Time dependence mean-squared displacement

Lattice vibrations are described as follows [9, 10]. If the deviation of an atom from its equilibrium position is u, then <(u2> is a measure for the average deviation of the atom (the symbol < ) represents the time average note that <(u> = 0). This so-called mean-squared displacement depends on the solid and the temperature, and is characteristic for the rigidity of a lattice. Lattice vibrations are a collective phenomenon they can be visualized as the modes of vibration... 

For the system in thennal equilibrium, one can compute the time-dependent mean square displacement (ICr)... 

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

Note that in this approximation the incoherent scattering measures the time-dependent thermally averaged, mean square displacement <(rd(t) — (O))2). 

Fig. 3.4 Time-dependent mean-square displacement of a PEP segment in the melt at 492 K. The solid line indicates the prediction of the Rouse model. (Reprinted with permission from [43]. Copyright 2003 The American Physical Society)...

We immediately realize that this function has exactly the Q-dependence predicted by the Gaussian approximation (Eq. 4.13). From the comparison between these two expressions (Eq. 4.14 and Eq. 4.13), the time-dependent mean square displacement of the hydrogens can easily be extracted ... 

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

The resulting trajectories are usually analyzed for the mode of motion executed by the particle as the motion provides information on the location and status of the particle as described in this review. The most common analysis starts with the calculation of the so-called mean-square displacement (MSD). Then, the time dependence of the MSD is plotted. This plot allows a mode of motion analysis [35]. A simplified way to calculate the MSD is depicted in Fig. 2b. The mean square displacement describes the average of the squared distances between a particle s start and end positions for all time-lags of certain length At within one trajectory. 

In the previous section we analyzed the random walk of molecules in Euclidean space and found that their mean square displacement is proportional to time, (2.5). Interestingly, this important finding is not true when diffusion is studied in fractals and disordered media. The difference arises from the fact that the nearest-neighbor sites visited by the walker are equivalent in spaces with integer dimensions but are not equivalent in fractals and disordered media. In these media the mean correlations between different steps (UjUk) are not equal to zero, in contrast to what happens in Euclidean space cf. derivation of (2.6). In reality, the anisotropic structure of fractals and disordered media makes the value of each of the correlations u-jui structurally and temporally dependent. In other words, the value of each pair u-ju-i-- depends on where the walker is at the successive times j and k, and the Brownian path on a fractal may be a fractal of a fractal [9]. Since the correlations u.juk do not average out, the final important result is (UjUk) / 0, which is the underlying cause of anomalous diffusion. In reality, the mean square displacement does not increase linearly with time in anomalous diffusion and (2.5) is no longer exact. 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

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)). 

Experimentally, the effective temperature of a colloidal glass can be determined by studying the anomalous drift and diffusion properties of an immersed probe particle. More precisely, one measures, at the same age of the medium, on the one hand, the particle mean-square displacement as a function of time, and, on the other hand, its frequency-dependent mobility. This program has recently been achieved for a micrometric bead immersed in a glassy colloidal suspension of Laponite. As a result, both Ax2(t) and p(co) are found to display power-law behaviors in the experimental range of measurements [12]. 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

The only characteristic of the carrier motion embodied in this equation is the mean square displacement per unit time, which gives D with a simple numerical factor dependent on the number of degrees of freedom summarized in x. All other detail of the motion is lost, and solutions of the diffusion equation represent the true evolution of p(x) only for times long compared with the jump time. 

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