Time correlation function velocity

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

Equation (6.14) associates the zero frequency component of the velocity time correlation function with the long-time diffusive dynamics. We will later find (see Section 6.5.4) that the high frequency part of the same Fourier transform, Eq. (6.15), is related to the short-time dynamics of the same system as expressed by its spectrum of instantaneous normal modes. 

Using Eqs (6.81) show that the velocity time correlation function of a classical harmonic oscillator in thermal equilibrium satisfies... 

As an application consider the velocity time correlation function for the simple Brownian motion. Using Eqs (7.76), (8.30), and (8.35) we get... 

Figure 11. Classical and quantum hydrogen velocity time correlation function for liquid water at 300 K. The classical result is obtained from classical MD and the quantum result is from CMD. The difference between the classical and quantum results is due primarily to the large zero-point energy in the O—H bonds. The simulation consisted of 125 periodically replicated flexible water molecules represented by site-site point-charge Coulomb and 0-0 Lennard-Jones interactions [47],...

In Paper IV, the self-diffusion process in fluid neon was also studied with CMD using the pairwise pseudopotential method. In Fig. 17 the centroid velocity time correlation function is plotted for quantum neon using the pseudopotential method and for classical neon. When the quantum mechanical nature of the Ne atoms is taken into account, the diffusion constant is reduced by a small fraction. In the gas phase and to some degree in liquids, the diffusion process can be viewed as a sequence of two-body collisions, the frequency of which depends on the collision cross section. Because the quantum centroid cross section is larger than the corresponding classical value, the quantum diffusion constant is found... 

G. J. Martyna, J. Chem. Phys. (in press, 1996). In this paper, an effective set of molecular dynamics equations are specified that provide an alternative path-integral approach to the calculation of position and velocity time correlation functions. This approach is essentially based on the Wigner phase-space function. For general nonlinear systems, the appropriate MD mass in this approach is not the physical mass, but it must instead be a position-dependent effective mass. 

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