Kinetic theory of classical liquids. I. Basic theory

A somewhat new approach to a kinetic theory of classical liquids is presented, and it is used to calculate the dynamical structure factor S(qω). It gives correctly the zeroth, second, and fourth frequency moments of S(qω), and it goes correctly over to the free particle value for large q. For small q and ω it goes over to proper hydrodynamics, including the coupling to heat diffusion. Results of numerical calculations on liquid argon are presented and they show very good agreement with available neutron scattering and molecular dynamics data.     

Q-dependent stoner factors and reformulated spin-fluctuation enhancements: V and Pd

Spin-polarized self-consistent LMTO band calculations are performed for supercells containing induced, “frozen” spin-waves. Momentum dependent Stoner factors S(q) are obtained as the local ratio between induced and applied spin splitting. The results are used to calculate the spin-fluctuation enhancement λ_sp on a similar basis as a calculation of the electron phonon coupling. The methods are tested on vanadium and palladium. For the latter a strong reduction of S(q) is found for large q. Obtained values for λ_sp are of reasonable magnitude.     

The correlation function of the Ising model in the MFA and their sum rule

It is shown that in the correctly performed molecular field approximation the correlation function 〈S(q) S(-q)〉 fulfills the sum rule N^-1Σ_q〈S(q) S(-q)〉 = 1. This can be proved for ferro- and antiferromagnets and the disordered phases of o-hydrogen.     

The structure factor and the transport properties of dense fluids having molecules with square well potential,a possible generalization

An analytical expression of the static structure factor S(k) has been presented, treating the square well potential as a perturbation on the hard sphere system incorporating the correct equation of state given by Carnahan and Starling (CS). Reasonable values of potential parameters are obtained from a fit with the experimental peak of S(k). The present calculation of transport coefficients for liquid sodium at 373 K, is found to be in closer agreement with the experimental data. The role of back scattering correction is also discussed.     

Consistent MFA correlation functions of Ising models for all temperatures

A correct calculation of the Ising model correlation function C(q) = 〈(S(q) ? 〈S(q)〉) (S(-q) ? 〈S(-q)〉)〉 in the MFA results in

$\text{C(q)=}\text{〈S}{}^2\text{〉?〈〉}{}^2\text{1?(〈S}{}^2\text{〉?〈S〉}{}^2\text{βJ(q}\text{1}\text{N}\text{∑}\text{q}\text{1}\text{1?〈S〉}{}^2\text{βJ(q)}{}^{\mathrm{?1}}\text{C(q) fulfills the exact sum rule N}{}^{-1}\text{Σ}{}_{\mathrm{q}}\text{C(q) = 〈S}{}^2\text{〉 ? 〈S〉}{}^2$

. Previous literature supposed a violation of this sum rule to be a characteristic disadvantage of this approximation.     

Principles of generalized spin-echo spectroscopy

The concept is proposed for determining the total dynamic scattering function of an object under study, representing a sum of odd and even parts measured by the generalized neutron spin-echo method in the form of the signals S _odd(q, t) ～ ΣS(ω, q)sin(ωt)dω and S_even(q, t) ～ ΣS(ω,q)cos(ωt)dω as functions of the momentum q transferred to the neutron and the time t corresponding to the frequency ω and the transferred energy ?ω. The principle of the generalized spin echo and the results of mathematical modeling are confirmed in experiments on inelastic scattering on magnetic fluids and polymer solutions. The developed method makes it possible to study the features of the dynamics of atomic and molecular systems, e.g., to analyze soft vibrational spectra of nanoparticle ensembles against the background of intense relaxation processes, which is inaccessible for classical spin-echo spectrometry.     

Slow dynamics in glycerol: collective de gennes narrowing and independent angstrom motion

The slow dynamics of microscopic density correlations in supercooled glycerol was studied by time-domain interferometry using ⁵⁷Fe-nuclear resonant scattering gamma rays of synchrotron radiation. The dependence of the relaxation time at 250 K on the momentum transfer q is maximum near the first peak of the static structure factor S(q) at q ～ 15 nm ^?1. The q-dependent behavior of the relaxation time known as de Gennes narrowing was confirmed in glycerol. Conversely, de Gennes narrowing around the second and third peaks of S(q) at q ～ 26 nm ^?1 and 54 nm ^?1 was not detected. The q dependence of the relaxation time was found to follow a power-law equation with power-law index of 1.9(2) in the q region well above the first peak of S(q) up to ～ 60 nm ^?1, which corresponds to angstrom scale, within experimental error. This suggests that in the angstrom-scale dynamics of supercooled glycerol, independent motions dominate over collective motion.     

