Autocorrelation function decay rate

Figure 4.6. Normalized autocorrelation function. Autocorrelation function for collagen single molecules. The autocorrelation function G(nAt) is normalized by dividing all points by the first experimental point G(l). The autocorrelation function decays to a value of the average squared intensity of scattered light divided by G(l). The average squared intensity is proportional to the weight average molecular weight, whereas the rate of decay is related to the translational diffusion coefficient. Reproduced from Silver, 1987.

Examine the autocorrelation function. The high autocorrelations will indicate the order of the autoregressive part if any. The rate of decay of the autocorrelations will indicate a need for differencing. 

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. 

The divergence between the rate of conformational transitions and the decay of the torsional autocorrelation functions (and hence local relaxations)... 

As the salt concentration continues to decrease, however, matters change dramatically Q). The total scattering intensity decreases more abruptly, and the QLS autocorrelation function, which has been a simple single-exponential decay, becomes markedly two-exponential. The two decay rates differ by as much as two orders of magnitude. The faster continues the upward trend of D pp from higher salt, and is thus assigned the term "ordinary . The slower, which is about 1/10 of Dapp high salt, and appears to reflect a new mode of solution dynamics, is termed "extraordinary . 

The first relevant quantity required to obtain the rates is the autocorrelation function which are shown in Fig.8 for the ground vibrational level of the two excited electronic states. The two cases present a very similar behavior. Simply, for the A case its decay seems much faster. What is notorious is the large difference between the EP halfwidths as a function of the energy for the two electronic states, of approximately 2-3 orders of magnitude, as shown in Fig.9. This is explain by the norm of the initial wavepackets, which is much smaller for the B state, because its well is at larger R and shorter r, where the non-adiabatic couplings are much smaller. 

Measurements at low Qs At low Qs, because of the large magnitudes of s , the measured intensity and its autocorrelation function are dominated by the cumulative diffusion of the particles. The measured decay rate thus represents the cumulative or mutual diffusion coefficient Dm given by... 

Equation (5.47) shows that the velocity autocorrelation function , v(t )-v(t), decays exponentially with time. The rate of decay is determined by the friction coefficient / (= 1 /b-m), that is, by particle mass and mobility. 

Figure 4.5. Autocorrelation function of scattered light. Schematic diagram showing the decay of the autocorrelation function to a baseline. The autocorrelation function is related to the product of two intensities separated by a time interval. As the time interval increases, the function decays to a baseline. The rate of decay is proportional to the translational diffusion coefficient. 

The particles must scatter independently otherwise the diffusion coefficient, and particle size, cannot be determined unambiguously from the decay rate of the autocorrelation function. The net effects of multiple scattering are that the instrument factor B/A decreases, and the autocorrelation factor decays faster, leading to too low an estimate for particle size. Thus, multiple scattering limits the application of the technique to low concentration dispersions (< 0.01% by volume), although techniques have been developed to overcome this condition. 

Fig. 24.2. Single-molecule recording of T4 lysozyme conformational motions and enzymatic reaction turnovers of hydrolysis of an E. coli B cell wall in real time, (a) This panel shows a pair of trajectories from a fluorescence donor tetramethyl-rhodamine blue) and acceptor Texas Red (red) pair in a single-T4 lysozyme in the presence of E. coli cells of 2.5mg/mL at pH 7.2 buffer. Anticorrelated fluctuation features are evident. (b) The correlation functions (C (t)) of donor ( A/a (0) Aid (f)), blue), acceptor ((A/a (0) A/a (t)), red), and donor-acceptor cross-correlation function ((A/d (0) A/d (t)), black), deduced from the single-molecule trajectories in (a). They are fitted with the same decay rate constant of 180 40s. A long decay component of 10 2s is also evident in each autocorrelation function. The first data point (not shown) of each correlation function contains the contribution from the measurement noise and fluctuations faster than the time resolution. The correlation functions are normalized, and the (A/a (0) A/a (t)) is presented with a shift on the y axis to enhance the view, (c) A pair of fluorescence trajectories from a donor (blue) and acceptor (red) pair in a T4 lysozyme protein without substrates present. The acceptor was photo-bleached at about 8.5 s. (d) The correlation functions (C(t)) of donor ((A/d (0) A/d (t)), blue), acceptor ((A/a (0) A/a (t)), red) derived from the trajectories in (c). The autocorrelation function only shows a spike at t = 0 and drops to zero at t > 0, which indicates that only uncorrelated measurement noise and fluctuation faster than the time resolution recorded (Adapted with permission from [12]. Copyright 2003 American Chemical Society)...

