Functions velocity autocorrelation

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. 

Fig. 7.10 Velocity autocorrelation functions for liquid argon at densities of l.i% gem and0.863gcm...

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. ... 

Fig. 1.20. Velocity autocorrelation function for the hard sphere fluid at a reduced density p/po = 0.65 as a function of time measured in t units [75].

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

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. 

Figure 3. Hydrogen and oxygen velocity autocorrelation function from two-body MCY with vibrations allowed (MCYL), and computed infrared spectrum for intramolecular bending modes and bond stretching.

For the analysis of the dynamical properties of the water and ions, the simulation cell is divided into eight subshells of thickness 3.0A and of height equal to the height of one turn of DNA. The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. From the study of the dipole orientational correlation function... 

An alternative approach is to use the fact that an MD calculation samples the vibrational modes of the polymer for a period of time, f, from 0 to fmax and to calculate from the trajectory, the mass weighted velocity autocorrelation function. Transforming this function from the time domain into the frequency domain by a Fourier transform provides the vibrational density of states g(v). In practice this may be carried out in the following way ... 

Thus, effects of the surfaces can be studied in detail, separately from effects of counterions or solutes. In addition, individual layers of interfacial water can be analyzed as a function of distance from the surface and directional anisotropy in various properties can be studied. Finally, one computer experiment can often yield information on several water properties, some of which would be time-consuming or even impossible to obtain by experimentation. Examples of interfacial water properties which can be computed via the MD simulations but not via experiment include the number of hydrogen bonds per molecule, velocity autocorrelation functions, and radial distribution functions. 

Stillinger and Rahman have also considered the diffusion coefficient, velocity autocorrelation function and scattering function for simulated water. For discussion of these interesting calculations the reader is referred to their papers 3>. 

It has been pointed out over the years that the simple exponential function of the form where / is travel time from the source, appears to approximate the Lagrangian velocity autocorrelation function R t) rather well (Neumann, 1978 Tennekes, 1979). If R(t) = exp(-l/r), then the mean square particle displacement is given by (Taylor, 1921)... 

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. 

As for the properties themselves, there are many chemically useful autocorrelation functions. For instance, particle position or velocity autocorrelation functions can be used to determine diffusion coefficients (Ernst, Hauge, and van Leeuwen 1971), stress autocorrelation functions can be used to determine shear viscosities (Haile 1992), and dipole autocorrelation functions are related to vibrational (infrared) spectra as their reverse Fourier transforms (Berens and Wilson 1981). There are also many useful correlation functions between two different variables (Zwanzig 1965). A more detailed discussion, however, is beyond the scope of this text. 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

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. 

Fig. 39. Velocity autocorrelation function, (u(0)u(t))/(u(0)u(0)), for carbon tetrachloride liquid at 301K from a molecular dynamics study by Steinhauser and Neumann [452a],...

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... 

Fig. 50. Velocity autocorrelation function for argon at 90 K. O, From the molecular dynamics calculation of Rahman [451] ——, from the Langevin approximation, exp — mDtfkuT D is taken as 2.72 X 10 9 m2 s"1.

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...

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