Specific heat in the nonextensive statistics: effective temperature and Lagrange parameter β

We analyse the specific heat and the fluctuation–dissipation theorem by considering the effective temperature, T_eff≡(Trρ_q^q)/β, in the Tsallis statistics. In particular, the results show that the specific heat is nonnegative for q∉[0,1). We also investigate how to obtain a family of entropies employing the condition C_q=−β²(∂U_q/∂β)⩾0 for q>0, S_q=S_q(Trρ_q^q) and the normalized constraints.     

Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map

Dynamic behaviours of the 2^∞ attractor at the accumulation of period doubling in the logistic map are studied by the sum of the local expansion rates S_n(x₁) of nearby orbits. The variance 〈[S_n(x)]²〉 and algebraic exponent ß_n(x₁) = S_n(x₁)/ln(n) exhibits self-similar structures. The critical bifurcations such as intermittency, band merging and crisis-sudden widening of the chaotic attractor are studied in terms of a q-weighted average Λ(q), (− ∞ < q < ∞) of the coarse-grained local expansion rates Λ of nearby orbitals.     

On a general analytic formula for U q(su(3)) Clebsch-Gordan coefficients

We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U _q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U _q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U _q(su(3)). We obtain a very compact general analytic formula for the U _q(su(3)) CGCs in terms of the U _q(su(2)) Wigner 3nj symbols.     

Representation of osp (2, 2) q (2) andS-matrix of super sine-Gordon theory

We investigate the representations of the osp (2, 2) _q ⁽²⁾ algebra, which leads to theS-matrix of super sine-Gordon theory. TheS-matrix has been derived from supersymmetric conformal field theory with some assumptions. We show that the conjecturedS-matrix can be derived from the representation theory using a correspondence between the representations of osp (1, 2) _q and those of sl(2) _q .     

Non-equilibrium structure factor for the disordering of adsorbed monolayers

We consider the non-equilibrium, time-dependent elastic-scattering structure factor S(q,t), for the disordering of an ordered overlayer, initially in equilibrium at temperature T_I and characterized by the structure factor S(q,0)=x(q,T_I, upon a sudden increase in temperature T_I→T_F at constant coverage, such that the adsorbates equilibrate at T_F in a disordered phase. The initial decay of a peak in x(q,T_I) proceeds exponentially in time, $\text{exp}\text{(−}\text{t}\text{τ}{}_{\mathrm{q}}\text{)}$, where τ_q is a wavevector-dependent lifetime, before it crosses over to a power-law, t⁻¹ decay. When x(q,T_I) is peaked at the boundaries of the Brillouin zone (BZ), the peak approximately maintains its shape in q-space as it decays exponentially. Except near the center of the BZ, after the peak has decayed sufficiently, the dependence of S(q,t) on q is as though the spins quasi-equilibrate to the equilibrium structure factor associated with T_F, x(q,T_F), in that the ratio $\text{S(q,t)}\text{x(q,T}{}_{\mathrm{<mtext>F</mtext>}}\text{)}$ is independent of q, is dependent on time, approaching unity as t⁻¹ for large t. For systems exhibiting an initial peak for q ≈ 0, the peak decays exponentially but does not preserve its shape, since τ_q strongly depends on q, diverging as q⁻² for q→0. For these systems too, away from the center of the BZ, $\text{S}\text{(q,t)}\text{x(q,T}{}_{\mathrm{<mtext>F</mtext>}}\text{)}$ rapidly evolves to a slowly decaying function of $\text{t}\text{t}{}_{\mathrm{w}}$, independent of q. In this case, however, the characteristic time scale, t_w, is anomalously long, proportional to ξ², where ξ is the correlation length associated with the initial state. This behavior of t_w can be related to the random walk of domain boundaries.     