Figure 5 (a) The structure of T4 lysozyme with the two dye labels schematically shown, (b) Fluorescence intensity trajectories of the TM R donor (blue) and the Texas Red acceptor (red) of a single T4 lysozyme in the presence of E coli B cell wall, (c) Distribution of the decay rate constants (/t) of the donor intensity autocorrelation functions. Reproduced with permission from Y. Chen D. Hu E. R. Vorpagel H. P. Lu, J. Phys. Chem. B. 2003,107, 7947-7956. Copyright (2003) American... 

Coming back to the timescale issue, it is clear that direct observation of signals such as shown in Fig. 13.2 cannot be achieved with numerical simulations. Fortunately an alternative approach is suggested by Eq. (13.26), which provides a way to compute the vibrational relaxation rate directly. This calculation involves the autocorrelation function of the force exerted by the solvent atoms on the frozen oscillator coordinate. Because such correlation functions decay to zero relatively fast (on timescales in the range of pico to nano seconds depending on temperature), its numerical evaluation requires much shorter simulations. Several points should be noted ... 

Although the eigenstate-to-eigenstate and autocorrelation function formulations of the spectrum, Ik(w), are mathematically equivalent, they focus attention on complementary features. The most readily interpretable features in the autocorrelation function picture are early-time features (initial decay rate, the times at which partial recurrences occur, the magnitudes of the earliest and largest partial recurrences) which primarily sample the potential surface at the highly localized and a priori known initial position of the wavepacket, I (O), before it has had time to explore the entire dynamically accessible region of the potential surface. This early time information is encoded in the broad envelope (low resolution) of the Ik(u>) spectrum (see Fig. 9.2). 

Thus by measuring the rate of decay of the autocorrelation function from the speckle fluctuations, and knowing the temperature of the solvent, the particle... 

Figure 19. (a) Decay of the fluorescence intensity autocorrelation function for a single penta-cene molecule in / -terphenyl at long times T = 1.4K). The solid line is a fit of Eq. 15 to the data, (b) Plot of X versus logarithmic intensity for three different pentacene molecules. The solid lines are fits of Eq. 16 to the data. For high intensities X —> 23/2) it can be seen that the population rate varies strongly from molecule to molecule (from Ref. 36). 

For strongly anisotropic particles, the decay rate of the intensity autocorrelation function depends also on the rotational diffusion coefficient (Pecora 1985) and Equation 1.33 is replaced ... 

The method of cumulants performs rather weakly for very broad distributions of the decay rate. In this case, the autocorrelation functions are better fitted by stretched exponentials (Williams and Watts 1970). The Williams-Watts analysis is mainly employed for phase transition in colloidal suspensions (Ruzicka et al. 2004 Katzel et al. 2007) and for polymer suspensions. 

The decay rate E in the autocorrelation function gi(T) for this cooperative mode is proportional to k. It means that the monomer density fluctuation within the blob is diffusional. There is a good reason for this dynamic mode to be called the cooperative diffusion. 

FIG. 8 Rate of decay of the bond autocorrelation function (b(t)) for cyclic (ring) chains of 100 sites, using local moves and topological configurational bias moves, as a function of CPU time. 

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