Diffusion velocity correlation function

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. 

Neumann, J. (1978). Some observations on the simple exponential function as a Lagrangian velocity correlation function in turbulent diffusion. Atmos. Environ. 12, 1965-1968. 

Such a decomposition of the diffusion coefficient has previously been noted by Pattle et al.(l ) Now we must evaluate >. The time-integrated velocity correlation function Aj j is due to the hydrodynamic interaction and can be described by the Oseen tensor. The Oseen tensor is related to the velocity perturbation caused by the hydrodynamic force, F. By checking units, we see that A is the Oseen tensor times the energy term, k T, or... 

The diffusion constant D is determined by the zero-frequency component of this spectral density of the velocity correlation function, by the relation... 

Recently a mode coupling theory study of diffusion and velocity correlation function of a one-dimensional LJ system was carried out [186]. This study reveals that the 1/f3 decay of the velocity correlation function could arise from the coupling of the tagged particle motion to the longitudinal current mode of the surrounding fluid. In this section a brief account of this study is presented. 

The mobility of a macromolecule, constrained by other macromolecules, can be also calculated as (5.1). In the linear approximation, the zeroth normal co-ordinates of the macromolecule (equation (4.1), at z/jj = 0) define diffusive mobility of macromolecule. The one-sided Fourier transform velocity correlation function is determined by expression (4.15), so that we can write down the Fourier transform... 

Therefore, before describing the modification of the equilibrium FDT, we need to study in details the behavior of D(t). Note, however, that the integrated velocity correlation function [, Cvv(/) df takes on the meaning of a time-dependent diffusion coefficient only when the mean-square displacement increases without bounds (when the particle is localized, this quantity characterizes the relaxation of the mean square displacement Ax2 t) toward its finite limit Ax2(oo)). 

Formula (164) shows that, when diffusion takes place in a thermal bath, the velocity correlation function is characterized by the same law as is the average velocity. This result constitutes the regression theorem, valid at equilibrium for any y(co). 

Moore, P., and Keyes, T. Normal mode analysis of liquid CS2 Velocity correlation functions and self-diffusion constants. 7. Chem. Phys 100, 6709 (1994). 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

The dynamics of the solvent in the region near a protein can be characterized by a number of properties (e.g., solvent velocity correlation functions, mean-square displacement correlation functions, dipole orientation correlation functions, etc.). These properties provide information on a range of phenomena from local solute-solvent interactions (velocity correlation functions) to solvent mobility (mean-square displacement correlation functions) and dielectric behavior (dipole correlation functions). Here we focus on the diffusion constant, which provides a convenient measure of mobility for water molecules near protein atoms. The diffusion constant for solvent molecules may be computed directly from the slope of the mean-square displacement correlation function,... 

This is called a Green-Kubo relation25 (see Chapters 10 and 11, andZwanzig, 1965). Thus we expect the results of the diffusion equation to be valid for times long compared to the velocity-correlation time, and the coefficient of self diffusion to be proportional to the area under the velocity correlation function. 

Note that since the diffusion constant is the zero-frequency component of the Fourier transform of the velocity correlation functions [65], Eq. 

Figure 17. Plot of the centroid velocity correlation function for liquid neon. The solid line is the CMD result calculated with the centroid pseudopotential approximation, while the dashed line is the classical MD result. The self-diffusion constant is proportional to the time integral of the centroid velocity correlation functions.

Diffusion eoeffieients can be calculated directly from the velocity correlation functions or from mean square displacements, as ... 

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