Time correlation functions diffusion coefficient

Analysis of the obtained configurations. Calculate g(r), time correlation functions, diffusion coefficients, etc. 

In Ref. [107] it has been demonstrated how, based on the scaling law for the diffusion coefficient, molar mass distributions can be calculated from time correlation functions obtained from scattering experiments. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

This initial attempt to compute the time-correlation function was followed by a study of the Gaussian memory function with no significantly new results.67 The Gaussian memory, adjusted to give the correct diffusion coefficient, is found in exactly the same way as the exponential memory. It turns out to be... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

The observation that the rate constant may be expressed in terms of an auto-time-correlation function of the flux, averaged over an equilibrium ensemble, has a parallel in statistical mechanics. There it is shown, within the frame of linear response theory, that any transport coefficients, like diffusion constants, viscosities, conductivities, etc., may also be expressed in terms of auto-time-correlation functions of proper chosen quantities, averaged over an equilibrium ensemble. 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

As a correlation function is recorded, the correlator offers the decay time (x), particles diffusion coefficient (D), and particles mean radius (R). The two latters are related with by Stokes-Einstein equation [6],... 

In a liquid crystal most properties are best expressed relative to a director based coordinate system. This is not a problem in a macroscopic system where the director is virtually fixed. However, it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. Thus a director based frame is not an inertial frame. Correction terms should therefore be added to time dependent properties. Time correlation functions with slowly decaying tails might also be affected by the director reorientation. Transport coefficient obtained from them will consequently be incorrect. When NEMD-simulation algorithms are applied, the fictitious external field exerts a torque that constantly twists the director, which could make it impossible to reach a steady state. 

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 ... 

The limit in front of the ratio means that the time t has to be much longer than the longest relaxation time of the chain. The resulting diffusion coefficients obtained by Monte Carlo simulation of the Evans-Edwards model of entangled polymers are presented in Fig. 9.33(a). The diffusion coefficient decreases with the number of monomers in the chain. Another quantity that can be extracted from the Monte Carlo simulations of the Evans-Edwards model is the relaxation time of the chain. It can be defined as the characteristic decay time of the time correlation function of the end-to-end vector R[t)R 0)) exp( t/Trep). Figure 9.33(b) presents the results of such simulations. 

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 time correlation function that involves the same observable at two different times is called an autocorrelation function. We have found that the self-diffusion coefficient is the time integral of the velocity auto-correlation function... 

In all the 2BSM calculations presented here, the diffusion coefficient Dj equals 1, which defines the unit of frequency (inverse time) whereas the diffusion coefficient for the solvent, Dj varied from 10 (very fast solvent relaxation) to 1, 0.1, 0.01 (very slow solvent relaxation). In the Dj = 10 case, one finds that the reorientation of the solute is virtually independent of the solvent a projection procedure could easily be adopted in this case to yield a one-body Smoluchowski equation for body 1 with perturbational corrections from body 2. The temporal decay of the first and second rank correlation functions is then typically monoexponential. When the solvent is relaxing slowly (i.e., Dj is in the range 1-0.01), the effect of the large cage of the rapid motion of the probe becomes... 

Time correlation function Time correlation functions are of great value for the analysis of dynamical processes in condensed phases. A time correlation function C(t) is obtained when a time-dependent quantity A(t) is multiplied by itself (auto-correlation) or by another time-dependent quantity B(t ) evaluated at time / (cross-correlation) and the product is averaged over some equilibrium ensemble. For example, the self-diffusion coefficient can be obtained... 

Most macromolecules when dissolved in salt solutions acquire charges that are shielded by an atmosphere of counterions. This ion atmosphere affects the diffusion coefficient of the macromolecule and hence the light-scattering time-correlation function. Electrolyte solutions are discussed in Chapters 9 and 13. Recent measurements of diffusion coefficients have been made by several groups. Lee and Schurr (1974) have studied poly-L-lysine-HBr. Schleich and Yeh (1973) have performed similar studies on poly-L-proline. Raj and Flygare (1974) have studied bovine serum albumin (BSA) and find that at high ionic strength and low pH the diffusion constant decreases. This they attribute to the expansion of the molecule. 

Various transport coefficients can also be related to time-correlation functions. For instance, as was shown in Section 5.9, the translational self-diffusion coefficient is proportional to the area under the time-correlation function of the velocity of the center of mass of the particle. 

The translational diffusion coefficient D is obtained from the first cumulant T of the electric field time correlation function (t), which is directly measured by photon correlation technique... 

On the other hand, we have, for non-equilibrium dynamic property, the time correlation function TCF, which is dynamic counterpart to g(r). One can define various TCP s for each purpose. However, at the present stage, no extensive theoretical relation has been derived between TCF and ([(r). Therefore, direct determination of self-diffusion coefficient, viscosity coefficient by the molecular simulation gives significant contribution in dynamics studies. 

Here is the appropriate microscopic current and tlim denotes the thermodynamic limit. (For reviews of this theory, see Zwanzig and Steele. ) We will here focus our attention on the self-diffusion process, both because it is the simplest case from a pedagogical standpoint and because it is the one that has been most extensively studied by means of molecular dynamics calculations. Calculations of the coefficients of viscosity and thermal conductivity and the associated time-correlation functions have been reported by Alder et for hard spheres. 

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