Entanglement,mixedness, and q-entropies

We revisit the relationship between entanglement and purity of states of two-qubits systems, using the q-entropies as measures of the degree of mixture. The q-entropies depend on the density matrix eigenvalues p_i through the quantity ω_q=∑_ip_i^q. Rényi's measures constitutes particular instances of these entropies. We pay particular attention to the case q=2 and to the limit case q→∞. We provide analytical support to numerical results recently reported in the literature.     

Measurement of Aggregate Fractal Dimensions Using Static Light Scattering

Aggregates formed from colloidal particles will vary in shape according to the aggregation regime prevalent. Compact structures are formed when the aggregation is slow, whilst loose tenuous structures are formed when rapid (or diffusion limited) aggregation prevails. These structures can be fractal in nature, that is, there is a relationship between porosity and the number of primary particles making up the aggregate, and is described by the fractal dimension, d_F. Fractal dimensions of hematite aggregates have been measured experimentally using the static light scattering technique. Fractal dimensions varied with aggregation regimes; for the rapid aggregation regime, d_F was found to be 2.8, whilst for conditions in which aggregation was slow (retardation forces prevail), d_F's of 2.3 were measured. For conditions which lead to aggregation in which both diffusion and retardation forces play a part, structures with fractal dimensions such that 2.3 < d_F < 2.8 were found. The effects of adsorbed fulvic acid, a naturally occuring organic acid, on the kinetics of hematite aggregation and on the resulting structure of hematite aggregates were also investigated. The study of aggregate structure shows that the fractal dimensions of hematite aggregates which are partially coated with fulvic acid molecules are higher than those obtained with no adsorbed fulvic acid. The scattering exponents obtained from static light scattering experiments of these aggregates range from 2.83 ± 0.08 to 3.42 ± 0.1. The scattering exponents of greater than 3 indicate that the scattering is the result of objects that contains pores which are bounded by surfaces with a fractal structure, and can be related only to surface fractal dimension. The high fractal dimensions are due to restructuring within the aggregates, which only occured at low coverage by the organic acid.     

Roughness and fractality of fracture surfaces as indicators of mechanical quantities of porous solids

The 3D profile surface parameter H _q and fractal dimension D were tested as indicators of mechanical properties inferred from fracture surfaces of porous solids. High porous hydrated cement pastes were used as prototypes of porous materials. Both the profile parameter H _q and the fractal dimension D showed capability to assess compressive strength from the fracture surfaces of hydrated pastes. From a practical point of view the 3D profile parameter H _q seems to be more convenient as an indicator of mechanical properties, as its values suffer much less from statistical scatter than those of fractal dimensions.     

Monte Carlo simulation of aggregate morphology for simultaneous coagulation and sintering

A model for simulation of the three-dimensional morphology of nano-structured aggregates formed by concurrent coagulation and sintering is presented. Diffusion controlled cluster–cluster aggregation is assumed to be the prevailing coagulation mechanism which is implemented using a Monte–Carlo algorithm. Sintering is modeled as a successive overlapping of spherical primary particles, which are allowed to grow as to preserve overall mass. Simulations are characterized by individual ratios of characteristic collision to fusion time. A number of resulting aggregate-structures is displayed and reveals structure formation by coagulation and sintering for different values of . These aggregates are described qualitatively and quantitatively by their mass fractal dimension D_f and radius of gyration. The fractal dimension increases from 1.86 for pure aggregation to 2.75 for equal characteristic time scales. As sintering turns out to be more and more relevant, increasingly compact aggregates start to form and the radius of gyration decreases significantly. The simulation results clearly reveal a strong dependence of the fractal dimension on the kinetics of the concurrent coagulation and sintering processes. Considering appropriate values of D_f in aerosol process simulations may therefore be important in many cases.     

