Correlation functions velocity autocorrelation function

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... 

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. 

This expression can be analytically inverted to yield the velocity autocorrelation function. The power spectrum, G (co), corresponding to this correlation function is... 

Figure 6. (a) 2-Gilorobutane at 50 K, 6x6 site-site potential, angular velocity autocorrelation functions. Crosshatdiing indicates computer noise difference between R and S enantiomers. (—) Racemic mixture, (b) As for (a), under the influence of a strong field E, producing a torque — 63 XE in each molecule of the molecular dynamics sample. (1) (—) Racemic mixture (2) (—) R enantiomer. Ordinate Normalized correlation function abscissa time, ps. 

Figure 1, The translational (thin) and rotational (heavy) velocity autocorrelation functions and power spectra at 273.15 K(solid line), 223.15 K(dashed line) and 173.15 K(dash-dot line). Left (time correlations), right (power spectra).

MD calculations readily relate the velocity, t), of a given atom, at time r = 0, to that of the same atom at some latter time, t. The development of this correlation as time passes provides access to the single-particle velocity autocorrelation function, VACF t). 

The second method is to compute the diffusion coefficient as the integral of the molecular CoM velocity autocorrelation function by using time correlation functions formalisms ... 

An attempt has been made to correlate some of the above results in hydrogen with the self-diffusion coefficient D following a suggestion by Zwanzig. The partial success of this attempt (a near-linear variation of Ti with D at a given temperature) can be understood by writing out the self-diffusion coefficient in terms of the velocity autocorrelation function... 

Diffusion. - Distribution of the diffusivitity of fluid in a horizontally oriented cylinder was demonstrated by NMR imaging in two papers on a granular flow system and in the earth s magnetic field. Correlation time (ic) and diffusion coefficient (D = Xc) imaging (CTDCI) was applied to a granular flow system of 2 mm oil-filled sphere rotated in a half-filled horizontal cylinder, ie. to an Omstein-Uhlenbeck process with a velocity autocorrelation function. Time dependent apparent diffusion coefficients are measured, and Tc... 

A from any lipid head group atom were considered to be bound, and any water more than 4 A away from all lipid head groups was considered to be bulk. Because the bound/bulk status of waters can change during the course of a simulation, the nonbonded atom list was updated every picosecond. Of the 553 waters used in the simulation, on average there were only 160 bulk waters. The velocity autocorrelation functions (VAF), the mean square displacements (MSD), and the orientational correlational functions (OCF) for the bound and bulk waters were calculated. VAFs were calculated as ... 

In many respects, at a superficial level, the theory for the chemical reaction problem is much simpler than for the velocity autocorrelation function. The simplifications arise because we are now dealing with a scalar transport phenomenon, and it is the diffusive modes of the solute molecules that are coupled. In the case of the velocity autocorrelation function, the coupling of the test particle motion to the collective fluid fields (e.g., the viscous mode) must be taken into account. At a deeper level, of course, the same effects must enter into the description of the reaction problem, and one is faced with the problem of the microscopic treatment of the correlated motion of a pair of molecules that may react. In the following sections, we attempt to clarify and expand on these parallels. 

The velocity autocorrelation function is an example of a single-particle correlation function, in which the average is calculated not only over time origins but also over all the atoms. Some properties are calculated for the entire system. One such property is the net dipole moment of the system, which is the vector sum of all the individual dipoles of the molecules in the system (clearly the dipole moment of the system can be non-zero only if each individual molecule has a dipole). The magnitude and orientation of the net dipole will change with time and is given by ... 

Other orientational correlation coefficients can be calculated in the same way as the correlation coefficients that we have discussed already. Thus, the reorientational correlation coefficient of a single rigid molecule indicates the degree to which the orientation of a molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelation function is the rotational equivalent of the velocity correlation function ... 

Finally, we mention the fact that the molecular dynamics calculation of Po t) for a gas of hard disks exhibited a vortex type of velocity correlation between the tagged molecule and the surrounding molecules that is very similar to the hydrodynamic flow field surrounding a moving volume element in a fluid initially at rest. This vortex pattern, illustrated in Fig. 26, suggests that a fraction of the momentum transferred by the tagged particle to the particles in front of it is eventually returned to it from behind. This process causes the velocity autocorrelation function to be larger than it would be if these vortices did not occur, and it is connected with the slow decay of the velocity autocorrelation function. 

To calculate the velocity autocorrelation function we therefore only need < sfor fc = 0. This simplifies matters considerably compared to a full calculation of the van Hove self-correlation function S ik, a>). 

