Correlation function equilibrium diffusion

We consider desorption from an adsorbate where surface diffusion is so fast (on the time scale of desorption) that the adsorbate is maintained in equilibrium throughout the desorption process. That is to say that, at the remaining coverage 9 t) at temperature T t), all correlation functions attain... 

The most important result of this work is that despite two different SRO patterns, we have found concentration independent EPI. The evolution of the diffuse intensity with composition is thus mainly due to the sensitivity of the equilibrium state (i.e. the correlation function) to the concentration. 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

Figure 2. The correlation function G(f), which measures the rate at which a step position anomalously diffuses away from a starting position. This diffusion is limited to the equilibrium width G(f —> oo ) = wi. The time, t is scaled by x t which is the equilibration time (Xe, defined in Eq. (17)) for the case = (perfect sticking). The curves are for (from the top) ...

Mode coupling theory of binary mixtures where the constituents are of rather different sizes is a challenging task, as we have already discussed while addressing the mass depenence of diffusion. In addition to the problem with proper formulation of mode coupling terms, there is an additional difficulty of the nonavailability of the equilibrium two-particle correlation functions The existing integral equation theories become unstable when the size ratio exceeds a certain (low) value, like 1.5 or so [195],... 

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. 

Equation (4.29) defines a correlation function due to the diffusive mechanism of relaxation. One can see that time dependence of the equilibrium... 

Results from DLS measurements [Figure 5 (left)], show the collective diffusion coefficient Dc = T/q2 for the PNIPA gels swollen in aqueous solutions at different equilibrium concentrations ce of phenols, where T is the characteristic relaxation rate of the DLS correlation function. With this technique systems can be measured only below the phase transition. Samples equilibrated with phloroglucinol cannot therefore be studied above ca 30 mM at room temperature. All the systems display a similar response, with a noticeable, albeit limited, downturn as the limiting concentration is approached. 

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). 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

It is evident that this is a way of rewriting the exact solution of Eq. (47). However, it is interesting to recover the fluctuation-dissipation prediction from a perspective that might lead to a free diffusion with no upper limit if an error is made that does not take into account the statistical properties of the fluctuation E,(f). Let us evaluate the correlation function of E,(f). Using the property of Eq. (48) and moving to the asymptotic time limit reflecting the microscopic equilibrium condition, we obtain... 

Here, Qa and Qb account for the different optical properties of the fluo-rophores that distinguish A and B as well as the laser intensity and other instrumental factors. K = kab/kba, and td = w jAD is the characteristic diffusion time for a Gaussian excitation intensity profile with exp(—2) radius w, and S is the area of the laser spot. It is readily shown that for equilibrium systems Gab (t) = Gba (t) due to the fact that kabCB = kbaC [25]. The NESS fluxes can be obtained from the initial slope of the correlation functions. 

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

By using the fitting function (16) and Eq. (15), we numerically reproduce Gg(f feq), and the reproduced curve well approximates the numerical result as shown in Fig. 9b. Note that cjg(f feq) is proportional to f2 in the limit of f 0, since Cp(s feq) in Eq. (15) goes to the constant Cp(0 feq). On the other hand, in the limit of f —> oo, cjg(f feq) is proportional to f, because both Cp(s feq) and sCp(s teq) are almost zeros in the long-time region, and hence their integrals become constants. The crossover from f2 to f is also observed if we assume an exponential correlation function, and hence we conclude that diffusion at equilibrium is normal as expected although a stretched exponential is present. 

To obtain the diffusion constant, D, we consider two alternative equilibrium time correlation function approaches. First, D can be obtained from the long time limit of the slope of the time-dependent mean square displacement of the electron from its starting position. The quantum expression for this estimator is... 

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