沉积在硅油表面上的Ag原子分形凝聚体

研究了沉积在硅油表面上的Ag原子团簇,经过随机扩散和转动,最终形成大尺度分形凝聚体的凝聚过程．研究结果表明：Ag原子团簇在这种液体基底上的转动为随机转动,转动角位移的方均值<(Δθ)²>和测量时间间隔Δt满足广义爱因斯坦关系<(Δθ)²>=4D_θΔt．随机转动系数D_θ与凝聚体面积S满足指数关系D_θ∝S^{－γ_θ,其中指数γ_θ＝2.4±0. 关键词：}     

On Different Approaches to Estimate the Mass Fractal Dimension of Coal Aggregates

Several methods to measure the structures of coal aggregates are compared. Loose and compact coal aggregates were generated through flocculation of ultrafine coal particles (mean volume diameter of 12 μm) under specific shearing conditions. Aggregate structure in terms of mass fractal dimension, D_f, was determined using various methods; namely 2D and 3D image analysis, interpretation of intensity patterns from small angle light scattering, changes in aggregation state through light obscuration, and settling behavior. In this study, the measured values of D_f ranged from 1.84–2.19 for coal aggregates with more open structures, and around 2.27–2.66 for the compact ones. All of these approaches could distinguish structural differences between aggregates, albeit with variation in D_f values estimated by the different techniques. The discrepancy in the absolute values for fractal dimension is due to the different physical properties measured by each approach, depending on the assumptions used to infer D_f from measurable parameters. In addition, image analysis and settling techniques are based on the examination of individual aggregates, such that a large number of data points are required to yield statistically representative estimations. Light scattering and obscuration measure the aggregates collectively to give average D_f values of the particulate systems; consequently ignoring any structural variation between the aggregates, and leaving possible small contaminations undetected (e.g. by dust particles or air bubbles). Appropriate utilization of a particular method is thus largely determined by system properties and required data quality.     

Positive-energy irreps of the quantum anti de Sitter algebra

We obtain positive-energy irreducible representations of theq-deformed anti de Sitter algebraU _q (so(3, 2)) by deformation of the classical ones. When the deformation parameterq isN-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations ofso(3, 2). We discuss in detail the singleton representations, i.e. the Di and Rac. WhenN is odd, the Di has dimension 1/2(N ²–1) and the Rac has dimension 1/2(N ²+1), while ifN is even, both the Di and Rac have dimension 1/2N ². These dimensions are classical only forN=3 when the Di and Rac are deformations of the two fundamental non-unitary representations ofso(3, 2).Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.On leave from Bulgarian Acad. Sci., Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria.On leave from Pennsylvania State University (Fulbright scholar).     

Classification of bicovariant differential calculi on quantum groups

Suppose thatq is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groupsGL _q (N),O _q (N) andSp _q (N) for which the differentials du _j ⁱ of the matrix entriesu _j ⁱ generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi onGL _q (N) forN3. In the limitq1 only two of these calculi give the ordinary differential calculus onGL(N). Our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties onO _q (N) andSp _q (N) forN4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi onO _q (3). We introduce two new 4-dimensional bicovariant differential calculi onO _q (3).     

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

The Rényi entropies R_p [ ρ], p> 0,≠ 1 of the highly-excited quantum states of the D-dimensional isotropicharmonic oscillator are analytically determined by use of the strong asymptotics of theorthogonal polynomials which control the wavefunctions of these states, the Laguerrepolynomials. This Rydberg energetic region is where the transition from classical toquantum correspondence takes place. We first realize that these entropies are closelyconnected to the entropic moments of the quantum-mechanical probability ρ_n(r)density of the Rydberg wavefunctions Ψ_{n,l, { μ}}(r); so, to the?_p-norms of the associated Laguerrepolynomials. Then, we determine the asymptotics n → ∞ of these norms by use of modern techniques ofapproximation theory based on the strong Laguerre asymptotics. Finally, we determine thedominant term of the Rényi entropies of the Rydberg states explicitly in terms of thehyperquantum numbers (n,l), the parameter order p and the universedimensionality D for all possible cases D ≥ 1. We find that (a) theRényi entropy power decreases monotonically as the order p is increasing and (b) thedisequilibrium (closely related to the second order Rényi entropy), which quantifies theseparation of the electron distribution from equiprobability, has a quasi-Gaussianbehavior in terms of D.     

