Computation and analysis of the dynamic structure factor S(k, ω) for small wave vectors

Molecular dynamics (MD) calculations have been performed for the Lennard-Jones (12-6) potential function using 2048 particles. Using conventional parameters the results may be compared with those for liquid argon.

The dynamic structure factor S(k, ω) has been determined both by Fourier inversion of the intermediate scattering function F(k, t) and from the longitudinal current-current correlation function C _∥(k, t). Particular attention was paid to the recurrence time of the system. The results for S(k, ω) by the two methods agree within 5 per cent for the whole region of small k-vectors considered. Double Fourier inversion of the van Hove function G(r, t) led to insufficiently accurate results for these small k-values. In view of the present data, the MD-results of Levesque et al. [1] for S(k, ω) have only a qualitative character. These latter data appear to contain truncation errors due to incomplete Fourier transformations.

Using a hydrodynamic assumption for F(k, t) we were able to extract the transport coefficients, the velocity of sound and the ratio of the specific heats in the limit of large wave lengths or small k. The velocity of sound was obtained by exploiting the MD generated anomalous dispersion curve of sound waves. Anomalous dispersion was found to set in for kσ ～ 0·25. A sound speed of 880 ms^-1 has been determined which is in excellent agreement with experimental values for liquid argon. The total error for the MD value amounts to about 5 per cent. In contrast, the ratio of the specific heats γ and the transport coefficients D _T and Γ (thermal diffusivity and sound attenuation) were determinable only with an accuracy of 15 per cent due to the need for larger extrapolations. Nevertheless, we found D _T, Γ and γ in agreement with experimental values within 5-10 per cent.     

On the normalization of the Gamov resonant wave function in the configuration space

The regularization of the normalization integral for the resonant wave function, proposed by Zeldovich, is valid only when |Req _res| > |Imq _res|. A new normalization procedure is proposed and implemented, which is valid when this condition fails. First, an arbitrarily normalized vertex function g(k) is calculated using the formula with the potential V(r) in the integrand. This Fourier integral converges for a potential with the asymptotics V(r) → constr ^?n exp(?μr) if |Imq _res| < μ/2. Then the function g(k) is normalized using the generalized normalization rule, which is independent of the resonance pole position. The proposed method is approved by the example of calculation for a virtual triton.     

The inverse backscattering problem in three dimensions

This article is a study of the mapping from a potentialq(x) onR ³ to the backscattering amplitude associated with the Hamiltonian –+q(x). The backscattering amplitude is the restriction of the scattering amplitudea(, , k), (, , k)S ²×S ²×₊, toa(,–, k). We show that in suitable (complex) Banach spaces the map fromq(x) toa(x/|x|, –x/|x|, |x|) is usually a local diffeomorphism. Hence in contrast to the overdetermined problem of recoveringq from the full scattering amplitude the inverse backscattering problem is well posed.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

A bound on the number of bound states in the Schrödinger equation

A boundS _l is given for the number of bound statesn _i in thelth partial wave corresponding to a spherically symmetric potential in non-relativistic quantum mechanics. This bound is given by whereV _a(l, r) is the attractive part of the effective potentialV(r)+l(l+1)/r ². Extensive comparative study ofS _i and the Bargmann inequality is made.     

Coordinate space form of interacting reference response function ofd-dimensional jellium

Summary The interacting reference response functionX _I ^[3](k) of three-dimensional jellium ink space was defined by Niklasson in terms of the momentum distribution of the interacting electron assembly. Here the Fourier transformF _I ^[d](r) ofX _I ^[d] (k) is studied for the jellium model withe ²/r interactions in dimensionalityd=1,2 and 3, in an extension of recent work by Holas, March and Tosi for the cased=3. The small-r and large-r forms ofF _I ^[d] (r) are explicitly evaluated from the analytic behaviour of the momentum distributionn _d(p). In the appendix, a model ofn _d (p) is constructed which interpolates between these limits.     

