Sixth Frequency Moments Research Articles

Self-diffusion coefficients of the metastable Lennard-Jones vapor

Self-diffusion coefficients of a metastable Lennard-Jones vapor were obtained using thememory function formalism and the frequency moments of the velocity autocorrelationfunction at reduced temperatures from 0.75 to 1.0. The radial density distribution functionsused to evaluate the second, fourth and sixth frequency moments of the velocityautocorrelation function were obtained from the restricted canonical ensemble Monte Carlosimulation (Corti and Debenedetti 1994 Chem. Eng. Sci. 49 2717). The self-diffusioncoefficients at reduced temperature 0.75 do not vary monotonically as the densityincreases, and for the other three temperatures the self-diffusion coefficients varynormally.

Self-diffusion coefficients of Lennard-Jones fluids

Expressions for the selfdiffusion coefficient have been obtained using the memory function formalism and the frequency moments of the velocity autocorrelation function. The fourth and the sixth frequency moments of the velocity autocorrelation function have been calculated using a low-order decoupling and the superposition approximation for the static quadruplet and the triplet correlation functions, respectively. These results have been used to calculate the diffusion coefficients of Lennard-Jones fluids over a wide range of densities and temperatures. The predicted results are found to be in good agreement with recent molecular dynamics data.

Frequency sum rules and charge fluctuation spectra in binary ionic mixtures

Abstract The explicit analytical expressions of the fourth and sixth frequency moments of the number density correlation functions of binary ionic mixtures in a uniform background are given for any concentration. Using the memory function formalism, we analyse the molecular dynamics results (MD) for charge fluctuation spectra of equimolar H + − He ++ mixtures for different coupling parameters. The use of the frequency moments allows us to determine the memory function in a systematic manner. The agreement is good except for long wavelength.

The longitudinal sixth moment for an anisotropic paramagnet having site or exchange-bond dilution

Analytic expressions are derived for the longitudinal sixth frequency moment of the spin correlation function for a dilute anisotropic Heisenberg paramagnet in the high-temperature limit. The results are valid for arbitrary spin and arbitrary range of the exchange interactions. Both site and exchange-bond dilution are treated. The latter is treated using two models introduced by Kawasaki and Tahir-Kheli (1978). In one the existence of longitudinal and transverse bonds depends only on a single random variable, whilst in the other they can fluctuate independently. As an application of the use of the moments, the extent to which pseudo-dipolar interactions can affect the high-temperature spin dynamics of a Heisenberg magnet is examined.

The sixth frequency moments for an anisotropic paramagnet with exchange-bond dilution

The sixth frequency moments of the spin correlation function for a bond-diluted anisotropic Heisenberg paramagnet in the high temperature limit are evaluated for arbitrary spin values. Results for both the transverse and longitudinal moments are presented for a model in which the transverse and longitudinal exchange bonds can fluctuate independently. The results are presented in a form which allows the moments for the undiluted, site diluted, and single-bond diluted moments to be also written down.

The frequency moments of the spin correlation function for a Heisenberg paramagnet having bond dilution

For a Heisenberg paramagnet having both uniaxial anisotropic exchange and single-ion uniaxial anisotropy, two types of dilution are considered: model (a) in which the existence of transverse and longitudinal bonds depends only on a single random variable, and model (b) in which these bonds can fluctuate independently. For both models analytic expressions for the moments up to the fourth are presented that are valid for arbitrary spin and arbitrary range of the magnetic integration. Both the longitudinal and transverse moments are considered. The results are specialised to a number of simple lattices (linear, square, simple cubic, body-centered cubic and face-centred cubic) having nearest-neighbour and second-nearest-neighbour interactions only. Finally, the sixth frequency moment in the case of the isotropic magnet is studied using model (a).

The sixth frequency moment of the relaxation-shape function for a Heisenberg paramagnet

A correction to the hitherto published expression for the sixth frequency moment of the relaxation-shape function of a Heisenberg paramagnet at elevated temperatures is presented. The results are in accord with a computer calculation for a face-centred cubic lattice with nearest- neighbour exchange interactions.

Sum rules and atomic correlations in classical liquids. III. Numerical estimates and an application

The expressions for the fourth and sixth frequency moments of current correlation functions are simplified by performing the angular integrations associated with the three-body contributions to these moments. The resultant expressions are applicable both in the long-wavelength limit as well as for any momentum transfer. These expressions are then numerically evaluated for a liquid-argon-like system near its triple point using the molecular dynamics data of Verlet for the static pair-correlation function. We have used Kirkwood superposition approximation for the triplet correlation function and a low-order decoupling approximation for the quadruplet correlation function. The Maxwell relaxation time for the longitudinal mode is calculated using our computed numbers for the fourth and sixth moments. The results for the relaxation time are compared with those of other authors as well as with the experimental observations. It is concluded that the approach of Machida and Murase is comparatively adequate and has to be modified in order to get a better wave-number dependence of the relaxation time.

