Autocorrelation function scaling dynamics

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

NMR 13C spin-lattice relaxation times are sensitive to the reorientational dynamics of 13C-1H vectors. The motion of the attached proton(s) causes fluctuations in the magnetic field at the 13C nuclei, which results in decay of their magnetization. Although the time scale for the experimentally measured decay of the magnetization of a 13C nucleus in a polymer melt is typically on the order of seconds, the corresponding decay of the 13C-1H vector autocorrelation function is on the order of nanoseconds, and, hence, is amenable to simulation. 

The autocorrelation function quantifies, on a unit scale, the degree to which a quantity is correlated with values of the same quantity at later times. The function can be meaningfully calculated for any dynamical simulation, in the sense defined earlier, and therefore including MC. We must consider a set of time-ordered values of the observable of interest, so that / — f(t — /At), with j — 1,2,..., N and At the time step between frames. (For MC simulations, one can... 

Polymer Backbone Motion. Alternate descriptions of molecular motion utilize an effectively non-exponential autocorrelation function to describe polymer dynamics. One formalism is the use of a log-/2 distribution of correlation times in place of a single correlation time(14). Such a description may simulate the various time scales for overall and internal motions in polymers. 

Another piece of information that we wanted to extract from our experiments was connected with the dynamic behavior of spatial variables. If we consider three successive particles in the chain and we denote by the distance of the middle one from the center of mass of the other two and by the distance between these two, we can compute the normalized autocorrelation function of these two variables. They are shown in Fig. 9 as can be immediately observed, they decay to zero on a time scale which is much greater than that of the velocity variable. Also, the center of mass decays faster than R . In the next section we shall argue that this suggests that the virtual potential characterizing the itinerant oscillator model has to be assumed to be fluctuating around a mean shape, which, moreover, will be shown to be nonlinear and softer than its harmonic approximation. 

The arrival times of fluorescence photons contain information about correlations in fluorescence signals. Eluorescence correlation spectroscopy (FCS) (26) exploits these correlations to measure the magnitude and time scales of fluctuations in fluorescence. These fluctuations contain information about the dynamic time scales of the system and the concentration of fluorescing molecules. Correlations may span time ranges from nanoseconds to milliseconds, which extends the dynamic time window for fluorescence measurements far beyond what is achievable in fluorescence lifetime measurements. The autocorrelation function is calculated as ... 

Our experimental measurements of the orientation autocorrelation function on sub-nanosecond time scales are consistent with the theoretical models for backbone motions proposed by Hall and Helfand(ll) and by Bendler and Yaris(12). The correlation functions observed in three different solvents at various temperatures have the same shape within experimental error. This implies that the fundamental character of the local segmental dynamics is the same in the different environments investigated. Analysis of the temperature dependence of the correlation function yields an activation energy of 7 kJ/mole for local segmental motions. 

From a conceptual standpoint, it is useful to have an understanding of the time scales for motions of particles near metal-water interfaces, to be able to better understand their nature, as well as how molecules and atoms near these interfaces differ from those of the bulk. The two most commonly calculated dynamic properties for metal surfaces are the mean-square displacement and the velocity autocorrelation functions, because these can be used to calculate diffusion constants and spectra. 

Most commonly it is assumed that Texp and are of the same order of magnitude, at least for reasonable observables A. But this is not true in general. In fact, one usually expects the autocorrelation function paa i) to obey a dynamic scaling law ° of the form... 

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