drift approximation for a toroidal plasma geometry with circular magnetic surfaces is presented. The method of solution introduces Jacobi functions as variables, making it possible to obtain results in a relatively simple manner. The analytic evaluation of the parallel components of the dielectric permittivity tensor _3j is shown and the resulting strong influence of the bounce resonance of untrapped electrons on the wave dissipation is discussed. Additionally, a numerical calculation of ₃₃, together with a comparison between the toroidal and the cylindrical (Landau damping) results, are presented.The authors wish to thank Katherina von Bülow, from the Mathematical Institute of UERJ, for fruitful comments and careful reading of this paper, and CNPq of Brazil for financial support.     

光学微盘腔与三能级量子点系统中的模耦合研究

用一种全量子理论方法研究了波导、光学微盘腔与三能级量子点耦合系统的动力学过程,求出其耦合后的透射模和反射模的解析解. 由于微腔表面粗糙引起反向散射,在微腔内形成两简并回音壁耦合共振模,其耦合率为β;量子点的两激发态分别以耦合率g₁,g₂与回音壁耦合共振模产生耦合. 在实数空间里,得出透射光谱和反射光谱的数值解,这些三能级模型结果比二能级模型结果更接近真实光学微盘腔系统,能更好地显示耦合系统的动力学特性. 关键词： 模耦合 光学微盘腔 三能级量子点 全量子理论     

Mass and stability of rigidly rotating relativistic polytropes by energy method

The energy of a rigidly rotating star is written in the first and second post-Newtonian approximation. Conditions for equilibrium and stability are derived, as well as evolutionary paths for stars shedding angular momentum or mass. For polytropic index n > 1, these analytic results agree with exact numerical results to within a few percent as long as the general relativity index (P/c²)_c < 0.1. The energy method is applied to low mass white dwarfs, to semirelativistic neutron stars and to supermassive stars.     

An analytic formula of one-dimensional Franck-Condon factors for linear harmonic oscillators

With the aid of an addition theorem for harmonic oscillator wave functions, the Franck-Condon factors between two displaced linear oscillators with unequal vibrational frequencies are transformed into an analytic form which is programmable for numerical evaluations. Franck-Condon factors for the A¹Σ_u⁺-X¹Σ_g⁺ system and the B¹Π_u-X¹Σ_g⁺ system of ⁷Li₂ are calculated via this analytic formula.     

幂函数叠加势的径向薛定谔方程的解析解

研究多种正幂势函数与逆幂势函数紧密耦合条件下薛定谔径向方程解析解的求解方法.对势函数为V（r）=α₁r⁸＋α₂r³+α₃r²+β₃r^-1＋β₂r^-3＋β₁r^-4的径向薛定谔方程存在解析解的条件以及精确的解析解进行了研究. 根据量子系统波函数必须满足单值、有界和连续的标准条件,首先求出径向坐标r→∞以及r→0时的渐近解,然后采用非正则奇点邻域附近的波函数级数解法与求得的渐近解相结合,通过幂级数系数比较法得到径向薛定谔方程在势函数系数紧密耦合条件下的一系列定态波函数解析解以及相应的能级结构,并作适当讨论与结论. 关键词： 级数解法 幂势函数 径向波函数 渐近解     

Dynamic and spectral mixing in nanosystems

In the framework of a simple spin-boson Hamiltonian we study an interplay between dynamic and spectral roots to stochastic-like behavior. The Hamiltonian describes an initial vibrational state coupled to discrete dense spectrum reservoir. The reservoir states are formed by three sequences with rationally independent periodicities 1; 1 ± δ typical for vibrational states in many nanosize systems (e.g., large molecules containing CH₂ fragment chains, or carbon nanotubes). We show that quantum evolution of the system is determined by a dimensionless parameter δΓ, where Γ is characteristic number of the reservoir states relevant for the initial vibrational level dynamics. When δΓ > 1 spectral chaos destroys recurrence cycles and the system state evolution is stochastic-like. In the opposite limit δΓ < 1 dynamics is regular up to the critical recurrence cycle k _c and for larger k > k _c dynamic mixing leads to quasi-stochastic time evolution. Our semi-quantitative analytic results are confirmed by numerical solution of the equation of motion. We anticipate that both kinds of stochastic-like behavior (namely, due to spectral mixing and recurrence cycle dynamic mixing) can be observed by femtosecond spectroscopy methods in nanosystems in the spectral window 10¹¹–10¹³ s⁻¹     

