Frequency spectrum of the velocity autocorrelation function in liquid rubidium

Calculations are reported of the memory function and frequency spectrum associated with the velocity correlations in a model of liquid rubidium. The spectrum shows some rather marked features associated with the shape of the potential beyond the core.     

Influence of collective modes in liquid rubidium on the velocity autocorrelation function

A brief derivation of a memory function is presented to show how the collective modes in liquid rubidium are coupled to the velocity auto-correlation function. The effect on the frequency spectrum is demonstrated and briefly discussed.     

High-frequency asymptotic behavior of the spectrum of the velocity autocorrelation function in a simple liquid

The effect of short range forces of interparticle repulsion on the formation of high-frequency asymptotic behavior of the velocity autocorrelation function in a simple liquid is studied. For this purpose, a method is developed of a group expansion of the evolution operator of a many-particle system. It is shown that only irreducible pair collisions make a contribution to the high-frequency asymptotic behavior. As an illustration, calculations are performed for interatomic potentials modeled by a power function. Application of the developed methods to the case of classical high-temperature plasma leads to the well-known Ginzburg formula for the high frequency conductivity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 59–65, July, 1986.The author considers it is to be his duty to express his sincere gratitude to N. P. Malomuzh for a thorough discussion of this work.     

The exact velocity autocorrelation function of a model system

The velocity autocorrelation function of a particle in a model system with realistic diffusion is calculated exactly and compared with the corresponding result in the one-dimensional case. The method employed yields the result of Lebowitz and Sykes in one dimension in a very simple manner.Research Supported by NSF Grant No. R019881001.     

The velocity autocorrelation function of a finite model system

We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particlesN and the length of the boxL approach infinity andN/L , the velocity autocorrelation function(t) is given simply by c² exp(–2ct@#@). For a finite system, the function _N(t) is periodic with period 2L/c. We also show that for more general velocity distribution functions (particles can have velocities ±c_i,i = 1,...), _N(t) is an almost periodic function oft. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keept fixed while letting the size of the system become infinite to obtain an auto-correlation function, such as(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, our _N (t) will be very close to(t) as long ast is small compared to the effective size of the system, which is 2L/c for the first model.Supported in part by the AFOSR under Contract No. F44620-71-C-0013.     

Complex singularities of the fluid velocity autocorrelation function

There are intensive debates regarding the nature of supercritical fluids: if their evolution from liquid-like to gas-like behavior is a continuous multistage process or there is a sharp well-defined crossover. Velocity auto-correlation function Z is the established detector of evolution of fluid particles dynamics. Usually, complex singularities of correlation functions give more information. For this reason, we investigate Z in complex plane of frequencies using numerical analytic continuation. We have found that naive picture with few isolated poles fails describing Z(ω) of one-component Lennard-Jones (LJ) fluid. Instead, we see the singularity manifold forming branch cuts extending approximately parallel to the real frequency axis. That suggests LJ velocity autocorrelation function is a multivalued function of complex frequency. The branch cuts are separated from the real axis by the well-defined “gap” whose width corresponds to an important time scale of a fluid characterizing crossover of system dynamics from kinetic to hydrodynamic regime. Our working hypothesis is that the branch cut origin is related to competition between one-particle dynamics and hydrodynamics. The observed analytic structure of Z is very stable under changes in the temperature; it survives at temperatures two orders of magnitude higher than the critical one.     

Does the velocity autocorrelation function oscillate in a hard-sphere crystal?

An approximate kinetic theory is used to compute the velocity autocorrelation function (VACF) in a hard-sphere crystal. In general the theory predicts that the VACF is oscillatory in time. However, in practice, near coexistence the oscillations will be difficult if not impossible to observe because by the time the oscillations occur the VACF has decayed almost to zero. At higher densities the theory predicts that the oscillations are probably just barely observable. In all cases the time integral of the VACF is zero.     

Short time behavior of the velocity autocorrelation function

A systematic method for evaluating velocity correlation functions of a hard sphere fluid for short times is presented The complete contributions ～ t and t² are obtained, formally valid for all densities.     