Clusters of cycles

A cluster of cycles (or (r,q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r,q)-polycyclic realization P(G) on the plane. An (r,q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q; (ii) all interior faces (denote their number by p_r) are combinatorial r-gons; (iii) all vertices, edges and interior faces form a cell-complex.An example of (r,q)-polycycle is the skeleton of (r^q), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r,q)=(3,3),(4,3),(3,4),(5,3),(3,5). Only for those five pairs, P((r^q)) is (r^q) without exterior face; otherwise, P((r^q))=(r^q).Here we give a compact survey of results on (r,q)-polycycles. We start with the following general results for any (r,q)-polycycle G: (i) P(G) is unique, except of (easy) case when G is the skeleton of one of the five Platonic polyhedra; (ii) P(G) admits a cell-homomorphism f into (r^q); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r^q) and a helicene, otherwise. In [ARS Comb. A 29 (1990) 5], all proper spheric polycycles are given. An (r,q)-helicene exists if and only if p_r>(q−2)(r−1) and (r,q)≠(3,3). We list the (4,3)-, (3,4)-helicenes and the number of (5,3)-, (3,5)-helicenes for first interesting p_r. Any outerplanar (r,q)-polycycle G is a proper (r,2q−2)-polycycle and its projection f(P(G)) into (r^2q−2) is convex. Any outerplanar (3,q)-polycycle G is a proper (3,q+2)-polycycle.The symmetry group Aut(G) (equal to Aut(P(G)), except of Platonic case) of an (r,q)-polycycle G is a subgroup of Aut((r^q)) if it is proper and an extension of Aut(f(P(G))), otherwise. Aut(G) consists only of rotations and mirrors if G is finite, so its order divides one of the numbers 2r, 4 or 2q. Almost all polycycles G have trivial AutG.Call a polycycle G isotoxal (or isogonal, or isohedral) if AutG is transitive on edges (or vertices, or interior faces); use notation IT (or IG, or IH), for short. Only r-gons and non-spheric (r^q) are isotoxal. Let T^*(l,m,n) denote Coxeter’s triangle group of a triangle on S², E² or H² with angles π/l, π/m, π/n and let T(l,m,n) denote its subgroup of index 2, excluding motions of 2nd kind. We list all IG- or IH-polycycles for spheric (r,q) and construct many examples of IH-polycycles for general case (with AutG being above two groups for some parameters, including strip and modular groups). Any IG-, but not IT-polycycle is infinite, outerplanar and with same vertex-degree, we present two IG-, but not IH-polycycles with (r,q)=(3,5),(4,4) and AutG=T(2,3,∞)PSL(2,Z), T^*(2,4,∞). Any IH-polycycle has the same number of boundary edges for each its r-gon. For any r≥5, there exists a continuum of quasi-IH-polycycles, i.e. not isohedral, but all r-gons have the same 1-corona.On two notions of extremal polycycles:

1. We found for the spheric (r,q) the maximal number n_int of interior points for an (r,q)-polycycle with given p_r; in general case, (p_r/q)≤n_int<(rp_r/q) if any r-gon contains an interior point.

2. All non-extendible (r,q)-polycycles (i.e. not a proper subgraph of another (r,q)-polycycle) are (r^q), four special ones, (possibly, but we conjecture their non-existence) some other finite (3,5)-polycycles, and, for any (r,q)≠(3,3),(3,4),(4,3), a continuum of infinite ones.

On isometric embedding of polycycles into hypercubes Q_m, half-hypercubes and, if infinite, into cubic lattices Z_m, : for (r,q)≠(5,3),(3,5), there are exactly three non-embeddable polycycles (including (4³)−e, (3⁴)−e); all non-embeddable (5,3)-polycycles are characterized by two forbidden sub-polycycles with p₅=6.     

Excitations in Heisenberg spin glasses

Within the RPA approach forT=0, the excitations of the Heisenberg spin glass system Eu _x Sr_1–x S are studied by numerical methods, using a continued fraction algorithm. Both the density of statesg(E) and also the spectral functionS(q,E) are calculated for systems with (16)³ sites, withx=0.4, 0.5, and 0.6 (spin glass phase), and also forx0.7 (ferromagnetic phase). Forq-vectors within the (1,1,1) plane,S(q,E) shows magnon peaks even in the spin glass phase, over the whole range ofq. However, these peaks are quite broad, and there is considerable intensity at small energies even for largeq, leading to a finite intercept ofg(E) forE0. Over a large temperature range, the specific heat is approximately linear inT forx0.7.     

