Autocorrelation functions

2 Autocorrelation Function The autocoirelator calculates the average of the product of two scattering intensities I t) and Kt + t) measured at the two times separated by t. Here t is called the delay time. The average I(t)I(f + t)) is a function of T and is called the autocorrelation function of I(t) or the intensity-intensity autocorrelation function. The autocoirelator converts l t) into I t)Kf + t)). 

This assumption, in general, is called ei odicity. It is one of the few hypotheses in statistical mechanics. We cannot prove it but believe it is correct. Note that, if the system is at equilibrium, the ensemble average does not change with time and therefore I(t)I(t + t)) = (/(0)/(t)). 

The autocorrelation function rp(r) thus specifies how the densities p u) and p(u ) in neighboring regions separated by r are correlated to each other on the average. When r equals 0, we obviously have 

If /(x) consists of islands of finite densities (representing particles ), islands also occur in T/(x) at positions that correspond to the interparticle distances in/(x), but the islands in T/(x) are more smeared (less sharply peaked) than the corresponding islands in /(x). 

The autocorrelation function is sometimes referred to simply as the correlation function. Among those working in crystal structure analysis, the autocorrelation function is known as the Patterson function. Many of the distribution functions obtained from scattering intensity data are in the nature of the correlation function, with possible differences in the normalization constant or a constant term. Functions in this vein include the pair correlation function or the radial distribution function (and its uniaxial variant cylindrical distribution function), discussed in Chapter 4. 

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Goldfisher, Autocorrelation function and power spectral density of laser-produced speckle pattern . J. Opt. Soc. Am., vol.55, p.247(1965). 

Day P N and Truhlar D G 1991 Benchmark calculations of thermal reaction rates. II. Direct calculation of the flux autocorrelation function for a canonical ensemble J. Chem. Phys. 94 2045-56... 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

Fig. 1. Comparison between the CID-CSP, CSP, TDSCF, and the numerically exact autocorrelation functions.

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

Figure 7.11 from Alder B J and T E Wainwright 1970. Decay of the Velocity Autocorrelation Function. Physical Review A 1 18-21. 

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 ... 

Fig. 7.10 Velocity autocorrelation functions for liquid argon at densities of l.i% gem and0.863gcm...

The slow decay of the velocity autocorrelation function towards zero can be explained in terms of the of a hydrodynamic vortex. (Figure adapted from Alder B J and T E Wainwright 1970. Decay of the Velocity tation Function. Physical Review A 1 18-21.)... 

One alternative approach to the calculation of the diffusion and other transport coefficier is via an appropriate autocorrelation function. For example, the diffusion coefficie... 

A powerful analytical tool is the time correlation function. For any dynamic variable A (it), such as bond lengths or dihedral angles, the time autocorrelation function Cy) is defined... 

An important property of the time autocorrelation function CaU) is that by taking its Fourier transform, F CA(t) a, one gets a spectral decomposition of all the frequencies that contribute to the motion. For example, consider the motion of a single particle in a hannonic potential (harmonic oscillator). The time series describing the position of the... 

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

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]. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

In order to study the vibrational properties of a single Au adatom on Cu faces, one adatom was placed on each face of the slab. Simulations were performed in the range of 300-1000"K to deduce the temperature dependence of the various quantities. The value of the lattice constant was adjusted, at each temperature, so as to result in zero pressure for the bulk system, while the atomic MSB s were determined on a layer by layer basis from equilibrium averages of the atomic density profiles. Furthermore, the phonon DOS of Au adatom was obtained from the Fourier transform of the velocity autocorrelation function. ... 

Step 4 - Calculate the Autocorrelation Function R(t) Assuming that cr ... 

The set of central second moments is related to the autocorrelation function by means of the following simple formula,... 

Since central second moments are called covariances, the function appearing on the left-hand side of Eq. (3-146) is called the autocovariance function of X(t). Autocorrelation functions will be studied in more detail in Sections 3.14 and 3.16. 

The last equation, which expresses the autocorrelation function of Y(t) in terms of h(t), is often referred to as Campbell s58 theorem. It is useful to note that the autocovariance function of Y(t) is given by the simpler expression... 

The function h(t ) i(t + r)dt is often referred to as the autocorrelation function of the Amotion h(t) however, the reader should be careful to note the difference between the autocorrelation function of h(t)—an integrable function—and the autocorrelation function of Y(t)—a function that is not integrable because it does not die out in time. With this distinction in mind, Campbell s theorem can be expressed by saying that the autocovariance function of a shot noise process is n times the autocorrelation function of the function h(t). 

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