Fourth Frequency Moments Research Articles

Ab initio study of collective eigenmodes in dynamics of molten salts

An effective fitting scheme within the thermo-viscoelastic model is proposed for estimation of dynamic eigenmodes of an ionic melt from ab initio molecular dynamics (AIMD) simulations. Dynamic eigenmodes are solutions of the generalized Langevin equation in matrix form, obtained on the basis set of eight dynamic variables. Fitting parameters are used only in the generalized hydrodynamic matrix T(k) for matrix elements involving heat density and heat current correlations only, because of huge computational efforts needed to calculate them directly in ab initio simulations. In the proposed scheme six AIMD-derived time correlation functions, three partial density-density and three partial current-current ones, are recovered by the proposed theoretical approach, which satisfies the exact sum rules up to the fourth frequency moments of partial dynamic structure factors. The suggested theoretical analysis was applied to estimation of propagating hydrodynamic and non-hydrodynamic eigenmodes in molten NaCl.

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.

Relaxation and Short Time Dynamics of Bulk Liquids and Fluids Confined in Spherical Cavities and Slit Pores

The density of states for bulk and confined fluids have been modeled using a recently proposed gamma distribution (Krishnan, S. H.; Ayappa, K. G. J. Chem. Phys. 2004, 121, 3197). The gamma distribution results in a closed form analytical expression for the velocity autocorrelation function and the relaxation time of the fluid. The two parameters of the gamma distribution are related analytically to the second and fourth frequency moments of the fluid using short time expansions. The predictions by the proposed gamma model are compared with the velocity autocorrelation functions obtained using the theory of instantaneous normal modes (INMs) and from molecular dynamics simulations. The model is applied to a bulk soft sphere liquid and fluids confined in a spherical cavity and slit-shaped pores. The gamma model is able to capture the resulting changes in relaxation time due to changes in density and temperature extremely well for both the bulk liquid and confined inhomogeneous fluid situations. In all cases, the predictions by the gamma model are superior to those obtained from the INM theory. In the case of the fluid confined in a slit pore, the loadings were obtained from a grand canonical Monte Carlo simulation where the pore is equilibrated with a bulk fluid. This is similar to a confinement situation in a surface force apparatus. The predicted relaxation times vs pore widths from the gamma model are seen to accurately capture the oscillations due to formation and disruption of layers within the slit pore.

Modeling velocity autocorrelation functions for confined fluids using gamma distributions.

We propose a model for the short-time dynamics of fluids confined in slit-shaped pores. The model has been developed from the observation that the real lobe of the instantaneous normal mode density of states (INM DOS) closely follows a gamma distribution. By proposing that the density of states of the confined fluid can be represented by a gamma distribution, the resulting velocity autocorrelation function (VACF) is constructed such that it is accurate upto the fourth frequency moment. The proposed model results in an analytical expression for the VACF and relaxation times. The VACFs obtained from the model have been compared with the VACFs obtained from molecular dynamic simulations and INM analysis for fluids confined in slit-shaped pores over a wide range of confinement and temperatures. The model is seen to capture the short-time behavior of the VACF extremely accurately and in this region is superior to the predictions of the VACF obtained from the real lobe of the INM DOS. Although the model predicts a zero self-diffusivity, the predicted relaxation times are in better agreement with the molecular dynamics results when compared with those obtained from the INM theory.

