Exponential autocorrelation time

Assuming the exponential decay of q, the autocorrelation function can be represented by T the time constant or correlation constant of the measured system as q, = exp (— /Tx), Tx = t for q, = 0. See for instance Unmixed, stratified obj ts such as soil and rock, usually cannot be represented by a simple correlation constant derived from the exponential autocorrelation function. Here another measure is introduced, the semivariance. The semivariance can be computed from 35-39)... 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

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. 

The autocorrelation function of autocorrelated time series (first or higher order) has an exponentially decreasing shape. An autocorrelation of the order unity means only a correlation between x(t) and x(t— 1). Because of the same correlation between x(t - 1) to x(t - 2) it seems that a correlation between x(t) and x(t - 2) is also detectable. For this reason, it is very difficult to find the correct order of the autocorrelation process. A useful tool in this case is the partial correlation coefficient. 

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. 

The fluorescence intensity trajectories of the donor (/d(f)) and acceptor (/a(t)) give autocorrelation times (Fig. 24.2b) indistinguishable from fitting an exponential decay to the autocorrelation functions, (A/d (0) A/d (t)) and (A/a (0) A/a (t)), where A/d(t) is /d(t) — (Id), (Id) is the mean intensity of the overall trajectory of a donor, and A/a(t) has the same definition for an intensity trajectory of an acceptor. In contrast, the cross-correlation function between the donor and acceptor trajectories, (A/d (0) A/d (t)), is anticorrelated with the same decay time (Fig. 24.2b) which supports our assignment of anticorrelated fluctuations of the fluorescence intensities of the donor and acceptor to the spFRET process. 

Figure 3. Dipolar J, for a CH carbon as a function of an effective correlation time for molecular tumbling, t. Key A, isotropic or pseudoisotropic tumbling characterized by an exponential autocorrelation function G(t) and B, complex behavior resulting from a nonexponential decay of G(t). This requires that motion be represented by a set of t s, or a distribution around some mean. (Reproduced, with permission, from Ref. 20. Copyright 1981, Adenine Press.)...

Figure 1.20 (A) Site-specific R relaxation rates of sRii from Anabaena sp. PCC7120 (ASR) determined at 12 kHz spinlock power and (B) motional correlation times estimated using single exponential autocorrelation function approximation. Repr/nfed from [250], Copyright 2014 American Chemical Society.

We see that the velocity autocorrelation function of a free Brownian particle falls exponentially with time. The time required for an arbitrary initial velocity distribution of the particle to settle down to the Maxwell-Boltzmann form corresponding to the temperature of the bath (thermaliza-tion of velocities) is on the order of... 

Inspection of Fig. 5.18 shows that the autocorrelation functions for this particular model decay exponentially with time, and that the rate constant for this decay is the sum of the rate constants for forward and backward transitions between the two states (kon + The upper curve in Fig. 5.18B, for example, decays to He (0.368) of its initial value in 16.61 At, which is the reciprocal of (0.05 -t 0.01 )Mt. In classical kinetics, if a system with first-order reactions in the forward and backward directions is perturbed by an abrupt change in the concentration of one of the components, a change in temperature, or some other disturbance, it will relax to equilibrium with a rate constant given by the sum of the rate constants for the forward and backward reactions. The fact that the autocorrelation functions in Fig. 5.18 decay with the relaxation rate constant of the system is a general property of classical time-correlation functions [259-262]. One of the potential strengths of fluorescence correlation spectroscopy is that the relaxation dynamics can be obtained with the system at equilibrium no perturbation is required. 

In this paper we have shown that the fluorescence anisotropy decay technique is a powerful tool to examine the orientation autocorrelation function of labelled chains in solution or in bulk polymers, as well as at of free probes in a polymer matrix. It clearly appears that OACF of polymer chains has a specific nature due to the chain connectivity. This implies a non exponential short time term characteristic... 

It has been pointed out over the years that the simple exponential function of the form where / is travel time from the source, appears to approximate the Lagrangian velocity autocorrelation function R t) rather well (Neumann, 1978 Tennekes, 1979). If R(t) = exp(-l/r), then the mean square particle displacement is given by (Taylor, 1921)... 

Fig. 51. A log—log diagram of the velocity autocorrelation function for a Brownian particle in nitrogen as at two different pressures. Both axes are scaled to make the velocity autocorrelation function normalised and the time dimensionless. The pressures were O, 0.1 MPa ", 1.135PMa. At short times, the experimental data fit an exponential...

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

In the simple cases worked out in the preceding section, the autocorrelation function always turned out to be a simple exponential. Its relaxation time x is therefore unambiguously defined by p(/, x) = < 1. In Table I we show the fluctuation functions F2(f t) obtained in the... 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

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