Inertial deposition of nanoparticle chain aggregates: Theory and comparison with impactor data for ultrafine atmospheric aerosols

Nanoparticle chain aggregates (NCAs) are often sized and collected using instruments that rely on inertial transport mechanisms. The instruments size segregate aggregates according to the diameter of a sphere with the same aerodynamic behavior in a mechanical force field. A new method of interpreting the aerodynamic diameter of NCAs is described. The method can be used to calculate aggregate surface area or volume. This is useful since inertial instruments are normally calibrated for spheres, and the calibrations cannot be directly used to calculate aggregate properties. A linear relationship between aggregate aerodynamic diameter and primary particle diameter based on published Monte-Carlo drag calculations is derived. The relationship shows that the aggregate aerodynamic diameter is independent of the number of primary particles that compose an aggregate, hence the aggregate mass. The analysis applies to aggregates with low fractal dimension and uniform primary particle diameter. This is often a reasonable approximation for the morphology of nanoparticles generated in high temperature gases. An analogy is the use of the sphere as an approximation for compact particles. The analysis is applied to the collection of NCAs by a low-pressure impactor. Our results indicate the low-pressure impactor collects aggregates with a known surface area per unit volume on each stage. Combustion processes often produce particles with aggregate structure. For diesel exhaust aggregates, the surface area per unit volume calculated by our method was about twice that of spheres with diameter equal to the aerodynamic diameter. Measurements of aggregates collected near a major freeway and at Los Angeles International Airport (LAX) were made for two aerodynamic cutoff diameter diameters (d _a,50), 50 and 75 nm. (Aerodynamic cutoff diameter refers to the diameter of particles collected with 50% efficiency on a low-pressure impactor stage.) Near-freeway aggregates were probably primarily a mixture of diesel and internal combustion engine emissions. Aggregates collected at LAX were most likely present as a result of aircraft emissions. In both measurements, the aggregate aerodynamic diameters calculated from the primary particle diameter were fairly close to the stage cutoff diameter. The number of primary particles per aggregate varied one order of magnitude for particles depositing on the same stage. The average aggregate surface area per unit volume was 2.41 × 10⁶ cm⁻¹ and 2.59 × 10⁶ cm⁻¹ (50 nm d _a,50) and 1.81 × 10⁶ cm⁻¹ and 1.68 × 10⁶ cm⁻¹ (75 nm d _a,50) for near-freeway and LAX measurements, respectively. These preliminary measurements are consistent with values calculated from theory.     

Analysis of fractals with dependent branching by box counting, P-adic coverages, and systems of equations of P-adic coverages

Concept of the dimension of space-time in the general relativity theory and quantum theory is discussed. It is emphasized that the dimension of a discrete space can be defined based on the Hausdorff measure. The noninteger dimension is a typical characteristic of a fractal. The process of hadron formation in interactions between high-energy particles and nuclei is supposed to possess fractal properties. The following methods for analyzing fractals are considered: box counting (BC), method of P-adic coverages (PaC), and method of systems of equations of P-adic coverages (SePaC), for determining the fractal dimension. A comparative analysis of fractals with dependent branching is performed using these methods. We determine the optimum values of parameters permitting one to determine the fractal dimension D _F , number of levels N _lev, and the fractal structure with maximal efficiency. It is noted that the SePaC method has advantages in analyzing fractals with dependent branching.     

A study of radiative properties of fractal soot aggregates using the superposition T-matrix method

We employ the numerically exact superposition T-matrix method to perform extensive computations of scattering and absorption properties of soot aggregates with varying state of compactness and size. The fractal dimension, D_f, is used to quantify the geometrical mass dispersion of the clusters. The optical properties of soot aggregates for a given fractal dimension are complex functions of the refractive index of the material m, the number of monomers N_S, and the monomer radius a. It is shown that for smaller values of a, the absorption cross section tends to be relatively constant when D_f<2 but increases rapidly when D_f>2. However, a systematic reduction in light absorption with D_f is observed for clusters with sufficiently large N_S, m, and a. The scattering cross section and single-scattering albedo increase monotonically as fractals evolve from chain-like to more densely packed morphologies, which is a strong manifestation of the increasing importance of scattering interaction among spherules. Overall, the results for soot fractals differ profoundly from those calculated for the respective volume-equivalent soot spheres as well as for the respective external mixtures of soot monomers under the assumption that there are no electromagnetic interactions between the monomers. The climate-research implications of our results are discussed.     

