Time correlation functions simple examples

A second problem with the GME derived from the contraction over a Liouville equation, either classical or quantum, has to do with the correct evaluation of the memory kernel. Within the density perspective this memory kernel can be expressed in terms of correlation functions. If the linear response assumption is made, the two-time correlation function affords an exhaustive representation of the statistical process under study. In Section III.B we shall see with a simple quantum mechanical example, based on the Anderson localization, that the second-order approximation might lead to results conflicting with quantum mechanical coherence. 

Here we describe two simple examples, one based on classical and the other on quantum mechanics, that demonstrate the power of time correlation functions in addressing important observables. 

The procedure described here is an example for combining theory (that relates rates and currents to time correlation functions) with numerical simulations to provide a practical tool for rate evaluation. Note that this calculation assumes that the process under study is indeed a simple rate process characterized by a single rate. For example, this level of the theory cannot account for the nonexponential relaxation of the v = 10 vibrational level of O2 in Argon matrix as observed in Fig. 13.2. 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

In multidimensional NMR studies of organic compounds, 2H, 13C and 31P are suitable probe nuclei.3,4,6 For these nuclei, the time evolution of the spin system is simple due to 7 1 and the strengths of the quadrupolar or chemical shift interactions exceed the dipole-dipole couplings so that single-particle correlation functions can be measured. On the other hand, the situation is less favorable for applications on solid-ion conductors. Here, the nuclei associated with the mobile ions often exhibit I> 1 and, hence, a complicated evolution of the spin system requires elaborate pulse sequences.197 199 Further, strong dipolar interactions often hamper straightforward analysis of the data. Nevertheless, it was shown that 6Li, 7Li and 9Be are useful to characterize ion dynamics in crystalline ion conductors by means of 2D NMR in frequency and time domain.200 204 For example, small translational diffusion coefficients D 1 O-20 m2/s became accessible in 7Li NMR stimulated-echo studies.201... 

This polarizability involves a set of characteristic times jlkT, Id(, illoil, and (IcIIodI), between any two of which the correlation function may take a fairly simple form. However, this example indicates that for a linear system which falls to show normal mode behaviour the frequency-dependent admittance may be much more straightforward than the corresponding correlation function. 

Figure 13 tests another prediction of the Rouse model, the time-temperature superposition property. Again, a representative example is shown, t.e., the correlation function of the third Rouse mode. As the theory anticipates, it is indeed possible to superimpose the simulation data, obtained at different temperatures, onto a common master curve by rescaling the time axis. The required scaling time, T3, is defined by the condition pp(r3) = 0.4. The choice of this condition is arbitrary. Since the Rouse model predicts that the correlation function satisfies equation (10) for all times, any other value of pp(t) could have been used to define T3. This scaling behavior is in accordance with the theory. However, contrary to the theory, the correlation functions do not decay as a simple exponential, but as... 

The time distribution of the fluorescence photons emitted by a single dye molecule reflects its intra- and intermolecular dynamics. One example are the quantum jumps just discussed which lead to stochastic fluctuations of the fluorescence emission caused by singlet-triplet quantum transitions. This effect, however, can only be observed directly in a simple fluorescence counting experiment when a system with suitable photophysical transition rates is available. By recording the fluorescence intensity autocorrelation function, i.e. by measuring the correlation between fluorescence photons at different instants of time, a more versatile and powerful technique is available which allows the determination of dynamical processes of a single molecule from nanoseconds up to hundreds of seconds. It is important to mention that any reliable measurement with this technique requires the dynamics of the system to be stationary for the recording time of the correlation function. 

Figure 12. Example of interaction of a molecule with a single tunneling system. There are two resonances in the laser scan range, showing exponential fluorescence correlation functions with the same decay times, and contrasts in inverse proportion to the peak intensies, in agreement with a simple kinetic model of a single molecule coupled to just one tunneling system. Occasional jumps of such pairs of resonances, preserving the spUtting and the correlation function confirm this interpretation.

The situation corresponding to the simple model discussed above occius rarely in a real disordered system. To begin with, molecules are always coupled to far-away TLS s, which will broaden their lines, or cause drifts on a broad range of timescales. Moreover, more than one TLS can be close enough to the molecule to cause, for example, two splittings and a quadruplet of lines, with particular intensity ratios [77]. However, such cases occur fairly seldom. If a TLS is too close to the molecule, the components are too far apart to be recorded in the same laser scan, or, with higher probability, if a TLS is too remote, the line is not split completely. This does not prevent the characteristic time of the TLS from appearing - with a low contrast - in the correlation function, if the frequency jump is not much smaller than the linewidth. 

More time-dependent structural correlation functions can be constructed, depending on the chemical nature of the system and of the phenomenon under investigation. For example, correlation functions over the positions and velocities of centers of mass yield information on translational diffusion. Molecular diffusion properties are often described by the self-diffusion coefiticient, D, using a simple formula that involves the mean square displacement of the centers of mass [9,10] ... 

---