Phonons,Diffusons, and the Boson Peak in Two-Dimensional Lattices with Random Bonds

Within the model of stable random matrices possessing translational invariance, a two-dimensional (on a square lattice) disordered oscillatory system with random strongly fluctuating bonds is considered. By a numerical analysis of the dynamic structure factor S(q, ω), it is shown that vibrations with frequencies below the Ioffe-Regel frequency ω_IR are ordinary phonons with a linear dispersion law ω(q) ∝ q and a reciprocal lifetime б ~ q³. Vibrations with frequencies above ω_IR, although being delocalized, cannot be described by plane waves with a definite dispersion law ω(q). They are characterized by a diffusion structure factor with a reciprocal lifetime б ~ q², which is typical of a diffusion process. In the literature, they are often referred to as diffusons. It is shown that, as in the three-dimensional model, the boson peak at the frequency ωb in the reduced density of vibrational states g(ω)/ω is on the order of the frequency ω_IR. It is located in the transition region between phonons and diffusons and is proportional to the Young’s modulus of the lattice, ω _b ≃E.

     

Integrable Equations of the Form q t =L 1(x,t,q,q x,q xx )q xxx +L 2(x,t,q,q x,q xx )

Integrable equations of the form q _t =L ₁(x,t,q,q _x ,q _xx )q _xxx +L ₂(x,t,q,q _x ,q _xx ) are considered using linearization. A new type of integrable equations which are the generalization of the integrable equations of Fokas and Ibragimov and Shabat are given.     

Thermal stability of nitro derivatives of hydrocarbon cubane

An original nonorthogonal tight-binding model is used to determine the structural and energetic characteristics of the family of polynitrocubanes C₈H_8−q (NO₂) _q , where q = 1−8. The mechanisms of isomerization are studied in detail and the minimum heights of the energy barriers to the decay of the metastable clusters are calculated. It is shown that nitro groups destabilize the cubic carbon skeleton. For octanitrocubane C₈(NO₂)₈, the temperature dependence of the characteristic decay time at temperatures 500–1000 K is obtained, and activation energy and frequency factor of the Arrhenius equation are calculated.     

Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra su_q(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.     

Integrable structure of melting crystal model with two -parameters

This paper explores integrable structures of a generalized melting crystal model that has two q-parameters q₁,q₂. This model, like the ordinary one with a single q-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight contains an infinite number of external potentials that depend on the shape of the diagonal slice of plane partitions. The partition function is thereby a function of an infinite number of coupling constants t₁,t₂,… and an extra one Q. There is a compact expression of this partition function in the language of a 2D complex free fermion system, from which one can see the presence of a quantum torus algebra behind this model. The partition function turns out to be a tau function (times a simple factor) of two integrable structures simultaneously. The first integrable structure is the bigraded Toda hierarchy, which determines the dependence on t₁,t₂,…. This integrable structure emerges when the q-parameters q₁,q₂ take special values. The second integrable structure is a q-difference analogue of the 1D Toda equation. The partition function satisfies thisq-difference equation with respect to Q. Unlike the bigraded Toda hierarchy, this integrable structure exists for any values of q₁,q₂.     

Nonstandard U q(so3) and U q(so4): tensor products of representations, oscillator realizations and roots of unity

Nonstandard q-deformed algebras U _q(so₃) and U _q(so₄), which can be embedded into U _q(sl₃) and U _q(sl₄) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U _q(so₃) when q is positive. Homomorphisms from U _q(so₃) and U _q(so₄) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U _q(so₃) and U _q(so₄) for q equal to a root of unity are obtained.     

On the dynamic structure factor for the electron gas

In this paper the well-known electron gas model is expressed as a path integral, and then its dynamic structure factor (DSF) is studied systematically. The study shows that q_c defined in the random phase approximation (RPA) not only is the threshold for a bare plasmon, but also in general is the threshold for any number of renormalized plasmons. The paper focuses on the large momentum transfer case (q > q_c). The conclusion that the self-energy correction appearing in the pertinent Green functions of the DSF vanishes has been reached. An explanation for the first peak in DSF observed in experiments, namely the strong correlation between the single-pair and multi-pair excitations involving only a few pairs, is suggested. An approximate, but nevertheless closed, formula for adding up a set of screened ladder diagrams, which describe the strongest correlation between the single pair and a few (two) pairs, is presented.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group L _q G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed.     

Onq-tensor operators for quantum groups

The fundamental theorem for tensor operators in quantum groups is proved using an appropriate definition forq-tensor operators. An example is discussed based on theq-boson realization of SU _q (2).Supported in part by the Department of Energy.     

Glassy dynamics of simulated polymer melts: Coherent scattering and van Hove correlation functions

Whereas the first part of this paper dealt with the relaxation in the β-regime, this part investigates the final relaxation (α-relaxation) of a simulated polymer melt consisting of short non-entangled chains in the supercooled state above the critical temperature of ideal mode-coupling theory (MCT). The temperature range covers the onset of a two-step relaxation behaviour down to a temperature merely 2% above . We monitor the incoherent intermediate scattering function as well as the coherent intermediate scattering function of both a single chain and the melt over a wide range of wave numbers q. Upon approaching the coherent α-relaxation time of the melt increases strongly close to the maximum q _max of the collective static structure factor S_q and roughly follows the shape of S_q for q q _max. For smaller q-values corresponding to the radius of gyration the relaxation time exhibits another maximum. The temperature dependence of the relaxation times is well described by a power law with a q-dependent exponent in an intermediate temperature range. Deviations are found very close to and far above , the onset of which depends on q. The time-temperature superposition principle of MCT is clearly borne out in the whole range of reciprocal vectors. An analysis of the α-decay by the Kohlrausch-Williams-Watts (KWW) function reveals that the collective KWW stretching exponent and KWW relaxation time show a modulation with S_q. Furthermore, both incoherent and coherent KWW times approach the large-q prediction of MCT already for q > q _max. At small q, a q^-3 power law is found for the coherent chain KWW times similar to that of recent experiments.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