Relaxation of quasi-stationary states in long range interacting systems and a classification of the range of pair interactions

Systems of particles interacting with long range interactions present generically ”quasi-stationary states” (QSS), which are approximately time-independent out of equilibrium states. In this proceedings, we explore the generalization of the formation of such QSS and their relaxation from the much studied case of gravity to a generic pair interaction with the asymptotic form of the potential v(r) ∼ 1/r ^γ with γ > 0 in d dimensions. We compute analytic estimations of the relaxation time calculating the rate of two body collisionality in a virialized system approximated as homogeneous. We show that for γ < (d − 1/2), the collision integral is dominated by the size of the system, while for γ > (d − 1/2), it is dominated by small impact parameters. In addition, the lifetime of QSS increases with the number of particles if γ < d − 1 (i.e. the force is not integrable) and decreases if γ > d − 1. Using numerical simulations we confirm our analytic results. A corollary of our work gives a ”dynamical” classification of interactions: the dynamical properties of the system depend on whether the pair force is integrable or not.     

Lüscher's μ-term and finite volume bootstrap principle for scattering states and form factors

We study the leading order finite size correction (Lüscher's μ-term) associated to moving one-particle states, arbitrary scattering states and finite volume form factors in (1+1)-dimensional integrable models. Our method is based on the idea that the μ-term is intimately connected to the inner structure of the particles, i.e., their composition under the bootstrap program. We use an appropriate analytic continuation of the Bethe–Yang equations to quantize bound states in finite volume and obtain the leading μ-term (associated to symmetric particle fusions) by calculating the deviations from the predictions of the ordinary Bethe–Yang quantization. Our results are compared to numerical data of the E₈ scattering theory obtained by truncated fermionic space approach. As a by-product it is shown that the bound state quantization does not only yield the correct μ-term, but also provides the sum over a subset of higher order corrections as well.     

生物组织的δ-P1近似漫反射光学模型

本文研究了生物组织的改进的δ-P₁近似漫反射光学模型,推导了含有等效光源一阶矩的双点源近似空间分辨漫反射解R_δ－P1(ρ).研究表明,考虑等效光源一阶矩的光学模型,较好地描述了具有强的前向散射特性和较大吸收系数的生物组织散射特性;与漫射近似下的漫反射率R_SDA(ρ)相比,新的光学模型能较好地描述光源附近的漫辐射强度分布,并且由于解析表达式中含有散射相函数的二阶参量γ,这对 关键词： 组织光学 1近似')" href="#">δ-P₁近似 等效光源 微区漫反射     

Characterization of ferrimagnetic Heisenberg chains according to the constituent spins

The low-energy structure and the thermodynamic properties of ferrimagnetic Heisenberg chains of alternating spins S and s are investigated by the use of numerical tools as well as the spin-wave theory. The elementary excitations are calculated through an efficient quantum Monte Carlo technique featuring imaginary-time correlation functions and are characterized in terms of interacting spin waves. The thermal behavior is analyzed with particular emphasis on its ferromagnetic and antiferromagnetic dual aspect. The extensive numerical and analytic calculations lead to the classification of the one-dimensional ferrimagnetic behavior according to the constituent spins: the ferromagnetic (S>2s), antiferromagnetic (S<2s), and balanced (S=2s) ferrimagnetism. Received 27 August 1999 and Received in final form 15 November 1999