Mobility in this region is dominated by short-time motion, typically < 2 ps. After that time, all correlation of molecular motion is lost due to frequent collisions with the cavity walls. The center-of-mass velocity autocorrelation function of the penetrant exhibits typical liquid-like behavior with a negative region due to velocity reversal when the penetrant hits the cavity wall [59]. This picture has recently been confirmed by Pant and Boyd [62] who monitored reversals in the penetrant s travelling direction when it hits the cavity walls. The details of the velocity autocorrelation function are not very sensitive to the force-field parameters used. On the other hand, the orientational correlation function of diatomic penetrants showed residuals of a gas-like behavior. Reorientation of the molecular axis does not have the signature of rotational diffusion, but rather shows some amount of free rotation with rotational correlation times of the order of a few tenths of a picosecond, although dependent in value on the Lennard-Jones radii of the penetrant s atoms. 

When the property calculated is a single particle property, such as the velocity autocorrelation function, or the particle s mean squared displacement, averaging over the particles does help a lot. Averaging over even a moderate number of particles of about 1000 decreases the error by more than an order of magnitude. Unfortunately, this is not possible for all properties. For instance, the calculation of viscosity calls for the stress correlation function [Eq. (67)], which is not a single particle property. This is why self-diffusion coefficients are usually estimated with much better accuracy than viscosity. 

The velocity autocorrelation function tells you how fast a particle forgets its initial velocity, owing to Brownian randomization. When the time t is short relative to the correlation time of the physical process, the particle velocity will be nearly unchanged from time 0 to t, and v(t) will nearly equal v(0) so (u(O)v(t)) greater than the system s correlation time. Brownian motion will have had time to randomize the particle velocity relative to its initial velocity, so v it) will be uncorrelated with u (0). This means that v(0)v(t) will be negative just as often as it is positive, so the ensemble average will be zero, (u(O)u(t)) = 0. 

EXAMPLE 18.6 Autocorrelation times. The correlation time m/ is the time required for the velocity autocorrelation function to reach 1/e of its initial value. The correlation time is short when the mass is small or when the friction coefficient is large. The correlation time for the diffusional motion of a small protein of mass m = 10,000 g moP is in the picosecond time range (see Equation (18.56)) ... 

In the diffusion limit it is foimd that the combined effects of particle inertia and shear flow modify the amphtude and the time-dependence of the particle-velocity autocorrelation functions, a result which is expressed in terms of the Stokes number, St = 7/fi. The shear flow breaks macroscopic time reversibility and stationarity the autocorrelation functions of the particle velocities are stationary and the velocity correlation along the shear is symmetric in the time difference t, but the cross correlation is non-symmetric in t function in the streamwise direction is non-stationary The time decay of the velocity correlation along the flow is not a pure exponential and the imderlying stochastic process is not an Omstein-Uhlenbeck process. 

Fig. 1 Random versus correlated jump diffusion velocity autocorrelation functions and corresponding real parts of the complex conductivity [42]...

The simplest approach to describe the ion dynamics in disordered materials is to assume completely uncorrelated, random ion movements [42]. In this case, the jump of an ion moving in a forward direction is only correlated to itself, thus the velocity autocorrelation function is proportional to a Dirac Delta function at = 0 (see Fig. la). The complex conductivity obtained by Fourier transform is then independent of frequency. This means that the real part of the conductivity shows no dispersion and at all frequencies the ac conductivity < (6 ) can be identified with the dc conductivity. By contrast, conductivity spectra of most ion-conducting materials show that o (o) varies with frequency. This is schematically illustrated... 

Fig. 7 Decay of translational (U) and rotational (12) velocity correlations of a suspended sphere. The time-dependent velocities of the sphere are shown as solid symbols the relaxation of the corresponding velocity autocorrelation functions are shown as open symbols (with statistical error bars). A sufficiently large fluid volume was used so that the periodic boundary conditions had no effect on the numerical results for times up to r = 1,000 in lattice units (h = b = 1). The solid lines are theoretical results, obtained by an inverse Laplace transform of the frequency-dependent friction coefficients [175] of a sphere of appropriate size (a = 2.6) and mass (pj/p = 12) the kinematic viscosity of the pure fluid = 1/6...

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(0 of the stochastic differaitial equation in Equation 1.1, which are summarized saying that v(0 is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specific results that follow from this simple mathanatical model regarding propo ties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 

The velocity autocorrelation functions and from them the self diffusion coefficients have been calculated from all versions for the normal masses as well as for the mass of the anion reduced by 26%. They are shown together in Figure 4. The velocity autocorrelation functions in version I do not behave qualitatively correct. If the anion mass decreases, the first zero of the anion correlation function should shift to smaller time, while the zero of the unchanged cation correlation function should slightly shift to larger time. The qualitatively correct behaviour of the velocity autocorrelation functions results from version II and III. 

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