The 2‐matrix of the spin‐polarized electron gas: contraction sum rules and spectral resolutions

The spin‐polarized homogeneous electron gas with densities ρ_↑ and ρ_↓ for electrons with spin ‘up’ (↑) and spin ‘down’ (↓), respectively, is systematically analyzed with respect to its lowest‐order reduced densities and density matrices and their mutual relations. The three 2‐body reduced density matrices γ_↑↑, γ_↓↓, γ_a are 4‐point functions for electron pairs with spins ↑↑, ↓↓, and antiparallel, respectively. From them, three functions G_↑↑(x,y), G_↓↓(x,y), G_a(x,y), depending on only two variables, are derived. These functions contain not only the pair densities according to g_↑↑(r) = G_↑uarr;(0,r), g_↓↓(r) = G_↓↓(0,r), g_a(r) = G_a(0,r) with r = | r ₁ ‐ r ₂|, but also the 1‐body reduced density matrices γ_↑ and γ_↓ being 2‐point functions according to γ_s = ρ_sf_s and f_s(r) = G_ss(r, ∞) with s = ↑,↓ and r = | r ₁ ‐ r ^′₁|. The contraction properties of the 2‐body reduced density matrices lead to three sum rules to be obeyed by the three key functions G_ss, G_a. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions n_↑(k) and n_↓(k), following from f_↑(r) and f_↓(r) by Fourier transform, are correctly normalized through f_s(0) = 1. In addition to the non‐negativity conditions n_s(k),g_ss(r),g_a(r) ≥ 0 [these quantities are probabilities], it holds n_s(k) ≤ 1 and g_ss(0) = 0 due to the Pauli principle and g_a(0) ≤ 1 due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin‐unpolarized homogeneous electron gas in terms of 2‐body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin‐polarized case and (ii) to the 2‐body reduced density matrix giving thus its spectral resolutions.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

Periodic and flat irreducible representations ofSU(3) q

We construct all the periodic irreducible representations ofU(SU(3)) _q forq am-root of unity. Their dimensions arek(2m) ² fork=1,...,m (onlyk=1,...,m/2 for evenm). Their interest is that they could be a tool to generalize the chiral Potts model. By truncation of these representations, we construct flat representations ofU(SU(3)) _q , in which all the multiplicities of the weights are set to 1.     

Renormalization of the Higgs model: Minimizers,propagators and the stability of mean field theory

We study the effective actionsS ^(k) obtained byk iterations of a renormalization transformation of the U(1) Higgs model ind=2 or 3 spacetime dimensions. We identify a quadratic approximationS _Q ^(k) toS ^(k) which we call mean field theory, and which will serve as the starting point for a convergent expansion of the Green's functions, uniformly in the lattice spacing. Here we show how the approximationsS _Q ^(k) arise and how to handle gauge fixing, necessary for the analysis of the continuum limit. We also establish stability bounds onS _Q ^(k) , uniformly ink. This is an essential step toward proving the existence of a gap in the mass spectrum and exponential decay of gauge invariant correlations.Dedicated to the memory of Kurt SymanzikSupported in part by the National Science Foundation under Grant PHY 82-03669     

Theoretical study of stereodynamics for the O（3P） ＋ H2 → OH ＋ H reaction

This paper employs the quasi-classical trajectory calculations to study the influence of collision energy on the title reaction on the potential energy surface of the ground ³A' triplet state developed by Rogers et al. (J. Phys. Chem. A 2000 104 2308). It calculates the product angular distribution of P(θ_r), P(φ_r) and P(θ_r, φ_r) which reflects vector correlation. The distribution P(θ_r) shows that product rotational angular momentum vectors j' of the products are strongly aligned along the relative velocity direction k. The distribution of P(φ_r) implies a preference for left-handed product rotation in planes parallel to the scattering plane. Four different polarisation-dependent cross-sections are also presented in the centre-of-mass frame. Results indicate that OH is sensitively affected by collision energies of H₂.     

Adiabatic theorem in axiomatic quantum field theory

It is shown that Møller matricesS _± and scattering matrixS in axiomatic field theory can be expressed through their adiabatic analogs. In particular, it is proved under certain conditions that \(S_ - = \mathop {s\lim }\limits_{\alpha \to 0} S_\alpha (0,\infty )W_\alpha \) whereW _α is a trivial phase factor [i.e. a unitary operator of the form exp i / α ∝r(k)a ⁺ (k)a(k)dk]. Corresponding results in Hamiltonian approach are discussed.     

