Combinatorial basis and non-asymptotic form of the Tsallis entropy function

Using a q-analog of Boltzmann's combinatorial basis of entropy, the non-asymptotic non-degenerate and degenerate combinatorial forms of the Tsallis entropy function are derived. The new measures – supersets of the Tsallis entropy and the non-asymptotic variant of the Shannon entropy – are functions of the probability and degeneracy of each state, the Tsallis parameter q and the number of entities N. The analysis extends the Tsallis entropy concept to systems of small numbers of entities, with implications for the permissible range of q and the role of degeneracy.     

Self-consistent treatment of a phase transition

A self-consistent treatment of a phase transition with a scalar order parameter in the ordered and disordered state is described. The factorization of the correlation functions in the disordered phase leads to a shift of the transition temperature, a linear divergence (=1) for the correlation length, a quadratic divergence (=2) for the susceptibility, and a finite value (=–1) for the specific heat. In the ordered phase the factorization of the correlation functions leads to no divergences in the correlation length and susceptibility. A study of the free energy shows that order persists above the transition temperature found by assuming disorder. The requirement of thermodynamic stability induces a first-order transition at a temperature which lies between the bare transition temperature and the shifted one.Supported in part by NSF grant No-GP-17560.This work is in partial fulfillment of Ph.D. requirements at Brandeis University.     

Monte Carlo study of hard spherocylinders. III. The angular correlation functions

Monte Carlo simulation method has been used to study angular correlations in the fluid of hard spherocylinders of the length/breadth ratio equal to 1.4,1.6, and 2.0, respectively. The correlation functions have been evaluated both by means of the spherical harmonic expansion and directly (at specific orientations) which enabled us to examine convergence of the former approach. The possibility of the bottleneck effect has been investigated as well using two different simulation schemes.     

Time and temperature in semi-classical gravity

It is shown how in the functional integral approach, time arises as a natural parametrization for the way matter adiabatically follows gravitational configurations where the latter are treated in the semi-classical approximation. In the presence of horizons this time becomes complex and if the motion is periodic behind the horizon such functional integrals become partition functions. So the functional integral approach naturally associates the very concept of time and horizon thermal radiation.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Relativistic scattering theory of two bound quark-antiquark systems in the low- and intermediate-energy region

Relativistic three-dimensional equations for meson-meson scattering amplitudes are constructed considering the mesons as bound quark-antiquark systems. The resulting equations have a form identical with the field-theoretical Low equations for the meson-meson scattering amplitude without quark degrees of freedom. Therefore the solution of these equations and their linearized representations satisfy the unitarity condition and the low-energy theorems. In order to build the effective potential of the proposed equations one uses meson-quark-antiquark bound-state wave functions and transition matrices. Also these quark-antiquark bound-state wave functions satisfy the relativistic three-dimensional integral equation, which contains all retardation effects.Supported by Deutsche Forschungsgemeinschaft under contract no. Fa. 67/10-5     

Persistence length for a model semirigid polyelectrolyte as seen by small angle neutron scattering: a relevant variation of the lower bound with ionic strength

In a SANS experiment, we have directly determined for the first time the conformation of hyaluronan, a model semirigid polyelectrolyte. At high ionic strength, this is completely possible, where the scattered intensity crosses over (when decreasing q) from a q(-1) rod variation to a q(-2) and, where fitting to the "wormlike" chain model gives the backbone, intrinsic, persistence length: L0 = 86.5 A. At low ionic strength, we can safely check that the measured persistence length appears increased by at least the amount predicted by Odijk for the electrostatic contribution, L(e) (approximately kappa(-2), square of the Debye screening length). However, the intensity at the lowest q is not only due to the single chain, since it crosses over from a q(-1) to a q(-4) variation, characteristic of polymer associations.     

Bacterium organizes hierarchical amorphous structure in microbial cellulose

A pellicle, a gel film of microbial cellulose, is a supermolecular system containing 99% of water by weight, which is closely related to an amorphous structure in it. Using ultra-small-angle neutron scattering, in order to cover over a wide range of length scales from nm to 10 microm, we examined the hierarchical amorphous structure in the microbial cellulose, which is synthesized by a bacterium (Acetobacter xylinum). The microbial cellulose swollen by water shows small-angle scattering that obeys a power law q -behavior according to q -alpha as a function of the magnitude of the scattering vector q . The power law, determined by scattering, is attributed to a mass fractal due to the distribution of the center of mass for the crystallite (microfibril) in amorphous cellulose swollen by water. As q increases, alpha takes the values of 2.5, 1, and 2.35, corresponding, respectively, to a gel network composed of bundles, a bundle composed of cellulose ribbons, and concentration fluctuations in a bundle. From the mass fractal q -behavior and its length scale limits, we evaluated a volume fraction of crystallite in microbial cellulose. It was found that 90% of the cellulose bundle is occupied by amorphous cellulose containing water.     

Evolution of particle-scale dynamics in an aging clay suspension

Multispeckle x-ray photon correlation spectroscopy was employed to characterize the slow dynamics of a suspension of highly charged, nanometer-sized disks. At wave vectors q corresponding to interparticle length scales, the dynamic structure factor follows a form f(q,t) approximately exp([-(t/tau)(beta)], where beta approximately 1.5. The relaxation time tau increases with the sample age t(a) approximately as tau approximately t(1.8)(a) and decreases with q as tau approximately q(-1). Such behavior is consistent with models that describe the dynamics in disordered elastic media in terms of strain from random, local structural rearrangements. The measured amplitude of f(q,t) varies with q in a manner that implies caged particle motion. The decrease in the range of this motion and an increase in suspension conductivity with increasing t(a) indicate a growth in interparticle repulsion as the mechanism for internal stress development implied by these models.     