Spin dynamics in lightly dopedLa2−xSrxCuO4: Relaxation function within thet−Jmodel

The relaxation function theory of doped two-dimensional $S=1/2$ Heisenberg antiferromagnetic (AF) system in the paramagnetic state is presented taking into account the hole subsystem as well as both the electron and AF correlations. The expression for fourth frequency moment of relaxation shape function is derived within the $t-J$ model. The presentation obeys rotational symmetry of the spin correlation functions and is valid for all wave vectors through the Brillouin zone. The spin diffusion contribution to relaxation rates is evaluated and is shown to play a significant role in carrier free and doped antiferromagnet in agreement with exact diagonalization calculations. At low temperatures the main contribution to the nuclear spin-lattice relaxation rate, $^{63}(1/T_1)$, of plane $^{63}$Cu arises from the AF fluctuations, and $^{17}(1/T_1)$, of plane $^{17}$O, has the contributions from the wavevectors in the vicinity of $(\pi,\pi)$ and small $q \sim 0$. It is shown that the theory is able to explain the main features of experimental data on temperature and doping dependence of $^{63}(1/T_1)$ in the paramagnetic state of both carrier free La$_2$CuO$_4$ and doped La$_{2-x}$Sr$_x$CuO$_4$ compounds.

Mass and size dependence of single ion dynamics in molten monohalides

This work is concerned with four molten monohalides with different ionic radii ratios (RbCl, NaI, AgCl, and CuCl) and ideal isotopic systems of these salts with different ionic mass ratios. The velocity autocorrelation functions of the two ionic species in each melt have been studied by both a theoretical approximation and molecular dynamics simulations. It is found that their main features may be qualitatively predicted by considering suitable combinations of the second and fourth frequency moments of their spectra. The analysis of these two parameters allows us to determine how the structure (strongly dependent on the ionic size difference) and the ionic masses contribute to the shape of the velocity autocorrelation functions. The results show that the averaged microscopic motion of the small ions is mainly determined by the first neighboring shell of unlike ions, whereas the nearest shell of like ions also affects the dynamics of the large ions. This effect is more pronounced as the size difference is greater. Furthermore, it is concluded that the size differences encourage the rattling motion of the large ions, whereas the mass difference encourages the backscattering and oscillations of the velocity autocorrelation function of the light ions. A simple rule is derived to determine the interplay between these two effects. Comparison between the mass and nearest distance ratios enables the prediction as to which species will experience a more pronounced backscattering motion. The size difference effects prevail in the hydrodynamics regime and the self-diffusion coefficient of the small ions is higher than that of the large ones. The difference between the self-diffusion coefficient increases as the size differences increases.

Viscoelastic behavior of helium at 39 K and 114 bar.

The dynamic structure factor S(k,\ensuremath{\omega}) of dense helium at 39.3 K (\ensuremath{\simeq}8${\mathit{T}}_{\mathit{c}}$) and 114 bar is measured by inelastic neutron scattering for wave numbers 3\ensuremath{\le}k\ensuremath{\le}24 ${\mathrm{nm}}^{\mathrm{\ensuremath{-}}1}$. In contrast to dense noble-gas fluids at T2${\mathit{T}}_{\mathit{c}}$, the present S(k,\ensuremath{\omega}) can be described completely by the viscoelastic model, in which S(k,\ensuremath{\omega}) is determined for each k by its zeroth, second, and fourth frequency moment and by one decay rate (that of the microscopic momentum flux).

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.

Sum rules and dynamical properties of two-dimensional classical electron liquid

Expressions for the fourth frequency moment sum rule for the longitudinal and transverse current correlation functions are obtained for a classical system of electrons trapped on the surface of liquid helium. The expressions involving static triplet correlation functions are considerably simplified so that the numerical results for the moments can be easily obtained for all momentum transfers. The results for the moments as a function of wavenumber k are presented for 2.23<or= Gamma <or=90.0. The authors make use of these results to analyse the recent molecular dynamics (MD) data on the dynamical structure function S(k, omega ) and the transverse current correlation function Ct(k, omega ) using the Mori formalism. It is found that a relaxation kernel essentially with single decay constant provides an overall good description of MD data over the entire range of Gamma .

Moments of the current-correlation functions and triplet correlations in liquid rubidium

We present here results for the fourth frequency moments of current-correlation functions for liquid rubidium obtained by simulating a system of 108 particles interacting through a potential due to Price et al. These are compared with the results obtained using superposition approximation for the static triplet correlation function involved in the molecular expressions for these moments. It is found that the superposition approximation is reliable as far as the calculation of moments is concerned.