The non-exponential behavior of the velocity autocorrelation function

The contribution to the velocity autocorrelation function from ring events is studied for low densities and t?2t₀ using a kinetic model. Agreement with computer experiments for $\text{V}\text{V}{}_0\text{= 18}$ is found to be good.     

The long time behavior of the velocity autocorrelation function in a plasma

The utilization of the Landau-Placzek method results in a velocity autocorrelation function for a one component plasma having oscillatory behavior (at the plasma frequency) in addition to the t^?(d/2) (d = dimension) behavior previously found in neutral gas systems.     

Evidence for slowly-decaying tails in the velocity autocorrelation function of a two-dimensional Lorentz gas

Molecular dynamics calculations on a two-dimensional hard-disk Lorentz gas indicate that the velocity autocorrelation function shows a slowly-decaying tail. The decay is consistent with a $\text{1}\text{τ}{}^2$ dependence, but the coefficient is significantly greater than predicted by Ernst and Weyland.     

On the derivation of the velocity autocorrelation function in liquids from the incoherent scattering function

The derivation of the velocity autocorrelation function from a knowledge of the self-correlation function is demonstrated for a liquid rubidium model. It is argued that the method will be important in the processing of neutron scattering data from liquids.     

Long-time behavior of the angular velocity autocorrelation function for a fluid of rough spheres

According to hydrodynamical and mode-coupling theories, the angular velocity autocorrelation function decays at long times as ₀(t/10^–14 sec)^–5/2. For rough spheres under the conditions reported here, the quantity ₀ is predicted to be 262. The molecular dynamics studies presented here yield a long-time tail of the form 230(t/10^–14 sec)^–2.38. The disagreement between theory and computer results probably arises from statistical error intrinsic to the computations.The authors are indebted to the National Science Foundation and the Computer Center of the University of Minnesota for financial support of the research reported here.     

A memory function model for the velocity autocorrelation function and the self-diffusion coefficient in simple dense fluids

A memory-function model is used to compute the velocity autocorrelation function and the self-diffusion coefficient of a dense Lennard-Jones fluid from the zero-time correlation functions of the molecular velocity and its first two time derivatives. It is shown that these zero-time correlation functions can be evaluated in terms of the radial distribution function and the pair potential only, i.e. without considering higher order correlation functions. Since molecular dynamics results are available for the radial distribution function as well as the velocity autocorrelation function and the self-diffusion coefficient, a rigorous test of the chosen memory function is possible. The agreement is reasonable, although generally not within the error bands of the molecular dynamics results.     

Connection of the velocity autocorrelation function to the mean-square-displacement and to the memory function of generalized master equations

General connections between the velocity autocorrelation function and the mean-square-displacement for a special initial condition are established and shown to reduce in the high-temperature limit to earlier such results obtained by Scher and Lax. A simple and useful relation, which is valid in the high temperature limit, between the velocity autocorrelation function and memory functions in generalized master equations is given.     

Velocity autocorrelation in liquid sodium

The expression for the memory function of Corngold et al. has been modified using an effective potential for liquid sodium to calculate the velocity autocorrelation function and the coefficient of self diffusion. The results are found in good agreement with computer experiment results of Schiff.     

The velocity autocorrelation function and self-diffusion coefficient of fluids with steeply repulsive potentials

We consider the velocity autocorrelation function, vacf, or C_v(t) and self-diffusion coefficients, D, of steeply repulsive inverse power fluids (SRP) in which the particles interact with a pair potential, ? (r) = ?(σ/r)ⁿ. The C_v(t) are calculated numerically by molecular dynamics simulations. Accurate expressions for the short time expansion of C_v(t) to order O(t⁴) for n large are derived for this fluid. We propose novel expressions for C_v (t) that, for n large, spans the transition from the short time regime (expandable in even powers of time) and the longer time exponential-like regime characteristic of hard spheres. Inter alia we introduce relaxation times that characterize the duration of a collision and the decay of the velocity correlation within the mean-collision or Enskog-like relaxation time, T_E.