On the p,q-binomial distribution and the Ising model

We employ p, q-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case p = q. We apply our investigation to the simple d-dimensional lattices for d = 1, 2, 3, 4, 5 and fit p, q-binomial distributions to our data, some of which are exact but most are sampled. For d = 1 and d = 5, the magnetisation distributions are remarkably well-fitted by p,q-binomial distributions. For d = 4 we are only slightly less successful, while for d = 2, 3 we see some deviations (with exceptions!) between the p, q-binomial and the Ising distribution. However, at certain temperatures near T _c the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results of the behaviour of the p, q-distribution and its moment growth exponents given a certain parameterisation of p, q. Since the moment exponents are known for the Ising model (or at least approximately for d = 3) we can predict how p, q should behave and compare this to our measured p, q. The results speak in favour of the p, q-binomial distribution's correctness regarding its general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution, however, is not settled.     

Absence of the partially ordered phase in theq-state models

We investigate theq-state models called (N ,N ) model using an infinitesimal Migdal-Kadanoff renormalization-group method. We distinguish two cases namely the isotropic model and the anisotropic model. The first one presents a critical value ofq,q _c such that forq _c we obtain an Ashkin-Teller phase diagrams while forq>q _c the partially ordered phase disappears then the model exhibits only phase transition between ferromagnetic phase and disordered one. The phase diagrams in the second case are qualitatively similar to one obtained forZ(6) model for all values ofq.     

Some observations related to intermittency and multifractality in Si and C-nucleus collisions at

An attempt is made to study the existence of dynamical fluctuations of relativistic particles using the methods of modified multifractal moments, G_q, and scaled factorial moments, F_q, in terms of new scaled variable X(η) suggested by Bialas and Gazdzicki. For this purpose analyses of experimental and UrQMD data involving interactions of ²⁸Si and ¹²C nuclei at with nuclear emulsion are used. The variation of lnG_q and lnF_q with lnM in pseudorapidity (η) phase space reveals power law behaviour. The values of slopes, τ_q and _q determined from the analyses of G_q and F_q moments are discussed. The generalized fractal dimensions, D_q, determined from the above methods are found to decrease with the order of the moments, q, indicating multifractality in multiparticle production. It is also observed that the spectral function f(α_q) for heavier projectile is much broader than for lighter beam due to larger number of participating nucleons present in heavier projectile.     

Entropy and specific heat as a measure of fluctuations in multiparticle production in relativistic nuclear collisions

An analysis to disentangle information about the occurrence of dynamical fluctuations in multiparticle production in high energy nucleus-nucleus collisions has been carried out in terms of fractal moments and entropy for the experimental and simulated data using FRITIOF, UrQMD and HIJING generators. Although there is a possibility to thermodynamically interpret the final state of multiparticle production by calculating the values of specific heat, c, using G _q - and F _q -moments. However, both these moments give markedly different values of c. The constancy observed in the values of specific heats calculated from either G _q - or F _q -moments is in accord with the predictions of constant specific heat approximation. Variations of multifractal and factorial moments and various other parameters calculated from these moments predict the presence of non-statistical fluctuations in high energy nuclear collisions.     

Real Forms of Quantum Orthogonal Groups,q-Lorentz Groups in any Dimension

We review known real forms of the quantum orthogonal groups SO _q (N). New *-conjugations are then introduced and we contruct all real forms of quantum orthogonal groups. We thus give an RTT formulation of the *-conjugations on SO _q (N) that is complementary to the U _q (g) *-structure classification of Twietmeyer. In particular, we easily find and describe the real forms SO _q (N-1,1) for any value of N. Quantum subspaces of the q-Minkowski space are analyzed.     

Kinetics of microphase segregation in one-dimensional symmetric diblock copolymer systems

An analytical one-dimensional model of the microphase separation in symmetric diblock copolymers is developed. Three stages of the process of the microphase segregation of a quenched diblock copolymer system into a lamella structure are predicted. The first stage involves a fast increase of the amplitude of the quasi-periodical lamella structure (with the average wave vector q₀) up to a nearly equilibrium value; the second stage is a slow phase diffusion process which is characterized by increasing coherency of the lamella structure; the third stage is the slow process of the lamella swelling, which is driven by the thermally activated process of the spontaneous deletion of excessive lamellas, the lamella size increasing from the initial value 2π/q ₀ to the equilibrium 2π/q _eq > 2π/q ₀ during the process. The last two stages are described with the specially introduced coarse-grained “amplitude-phase” approximation. It is shown that the relaxation of the gradient of the phase of the lamellar structure is the slowest relaxation process and, thus, can be used as an effective order parameter of the lamellar structure at the later stages of the microphase segregation. Received 10 March 2000 and Received in final form 5 June 2000     

On a nonstandard two-parametric quantum algebra and its connections withU p,q (gl(2)) andU p,q(gl(1/1))

A quantum algebraU _{p, q} (,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal -matrix.     

A(SLq(2)) at Roots of Unity is a Free Module over A(SL(2))

It is shown that when q is a primitive root of unity of order not equal to 2 mod 4, A(SL_q(2)) is a free module of finite rank over the coordinate ring of the classical group SL(2). An explicit set of generators is provided.