Correlation function friction

In this report problem of information processing in MIA pulse flaw detectors by means of cross correlation function is considered. Such processing promises to increase the sensitivity and to reduce the noises, mainly the frictional ones. 

VER occurs as a result of fluctuating forces exerted by the bath on the system at the system s oscillation frequency O [5]. Fluctuating dynamical forces are characterized by a force-force correlation function. The Fourier transfonn of this force correlation function at Q, denoted n(n), characterizes the quantum mechanical frequency-dependent friction exerted on the system by the bath [5, 8]. 

Equation (C3.5.2 ) is a function of batli coordinates only. The VER rate constant is proportional to tire Fourier transfonn, at tire oscillator frequency Q, of tire batli force-correlation function. This Fourier transfonn is proportional as well to tire frequency-dependent friction q(n) mentioned previously. For example, tire rate constant for VER of tire Emdamental (v = 1) to tire ground (v = 0) state of an oscillator witli frequency D is [54]... 

From Eq. (70) we see that the time-dependent friction coefficient is given in terms of the force correlation function with projected dynamics. Instead, in MD simulations the time-dependent friction coefficient is computed using ordinary dynamics. 

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

The dimensionless group hD/k is called the Nusselt number, /VNu, and the group Cp i/k is the Prandtl number, NPl. The group DVp/p is the familiar Reynolds number, NEe, encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as... 

The time dependent friction coefficient, per solute mass p, is related to the fluctuating forces exerted by the solvent on the solute coordinate x through their time correlation function ... 

We should point out here the great analogy between and the friction coefficient studied in the Brownian motion problem of Section IV (see Eq. (242)) instead of having the time autocorrelation function of the force F , we now have the time correlation function between F and Fe. 

In this generalized oscillator equation, the frequency is related to the restoring force acting on a particle and Q is a friction constant. The key quantity of the theory is the memory kernel mq(l — t ), which involves higher order correlation functions and hence needs to be approximated. The memory kernel is expanded as a power series in terms of S(q, t)... 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

Fix the proton at some position s and run a MD simulation. The friction kernel is calculated from the force-force correlation function. 

The osmotic modulus, K, the frictional coefficient, f, and the diffusion coefficient, D, are related to density-density correlation function of the network, g(r), by [62]... 

Physically this description corresponds to putting an atom (mass M) in an external time-dependent harmonic potential (frequency co0). The potential relaxes exponentially in time (time constant l/x0) so that eventually the atom experiences only a frictional force. Compared with other models2 which have been proposed for neutron scattering calculation, the present model treats oscillatory and diffusive motions of an atom in terms of a single equation. Both types of motion are governed by the shape of the potential and the manner in which it decays. The model yields the same velocity auto-correlation function v /(r) as that obtained by Berne, Boon, and Rice2 using the memory function approach. 

Larachi et al. [37] presented a simplified version of Ellman s correlation. A friction factor, fiGG, is represented as a function of dimensionless groups which takes inertia, viscosity and surface-tension effects into account by using, respectively, %g, Rei, and Wee. 

Let the factor multiplying tc1/2/2 be called p. Thus we see that if we assume a functional form for the memory function, then it is possible to determine the parameters in the functional form by using the moment theorems of Eq. (162) and to determine, thereby, the transport coefficients, such as the friction coefficient. Moreover, the time correlation function, i /(t), can also be determined. 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

The friction is given in terms of the force-force time correlation functions and in the frequency plane can be written as... 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

It is worthwhile to discuss the relative contributions of the binary and the three-particle correlations to the initial decay. If the triplet correlation is neglected, then the values of the Gaussian time constants are equal to 89 fs and 93 fs for the friction and the viscosity, respectively. Thus, the triplet correlation slows down the decay of viscosity more than that of the friction. The greater effect of the triplet correlation is in accord with the more collective nature of the viscosity. This point also highlights the difference between the viscosity and the friction. As already discussed, the Kirkwood superposition approximation has been used for the triplet correlation function to keep the problem tractable. This introduces an error which, however, may not be very significant for an argon-like system at triple point. 

Another factor that contributes to the decoupling is the two particle direct correlation function. The product cn q)F q, t) defines the modified structure of the solvent probed by the solute. The value of the direct correlation function is less for smaller solutes at all wavevectors. The smaller the value of the two-particle direct correlation function, the lesser will be the contribution of the density mode to the total friction. 

Following Oxtoby s method, the bond friction here is calculated in terms of the friction (or the force-force time correlation function) on the individual solute atoms connected by the bond. The calculation of the friction on a solute atom in a medium of solvent spheres is performed following the method described in Section IX. 

Note that the angular momentum-momentum correlation function related to the vibrational-rotation friction too is highly nonexponential. This again could significantly alter the 2 dependence of the rate. However, for the systems studied here, the contribution of vibration-rotation coupling is negligible. 

The frequency (z)-dependent velocity correlation function Cv(z) is related to the frequency dependent friction by the well-known generalized Einstein relation,... 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

The Second ingredient is the expression of the rotational friction in terms of the orientational time correlation functions. We have earlier derived an expression for this which was based on Kirkwood s formula [190]. The full expression should be derived by following an approach similar to that of Sjogren and Sjolander [9]. In addition, the coupling to rotational currents (the vortices) have not been touched upon. 

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