Dynamics of mixed-spin paramagnet

Second and fourth frequency moments of the wave-vector dependent correlation functions of the A−A, B−B, and A−B spins are computed at high temperatures for a mixed spin paramagnet with nearest neighbour Heisenberg exchange couplings J AA v , J BB v , J AB v with unaxial symmetry, v ≡ 0, +. Expression for spin diffusion coefficient is given.

Dynamical correlations in binary liquids. IV. Fourth moment of current and its application

For pt.III see ibid., vol.11, p.1269 (1978). The expressions for the fourth frequency moments of longitudinal and transverse current correlation functions are derived for binary liquids. The numerical calculations are done for a model of molten sodium chloride using the molecular dynamics data of Hansen and McDonald. The results are used to estimate the relaxation times corresponding to number and charge current fluctuations and are in reasonably good agreement (up to q<2.5) with those of Hansen and McDonald (1975). At large wavevectors the agreement is only qualitative.

Computer "experiments" on classical fluids: Triplet correlations and moments of current correlation functions

We present here results for the fourth frequency moments of current-correlation functions obtained by simulating a Lennard-Jones system of 256 particles near its triple point. A comparison with the results of Bansal and Pathak based on superposition approximation (SA) for the triplet correlation function ${g}_{3}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ leads us to a conclusion that the errors involved in the calculation of moments due to use of SA are not of much significance, in contrast to what is generally expected.

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(q ω). It gives correctly the zeroth, second, and fourth frequency moments of S(q ω), 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.

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.

Damping of collective modes in liquid metals. II

For pt.I see ibid., vol.8, p.89 (1975). Calculations are reported of the memory function associated with the longitudinal current correlations along with some results for the density fluctuation spectrum for a model of liquid rubidium. The latter are compared with the information about the dynamics of density fluctuations obtained from molecular dynamics computations and neutron scattering experiments on the liquid near its melting point. The fourth frequency moment has played an essential role in generalized hydrodynamic or viscoelastic theories of the dynamic structure factor. However, the fourth moment is infinite for the rigid core potential and yet the rigid sphere fluid is frequently used as a model for real liquids. It is argued that a theory of the density fluctuation spectrum of a liquid should accurately describe the short-range collision processes, so that the rigid core limit can be recovered, even if this can only be achieved, as with the present approach, at the expense of the fourth moment condition.

Ionic correlations in fused salts

The expression for the fourth frequency moment of current correlation function in a two-component fluid is derived. This result is used to obtain certain exact sum rules, in the long-wavelength limit, for charge current correlations in fused salts.

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.

Addendum to "Sum rules and atomic correlations in classical liquids"

An explicit expression for the fourth frequency moment of the spectral function of the energy-density fluctuation correlation function, and hence for the energy-density correlation f0323;nction, has been derived for a classical system of particles interacting through a two-body potential. It has been found to contain correlations up to four particles only. An expression in the long-wavelength limit for the eighth frequency moment of $S(q, \ensuremath{\omega})$ has also been given.

Damping of collective modes in liquid metals

For pt.I see ibid., vol.8, p.89 (1975). Calculations are reported of the memory function associated with the longitudinal current correlations along with some results for the density fluctuation spectrum for a model of liquid rubidium. The latter are compared with the information about the dynamics of density fluctuations obtained from molecular dynamics computations and neutron scattering experiments on the liquid near its melting point. The fourth frequency moment has played an essential role in generalized hydrodynamic or viscoelastic theories of the dynamic structure factor. However, the fourth moment is infinite for the rigid core potential and yet the rigid sphere fluid is frequently used as a model for real liquids. It is argued that a theory of the density fluctuation spectrum of a liquid should accurately describe the short-range collision processes, so that the rigid core limit can be recovered, even if this can only be achieved, as with the present approach, at the expense of the fourth moment condition.