Fractal dimensions and ƒ(α) spectrum of the Hénon attractor

We measure the generalized fractal dimensions D_q(q⩾0) of the Hénon attractor by the box counting and spatial correlation methods. The technique of virtual memory is exploited to handle the extremely large numbers of iterates needed for the convergence of the algorithms. We study quantitatively the oscillations which appear in the usual linear regressions of the log-log plot and which are inherent in lacunar fractal sets. These oscillations are the cause of previous underestimates of the Renyi dimensions and in fact make accurate dimension estimates an elusive goal. The Legendre transform of the D_q yields the ƒ(α) spectrum which characterizes the multifractal structure of the attractor. We point out that this spectrum of singularities can be extracted directly from the computed invariant measure, avoiding the log-log regression procedure.     

Ecriture de l'énergie cinétique à l'aide d'opérateurs quasi-moments

Cet article concerne l'utilisation des quasi-moments ?π _m , définis par

,

pour exprimer l'opérateur correspondant à l'énergie cinétique de N particules en Mécanique Quantique. La condition de Wilson-Howard portant sur les coefficients s^ml est interprétée comme la condition pour que les opérateurs ?π _m soient hermitiques quand on utilise l'élément de volume s dq ₁ … dq ₃ N (s=[dét {s^ml }]^-1). La condition générale pour qu'il soit possible de trouver un élément de volume avec lequel les opérateurs ?π _m sont hermitiques est donnée et différentes expressions de l'opérateur énergie cinétique sont établies quand cette condition est remplie et quand elle ne l'est pas.     

Variational Matrix Padé Approximants in Two Body Scattering

The convergence and bounding properties of the variational matrix Padé approximants are investigated for non relativistic two body interactions. Selecting L – 1 discrete values qi, i = 1, …, L – 1 and the physical momentum q₀ the off shell scattering amplitudes are L X L matrices. The [N/N] Padé approximants to the Born series of these matrices are the variational solution of the Schwinger principle and the corresponding physical amplitude has variational properties in the off shell momenta. For positive interactions the best approximants to the phase shift is an absolute minimum on the q_i and monotonic convergence to the exact result for N → ∞ or L → ∞ ca be proved. Similar properties are shown for the bound states using the Ritz variational principle. The required mathematical background is extensively worked out, the extensions to non positive, singular and long range potentials are considered and some numerical examples are presented.     

A highly efficient algorithm for the generation of random fractal aggregates

A technique to generate random fractal aggregates where the fractal dimension is fixed a priori is presented. The algorithm utilizes the box-counting measure of the fractal dimension to determine the number of hypercubes required to encompass the aggregate, on a set of length scales, over which the structure can be defined as fractal. At each length scale the hypercubes required to generate the structure are chosen using a simple random walk which ensures connectivity of the aggregate. The algorithm is highly efficient and overcomes the limitations on the magnitude of the fractal dimension encountered by previous techniques.     

Effects of Monomer Size Distribution on the fractal dimensionality of diffusion-limited aggregates

Computer simulations of diffusion-limited aggregation (DLA) for monomers to investigate the effects of size and of lognormal distribution on the fractal dimensionality of the aggregates were conducted on a two-dimensional lattice. The results show the DLA clusters posses multifractal characteristics. For clusters consisting of monodisperse monomers, the bifurcation point on the graph of the pair correlation function (PCF) for each cluster is located right at the monomers size under investigation The textural dimension (D_f1) has a stable value of about 1.65, whereas the structural dimension (D_f2) decreased with increase in monomer size. For the cases with monomers in log-normal distributions, the textural dimension is around 1.67; however, the structural dimension decreases with increasing polydispersity of monomer size.     

Diffusion on two-dimensional percolation clusters with multifractal jump probabilities

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension D_F and a multifractal distribution of probabilities of dimensionsD _q (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD' _q =D _q +D _F -d, for a finite interval ofq.     

The infinite number of generalized dimensions of fractals and strange attractors

We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions D_q, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities D_q. We prove that lim _q→0D_q = fractal dimension (D), lim_q→1D_q = information dimension (σ) and D_q=2 = correlation exponent (v). D_q with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > D_q for any q′ > q. For homogeneous fractals D_q = D_q. A particularly interesting dimension is D_q=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D_∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D_∞.