Dynamics and sedimentation velocity in charge-stabilized colloidal suspensions

Summary Charge-stabilized suspensions are characterized by the strong electrostatic interactions between the particles so that rather dilute systems may exhibit strong correlation resulting in a well-developed short-range order. This microstructure, quantitatively described by the pair distribution functiong(r), is rather different from that of (uncharged) hard spheres. It is shown how this difference affects the ?hydrodynamic function?H(k), which appears in the expression for the first cumulant Γ(k)=k ² D _eff(k)=k ² H(k)/S(k) of the dynamic autocorrelation function. Without hydrodynamic interaction,H(k)=D ⁰, which is the free-diffusion coefficient. Using pairwise additive hydrodynamic interaction and the lowest-order many-body theory of hydrodynamic interaction, it is found thatH(k) can deviate considerably fromD ⁰ even for systems of volume fractions ϕ as low as 10⁻³. These effects are more pronounced for collective diffusion than for self-diffusion. SinceH(k=0) is closely related to the sedimentation velocity, we have studied this quantity as a function of volume fraction. It is found that (H(0)/D ⁰) −1 scales asφ ^1/3 at low ϕ in salt-free suspensions. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Static charged spheres with anisotropic pressure in general relativity

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+ε)r ², u=4πξr ², v _r=4πp _r r ², v _⊥=4πp _⊥ r ²[ρ, ξ(=−(1/2)F ₁₄ F ¹⁴), p _r, p _⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas ε denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=v _r=(a ²/2κ)r ⁿ⁺², v _⊥=k ₁ v _r, w=k ₂ v _r; a ², n(>0), k ₁, k ₂ being constants with κ=((k ₁+2)/3+k ₂) and (ii) w+u=(b ²/2)r ⁿ⁺², u=v _r, v _⊥−v _r=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r ² since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary. Dedicated to Prof. F A E Pirani.     

Saturation of Coulomb sum rules in the <Superscript>6</Superscript>Li case

The Coulomb sums S _L(q) of the ⁶Li nucleus have been obtained from electron scattering measurements at 3-momentum transfers q = 1.125–1.625 fm⁻¹. It is found that at q > 1.35 fm⁻¹ the Coulomb sum of the nucleus becomes saturated: S _L(q) = 1 .     

Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

Three-dimensional nuclear dynamics in the quantized ATDHF approach

The quantized adiabatic time-dependent Hartree-Fock theory is numerically applied to the low energy large amplitude collective dynamics of heavy ion systems ranging from α + α to ¹⁶O + ¹⁶O. The problem is reduced to three successive steps. First, for the lowest mode the optimal, i.e., maximally decoupled, collective path {φ_q} is evaluated by solving a coupled set of nonlinear differential equations for the single-particle wave functions _q^(a)(r) of φ_q, depending on the collective coordinate q and three spatial coordinates. A density-dependent interaction with a direct finite range Yukawa-term is employed and three-dimensional coordinate- and momentum-grid techniques are used, including fast Fourier methods. In a second step the quantized collective Hamiltonian H_c(q, d/dq) is extracted from {φ_q} by means of generator coordinate techniques involving, besides q, a conjugate variable p. Starting from {φ} this procedure includes the numerical evaluation of the classical potential, (q), of the intertia parameter, (q), of the quantum corrections with regard to rotation, translation and collective q-motion, (q), and of the centrifugal potential. The third step consists of actually calculating the subbarrier fusion cross section by means of WKB methods applied to the collective Hamiltonian H_c(q, d/dq). The theoretical numbers are compared with results from Hartree-Fock calculations with quadrupole constraint, and with experimental data. The microscopic aspects of the dynamics, the relation to other theories, and the practical and conceptual problems arising from the quantized ATDHF theory are discussed in detail.     

The Bethe Equation at q = 0, The Möbius Inversion Formula,and Weight Multiplicities: III. The X

It is shown that the numbers of off-diagonal solutions to the U _q (X ^(r) _N ) Bethe equation at q = 0 coincide with the coefficients in the recently introduced canonical power series solution of the Q-system. Conjecturally, the canonical solutions are characters of KR (Kirillov–Reshetikhin) modules. This implies that the numbers of off-diagonal solutions agree with the weight multiplicities, which is interpreted as a formal completeness of the U _q (X ^(r) _N ) Bethe ansatz at q = 0.