The velocity autocorrelation function

If we take the scalar product of an atom s velocity with its velocity a short time later, t, and take an average over all the atoms of the same t)q)e then we may represent it as a convolution v(0) v(i), also commonly written (v(O).v(t)). The Fourier transform of a convolution is equivalent to product of the Fourier transforms of the two functions taken separately. Self convolution, autocorrelation, yields the square of the functions imder Fourier transformation and only the real part of the transformation is available, the power spectrum, p (a). 

An atom s average position, squared, r tf in a solid, measured after a long time, t, tends to a constant, this is the total vibrational mean squared displacement, A. The contribution to A arising firom a specific vibrational mode, H at cOy, is the strength of the power spectrum at (Oy. 

---

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

Figure 7.11 from Alder B J and T E Wainwright 1970. Decay of the Velocity Autocorrelation Function. Physical Review A 1 18-21. 

The slow decay of the velocity autocorrelation function towards zero can be explained in terms of the of a hydrodynamic vortex. (Figure adapted from Alder B J and T E Wainwright 1970. Decay of the Velocity tation Function. Physical Review A 1 18-21.)... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

In order to study the vibrational properties of a single Au adatom on Cu faces, one adatom was placed on each face of the slab. Simulations were performed in the range of 300-1000"K to deduce the temperature dependence of the various quantities. The value of the lattice constant was adjusted, at each temperature, so as to result in zero pressure for the bulk system, while the atomic MSB s were determined on a layer by layer basis from equilibrium averages of the atomic density profiles. Furthermore, the phonon DOS of Au adatom was obtained from the Fourier transform of the velocity autocorrelation function. ... 

Let us now consider the velocity autocorrelation function (VACF) obtained from the MCYL potential, (namely, with the inclusion of vibrations). Figure 3 shows the velocity autocorrelation function for the oxygen and hydrogen atoms calculated for a temperature of about 300 K. The global shape of the VACF for the oxygen is very similar to what was previously determined for the MCY model. Very notable are the fast oscillations for the hydrogens relative to the oxygen. 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

The calculations of g(r) and C(t) are performed for a variety of temperatures ranging from the very low temperatures where the atoms oscillate around the ground state minimum to temperatures where the average energy is above the dissociation limit and the cluster fragments. In the course of these calculations the students explore both the distinctions between solid-like and liquid-like behavior. Typical radial distribution functions and velocity autocorrelation functions are plotted in Figure 6 for a van der Waals cluster at two different temperatures. Evaluation of the structure in the radial distribution functions allows for discussion of the transition from solid-like to liquid-like behavior. The velocity autocorrelation function leads to insight into diffusion processes and into atomic motion in different systems as a function of temperature. 

The velocity autocorrelation function (VAF) may be used to investigate the possibility of coupling between translational and rotational motions of the sorbed molecules. The VAF is obtained by taking the dot product of the initial velocity with that at time t. It thus contains information about periodic fluctuations in the sorbate s velocity. The Fourier transform of the VAF yields a frequency spectrum for sorbate motion. By decomposing the total velocity of a sorbate molecule into translational and rotational terms, the coupling of rotational and translational motion can be investigated. This procedure illustrates one of the main strengths of theoretical simulations, namely to predict what is difficult or impossible to determine experimentally. 

D only assumes a constant value over times long compared with rc( 10rc), such that (u(O)u(f)), the velocity autocorrelation function, is nearly zero. The diffusion equation is not valid over times < 10rc(i.e. a few picoseconds at least). A better approach would be to use a generalised Langevin equation with a friction coefficient which has a memory and such that the velocity autocorrelation takes 0.5ps to decay to insignificant levels (see Chap. 11) [453]. 

Some qualitative comments were made about the velocity autocorrelation function in Chap. 8, Sect. 2.1. In this section, it is considered in more quantitative detail. One of the simplest expressions for the diffusion coefficient is that due to Einstein [514]. He found that a particle executing a random walk has an average mean square displacement of (r2 > after a time t, such that... 

It shows the clear link between the change of motion of the particle and its diffusion coefficient. In Fig. 50, the velocity autocorrelation function of liquid argon at 90 K (calculated by computer simulation) is shown [451], The velocity becomes effectively randomised within a time less than lps. Further comments on the velocity autocorrelation functions obtained by computer simulation are reserved until the next sub-section. Because the velocity autocorrelation function of molecular liquids is small for times of a picosecond or more, the diffusion coefficient defined in the limit above is effectively established and constant. Consequently, the diffusion equation becomes a reasonable description of molecular motion over times comparable with or greater than the time over which the velocity autocorrelation function had decayed effectively to zero. Under... 

The velocity autocorrelation function has been measured by Fedele and Kim [515]. A charged Brownian particle of radius 0.11 0.01 jam... 

Fig. 51. A log—log diagram of the velocity autocorrelation function for a Brownian particle in nitrogen as at two different pressures. Both axes are scaled to make the velocity autocorrelation function normalised and the time dimensionless. The pressures were O, 0.1 MPa ", 1.135PMa. At short times, the experimental data fit an exponential...

The effect of density on the velocity autocorrelation function was studied by Verlet and Levesque [519]. In Fig. 52, two velocity autocorrelation functions are shown which correspond to two densities of the Lennard—Jones spheres used in the numerical study (see also Gubbins 1520]. Rdsibois and De Leener [490] have made the following observations on these results. 

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

On multiplying eqn. (284) by u0, integrating over time, and taking the ensemble average, the velocity autocorrelation function is obtained [271]. 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

The rate coefficient kernel of eqn. (369) and the initial condition term of eqn. (370) are those given by Northrup and Hynes [ 103]. These expressions are cast into the form of correlation functions (cf. the velocity autocorrelation function) and have a close similarity to the matrix elements in quantum mechanic applications. While they are quite easy to derive and... 

As an example, we consider these error bounds for the cumulative distribution of the spectral density of the velocity autocorrelation function,... 

Fig. 2. Error bounds for the cumulative frequency distribution of the spectral density for the velocity autocorrelation function using jU.0, /n2, and ja evaluated for a classical model of liquid argon.29...

Fig. 4. Spectral density for the velocity autocorrelation function for vibrations of abody-centered cubic lattice, extrapolated from 7 even moments, by the method of Section IV.

U. Buontempo, S. Cunsolo, and P. Dore. Intercollisional memory effects and short-time behavior of the velocity-autocorrelation function from translational spectra of liquid mixtures. Phys. Rev., A 10 913, 1974. 

Equation (5.47) shows that the velocity autocorrelation function , v(t )-v(t), decays exponentially with time. The rate of decay is determined by the friction coefficient / (= 1 /b-m), that is, by particle mass and mobility. 

We mention this result here in order to assert that the spectral distribution of B(jf is the Fourier transform of the (force) autocorrelation function 0(t). In view of Eqn. (5.45), we can restate this result in terms of the velocity t>(/). The spectral distribution of the velocity autocorrelation function is directly related to the Fourier transform of 0 j), the force autocorrelation function. Thus, we see that the classical equation of motion when properly averaged over many particles provides insight into the relation between transport kinetics and particle dynamics [R. Becker (1966)]. 

Since j-c-v, the electrical current density autocorrelation function and the velocity autocorrelation function are proportional to each other. The latter function, however, can be expressed with the help of the time derivative of the decaying pro-... 

Let us summarize by modeling the velocity autocorrelation function using Debye-Huckel type interactions between charged point defects in ionic crystals, one can evaluate the frequency-dependent conductivity and give an interpretation of the universal dielectric response. 

---