Many-body hydrodynamic interactions in charge-stabilized suspensions

In this joint experimental-theoretical work we study hydrodynamic interaction effects in dense suspensions of charged colloidal spheres. Using x-ray photon correlation spectroscopy we have determined the hydrodynamic function H(q), for a varying range of electrosteric repulsion. We show that H(q) can be quantitatively described by means of a novel Stokesian dynamics simulation method for charged Brownian spheres, and by a modification of a many-body theory developed originally by Beenakker and Mazur. Very importantly, we can explain the behavior of H(q) for strongly correlated particles without resorting to the controversial concept of hydrodynamic screening, as was attempted in earlier work by Riese [Phys. Rev. Lett. 85, 5460 (2000)].     

From equilibrium spin models to probabilistic cellular automata

The general equivalence betweenD-dimensional probabilistic cellular automata (PCA) and (D+1)-dimensional equilibrium spin models satisfying a disorder condition is first described in a pedagogical way and then used to analyze the phase diagrams, the critical behavior, and the universality classes of some automata. Diagrammatic representations of time-dependent correlation functions of PCA are introduced. Two important classes of PCA are singled out for which these correlation functions simplify: (1) Quasi-Hamiltonian automata, which have a current-carrying steady state, and for which some correlation functions are those of aD-dimensional static model. PCA satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes. (2) Linear (and more generally affine) PCA for which the diagrammatics reduces to a random walk problem closely related to (D+1)-dimensional directed SAWs: both problems display a critical behavior with mean-field exponents in any dimension. The correlation length and effective velocity of propagation of excitations can be calculated for affine PCA, as is shown on an explicitD=1 example. We conclude with some remarks on nonlinear PCA, for which the diagrammatics is related to reaction-diffusion processes, and which belong in some cases to the universality class of Reggeon field theory.     

A simple treatment of color screening in a $q\bar{q}$ plasma

The formation of Hückel-Debye screening in a plasma is studied following as close as possible the procedures used in the usual electric case. The equivalent of the Debye length is found, but the screening as a result is found to be less efficient than for the Coulomb interaction, since the correlation functions decay as the exponential of the distance raised at a power less than one.Received: 9 February 2004, Revised: 1 April 2004, Published online: 2 July 2004     

Configuration ray representations in non-Abelian quantum kinematics and dynamics

Ray extensions of the regular representation of noncompact non-Abelian Lie groups are examined as generalizations of the Cartesian coordinate representation of ordinary quantum mechanics to the case of generalized non-Cartesian coordinates and generalized noncommuting momenta. (The momenta are in fact the generators of the representation, and so they satisfy the Lie algebra of the group.) The concept of configuration ray representation is introduced within this new kinematic formalism as subrepresentations of the regular representation which are embossed with the relativity theory of a given system. The main features of the mathematical formalism leading to these representations in configuration spacetime are discussed, and their importance for non-Abelian quantum kinematics and dynamics is emphasized. Two miscellaneous examples on the calculus of phase functions for configuration ray representations are given.     

红宝石R'、B线及U、Y带压力移位的理论计算

基于电子径向波函数在加压下扩展的概念,把B、C、Dq、ξ等参量表示成线性压缩κ\的函数,并进一步考虑三角场作用,理论计算了红宝石R''、B线及U、Y带的光谱压力移位,结果与实验数据符合很好.     

Hyperfunction quantum field theory

The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not containC functions with compact support. In spite of this defect the support concept ofH-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory.     

Momentum dependence of magnetic response in heavy-fermion systems

The dynamical magnetic response of a heavy-fermion system CeCu₆ is analyzed with the concept of nearly localized quasi-particles. It is explained why the product χ(q)Г(q) of the susceptibility χ(q) and the magnetic relaxation rate Г(q) is independent of the momentum q although both χ(q) and Г(q) depend on q. A generalization of the Korringa relation between χ(q) and Г(q) is formulated.     

Finite transformations of SU(4)

The problem of determining the representation matrices of SU(4) is investigated. A convenient set of parameters is first introduced by writing the general element of the group as a product of exponential functions of the generators, and the generators are expressed as differential operators involving these parameters. Special matrix elements of finite transformations with a SU(3) singlet as the initial state are then obtained by solving the eigenvalue equation of the quadratic Casimir operator of SU(4). The solution has the form of a product of elementary functions and threed _mm ^j functions of SU(2) and is free from summation over intermediate states. By expanding one of thed _mm ^j functions in an appropriate series a sum rule for the special matrix elements of the permutation operator 12343412 is obtained. The discussions are strictly confined to SU(4), but, some of the results given here can be extended to unitary groups of higher dimensions.     

General Solution and Localized Coherent Soliton Structures of the (2+1)-Dimensional Generalized Davey-Stewarson Equations

In this paper, the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations: iqt 1/2(qxx qyy) (R S)q = O, Rx=-σ/2|q|2y Sy = -σ/2|q|2/x.Applying a special Backlund transformation and introducing arbitrary functions of the seed solutions, an abundance of the localized structures of this model is derived. By selecting the arbitrary functions appropriately, some special typesof localized excitations such as dromions, dromion lattice, breathers, and instantons are constructed.     

q,k-Generalized Gamma and Beta Functions

Abstract

We introduce the q,k-generalized Pochhammer symbol. We construct Γ _q,k and B _q,k , the q,k-generalized gamma and beta functions, and show that they satisfy properties that generalize those satisfied by the classical gamma and beta functions. Moreover, we provide integral representations for Γ _q,k and B _q,k .