High-Frequency Expansion of Memory Function in Classical Simple Liquids

Transverse collective motion in classical simple liquids is studied by using a high-fre· quency expansion of memory functions for the transverse current correlation function. Ex­ plicit expressions for the 'first, second, fourth and sixth frequency moments of the spectrum of the correlation function are obtained, in which atomic correlations in the liquids are taken into account up to quartets. As an application, the dispersion curve of transverse collective modes in liquid argon is calculated numerically. As in the case of longitudinal collective modes, fairly good agreement with the result obtained by Rahman using molecular dynamics calculations is obtained, provided that an approximate cancellation is assumed to take place between a term containing a three-body correlation function and those with two-body corre­ lation functions which appear in an expansion term giving the fourth frequency moment of the spectrum. By applying Kirkwood's superposition approximation, discussion is given on the role of the three-body correlation function in determining the dispersion of transverse collective modes.

Sum rules and atomic correlations in classical liquids. II

The sixth and eighth frequency moments of the dynamical structure factor for a classical liquid are examined in some recent approximate models. These are compared with their exact expressions. Using the approximate models, the moments are calculated for liquid argon and the results are compared with those obtained from the molecular-dynamics data.

Relaxation-shape function of a Heisenberg paramagnet based on the method of moments

The sixth frequency moment of the frequency-wavevector dependent relaxation-shape function F(k, omega ), at elevated temperatures is presented for a model applicable to RbMnF3. The exact value of this moment is compared to the approximate values predicted by the semi-phenomenological representations of the relaxation-shape function based on a knowledge of only the second and fourth moments. Finally, a Gaussian choice for the memory function associated with the spin-velocity relaxation function is made which preserves all the frequency moments of F(k, omega ), up to and including the sixth. The form of F(k, omega ) obtained with this memory function is compared to the first principle calculations of Blume and Hubbard and of Reiter. The value of the diffusion constant is found to be very close to that given by Mori and Kawasaki and by Bennett and Martin.

The sixth frequency moment of the frequency-wave-vector-dependent relaxation-shape function for an isotropic Heisenberg paramagnet at infinite temperature

An analytic expression for the sixth frequency moment of the frequency-wave-vector-dependent relaxation-shape function for an isotropic Heisenberg paramagnet at infinite temperature is presented. The expression is valid for any lattice (centrosymmetric), and for general spin and arbitrary range of exchange interactions.

Sixth frequency moment of two-spin correlation functions in a linear chain of spins at high temperature

We have computed for an arbitrary spin the sixth frequency moments of the two-spin correlation function for an anisotropic Heisenberg Hamiltonian in a linear chain of spins at high temperature, and we compare our results with those previously known.

Sum rules and atomic correlations in classical liquids

The sixth frequency moment of the spectral functions of both the longitudinal and transverse current correlations along with their self-parts has been derived for a system of particles interacting through a two-body potential. The first two frequency moments of the spectral function of the energy-density fluctuation, and the first four frequency moments of the spectral function of the kinetic-energy-density fluctuation-correlation functions have also been derived. Numerical estimates for the fourth and sixth frequency moments of the spectral function of the longitudinal current correlations have been made for liquid argon. From this some information about the behavior of the contributions of triplet and quadruplet correlation functions to the frequency moments of liquid argon have been obtained. Numerical estimates have also been made for the first four frequency moments of the kinetic-energy-density fluctuation-correlation function in the long-wavelength limit for liquid argon.

Sixth Frequency Moment of the Frequency-Wave-Vector-Dependent Correlation Function for Isotropic Heisenberg Paramagnets at Elevated Temperatures

We calculate the sixth frequency moment ${〈{\ensuremath{\omega}}^{6}〉}_{\mathrm{K}}$ of the frequency-wave-vector-dependent spectral function for the isotropic Heisenberg paramagnet at elevated temperatures with arbitrary spin and arbitrary range of exchange interaction. These exact results are compared with the approximate sixth moment predicted by the two-parameter Gaussian representation used in the preceding paper for the generalized diffusivity. The general agreement of the approximate and the exact results for ${〈{\ensuremath{\omega}}^{6}〉}_{\mathrm{K}}$ is satisfactory. For the exactly soluble spin-\textonehalf{} nearest-neighbor one-dimensional $\mathrm{XY}$ model, we compare the predictions of the two-parameter Gaussian approximation for ${〈{\ensuremath{\omega}}^{6}〉}_{{\mathrm{K}}^{\mathrm{zz}}}$ and ${〈{\ensuremath{\omega}}^{8}〉}_{{\mathrm{K}}^{\mathrm{zz}}}$ against the exact results for these moments (which are derived by using the corresponding exact spectral function given by Katsura et al. The agreement is again found to be satisfactory. This reinforces our confidence in the qualitative validity of the simple Gaussian approximation for the generalized diffusivity.