Pole expansion determination of the pair correlation function of a simple fluid

An explicit expression for the pair correlation function is given in terms of an expansion in the poles of the structure factor S(q) in the complex q plane. This is derived when the direct correlation function or structure factor is of known analytical form. An illustrative example is given for a hard-sphere-Yukawa fluid in the mean spherical approximation.     

Duality and quantum-algebra symmetry of the AN−1(1) open spin chain with diagonal boundary fields

We show that the transfer matrix of the A_N−1⁽¹⁾ open spin chair with diagonal boundary fields has the symmetry U_q(SU(l)) × U_q(SU(N−l)) × U(1), as well as a “duality” symmetry which maps l ↔ N − l. We exploit these symmetries to compute exact boundary S-matrices in the regime with q real.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

High frequency sound and the boson peak in amorphous silica

We present the results of extensive molecular dynamics computer simulations in which the high frequency dynamics of silica, i.e. for frequencies ν > 0.5 THz, is investigated in the viscous liquid state as well as in the glass state. We characterize the properties of high frequency sound modes by analyzing J _l(q,ν) and J _t(q,ν), the longitudinal and transverse current correlation function, respectively. For wave-vectors q > 0.4 ?^-1 the spectra are sitting on top of a flat background. The dynamic structure factor S(q,ν) exhibits for q > 0.23 ?^-1 a boson peak which is located nearly independent of q around 1.7 THz and for which the intensity scales approximately linearly with temperature. We show that the low frequency part of the boson peak is mainly due to the elastic scattering of transverse acoustic modes with frequencies around 1 THz. The strength of this scattering depends on q and is largest around q = 1.7 ?^-1, the location of the first sharp diffraction peak in the static structure factor. By studying S(q,ν) for different system sizes we show that strong finite size effects are present in the low frequency part of the boson peak in that for small systems part of its intensity is missing. We discuss the consequences of these finite size effects for the structural relaxation. Received 27 June 2000 and Received in final form 9 January 2001     

Quantum integrable systems related to lie algebras

Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g²v(q) of the following 5 types: v_I(q) = q^?2, v_II(q) = sinh^?2q, v_III(q) = sin^?2q, $\text{v}{}_{\mathrm{<mtext>IV</mtext>}}\text{(q) =}\text{P}\text{(q)}$, v_V(q) = q^?2 + ω²q². Here $\text{P}\text{(q)}$ is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp(q_jq_{j+ 1}) is moreover considered.This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.     

On asymptotical behaviour of the total γp- and γγ- cross sections off the mass shell

It is shown that the Regge type behaviour and the scaling law Sσ_γp ∼ f(−S/q₂) give for Sσ_γγ the asymptotics f (SS₀/q₁²q₂²) at large S, q₁² and q₂². This result is confirmed by the summation of the diagrams in ?³-theory. It is announced also that the introduction of the spin (γ⁵-theory) results in a slight breaking of the scaling law and in the suppression of the scalar photon contribution to these processes compared with the contribution of the transversal photon.     

Strictly two-dimensional self-avoiding walks: Thermodynamic properties revisited

The density crossover scaling of various thermodynamic properties of solutions and melts of self-avoiding and highly flexible polymer chains without chain intersections confined to strictly two dimensions is investigated by means of molecular dynamics and Monte Carlo simulations of a standard coarse-grained bead-spring model. In the semidilute regime we confirm over an order of magnitude of the monomer density ?? the expected power law scaling for the interaction energy between different chains e _int ?? ?? ^21/8, the total pressure P ?? ?? ³ and the dimensionless compressibility gT = lim _q??0 S(q) ?? 1/?? ². Various elastic contributions associated to the affine and non-affine response to an infinitesimal strain are analyzed as functions of density and sampling time. We show how the size ??(??) of the semidilute blob may be determined experimentally from the total monomer structure factor S(q) characterizing the compressibility of the solution at a given wave vector q . We comment briefly on finite persistence length effects.      

Calculation of structure factors for a simple model of adsorption on a surface by real-space renormalization-group method

The structure factor S(q) of a simple model describing adsorption on a two-dimensional surface is computed by real-space renormalization-group technique for T ? T_c. The method used is based on first-order cumulant expansion implemented with an ad hoc recursion relation for the coupling constants. The model is equivalent to an antiferromagnetic Ising model in an external field.