Correlation function velocity

Kuwabara function for particle deposition in porous media particle radial distribution function pair correlation function for particles of types a and / constant appearing in the pair correlation function velocity parameters used in conjunction with EQMOM mean velocity difference used to approximate vi - Vp I... 

Very important differences emerge if we attempt to use the stochastic integrators to compute dynamics, e.g. a time-correlation function. Velocity auto-correlation functions are shown in Fig. 8.4 for various choices of the parameters. 

Zhu S-B, Lee J, Robinson G Wand Lin S H 1989 Theoretical study of memory kernel and velocity correlation function for condensed phase isomerization. I. Memory kernel J. Chem. Phys. 90 6335-9... 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

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

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

Fig. 1.14. Comparison of the MD calculations of the correlation functions of the translational velocity and angular momentum in liquid nitrogen [65]. The time is in units of 10-13 s.

Fig. 1.15. Translational and angular velocity correlation functions for nitrogen. MD simulation data from [70], T = 122 K, densities are indicated in the figure. Reduced units for time t = (e/cr2), for density p" = p

Anderson J. E., Ullman R. Angular velocity correlation functions and high-frequency dielectric relaxation, J. Chem. Phys. 55, 4406-14, (1971). 

Kluk E., Monkos K., Pasterny K., Zerda T. A means to obtain angular velocity correlation functions from angular position correlation functions of molecules in liquid. Part I. General discussion and its application to linear and spherical top molecules, Acta Physica Polonica A 56, 109-16 (1979). 

Taylor series 260 torque, correlation functions 28 transfer time, rotational relaxation 51 transitions dipole moment 30 forbidden 30 non-adiabatic 130 translational velocity v 6... 

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... 

Thus, U -, U2-, and K can be evaluated from the velocity group correlation functions C p ... 

That the time correlation function is the same using the terminal velocity or the coarse velocity in the intermediate regime is consistent with Eqs (53) and (54). 

To simplify the discussion, a scalar even variable x will be used. In this case the most likely terminal position is x (x, x) = —Q(x)Sx, where the correlation function is <2(x) = 1 (x(t + x)x(f))0. The most likely terminal velocity is... 

The last issue we address concerns the existence of long-time tails in the discrete-time velocity correlation function. The diffusion coefficient can be written in terms of the velocity correlation function as... 

Since the diffusion coefficient is the infinite-time integral of the velocity correlation function, we have the Einstein relation, D = kBT/Q. 

Discrete-time velocity correlation function, multiparticle collision dynamics, macroscopic laws and transport coefficients, 103-104 Dissipative structures ... 

Using the statistical mechanical theory, Longuet-Higgins and Pople obtained a correlation function for the velocities in a liquid consisting of... 

The pair correlation function of the velocities and the pair correlation functions of some time derivatives of the velocity are sometimes taken into account.75 However, the validity of this description in the nonadiabaticity regions also has to be proved. The dynamic description or the description using the differentiable random process is more rigorous in this region.76... 

The single matched filter can be used only when the target s position and velocity are known. To detect a target at an unknown position, it is necessary to utilize a bank of filters matched to all possible target ranges and velocities. This approach leads directly to the range-Doppler correlation function, described by the formula... 

In homogeneous isotropic turbulence, the two-point velocity correlation function can be expressed (Pope 2000) in terms of the longitudinal (/) and transverse (g) auto-correlation functions ... 

The spatial correlation functions are computed from the two-point joint velocity PDF based on two points in space. Obviously, the same idea can be extended to cover two points in time. Indeed, the Eulerian9 two-time joint velocity PDF /Lf (J (V, V x, t, t ),... 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

This relation shows that for homogeneous turbulence, working in terms of the two-point spatial correlation function or in terms of the velocity spectrum tensor is entirely equivalent. In the turbulence literature, models formulated in terms of the velocity spectrum tensor are referred to as spectral models (for further details, see McComb (1990) or Lesieur (1997)). 

The Fourier transform introduces the wavenumber vector , which has units of 1 /length. Note that, from its definition, the velocity spatial correlation function is related to the Reynolds stresses by... 

---