Autocorrelation function for

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... 

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

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

As a further application of the Wiener-Khinchine theorem, we shall now calculate the power density spectrum of the shot noise process. The autocorrelation function for such a process is given by Campbell s theorem, Eq. (3-262), repeated below... 

Fig. 1.20. Velocity autocorrelation function for the hard sphere fluid at a reduced density p/po = 0.65 as a function of time measured in t units [75].

Let us now consider the velocity autocorrelation function (VACF) obtained from the MCYL potential, (namely, with the inclusion of vibrations). Figure 3 shows the velocity autocorrelation function for the oxygen and hydrogen atoms calculated for a temperature of about 300 K. The global shape of the VACF for the oxygen is very similar to what was previously determined for the MCY model. Very notable are the fast oscillations for the hydrogens relative to the oxygen. 

Fig. 5.2.8 (a) Standard deviation map, and temporal autocorrelation functions for data recorded at a gas velocity of 27 mm s arid liquid velocities of (b) 0.40 and (c) 0.79 mm s ]. The temporal autocorrelation functions shown by the dashed and solid line styles are... 

The character of an FCS autocorrelation function for a chemical reaction system depends on the relative rates of reaction and diffusion. It is useful to illustrate this dependence by calculating the autocorrelation functions to be expected for a simple one-step reaction system (Elson and Magde, 1974). We take as an example the simplest possible isomerization within the unfolded state, a single-step isomerization ... 

Experimental considerations Frequently a numerical inverse Laplace transformation according to a regularization algorithm (CONTEST) suggested by Provencher [48,49] is employed to obtain G(T). In practice the determination of the distribution function G(T) is non-trivial, especially in the case of bimodal and M-modal distributions, and needs careful consideration [50]. Figure 10 shows an autocorrelation function for an aqueous polyelectrolyte solution of a low concentration (c = 0.005 g/L) at a scattering vector of q — 8.31 x 106 m-1 [44]. 

Fitzgerald et al. (1984) measured pressure fluctuations in an atmospheric fluidized bed combustor and a quarter-scale cold model. The full set of scaling parameters was matched between the beds. The autocorrelation function of the pressure fluctuations was similar for the two beds but not within the 95% confidence levels they had anticipated. The amplitude of the autocorrelation function for the hot combustor was significantly lower than that for the cold model. Also, the experimentally determined time-scaling factor differed from the theoretical value by 24%. They suggested that the differences could be due to electrostatic effects. Particle sphericity and size distribution were not discussed failure to match these could also have influenced the hydrodynamic similarity of the two beds. Bed pressure fluctuations were measured using a single pressure point which, as discussed previously, may not accurately represent the local hydrodynamics within the bed. Similar results were... 

The basic Robertson and Yarwood autocorrelation function, for the underdamped situation, may be written within our present notations [8] ... 

Figure 16 Second Legendre polynomial of the CFI vector autocorrelation function for the sp3 cis-carbon (dashed lines) and the sp2 carbon in a trans-group next to a transgroup (dashed-dotted lines) for two different temperatures. The fit curves to the cis-correlation functions are a superposition of exponential and stretched exponential discussed in the text.

Fig. 11.12. Autocorrelation function for rhodamine 6G 10 9 M in ethanol. The best fit (solid line) yields Dt = 3 x 10-6 cm1 2 s-1 (reproduced with permission from Thompson, 1991).

Figure 37. Modulus of the autocorrelation function for the four-mode model of pyrazine. The full line is the quantum result, and the dotted line is the semiclassical result.

The PRBS is to be preferred to other random inputs with approximately impulse-shaped autocorrelation functions for the following reasons ... 

Figure 5 State autocorrelations computed from 100 ns butane simulations. The central torsion was labeled as either trans, g+, or g—, and the autocorrelation function for presence in each state was computed.

One finds that the coarse-graining with 200 bins is insufficient for correct representation of the autocorrelation functions. On the other hand, using 1000 bins results in autocorrelation functions that are practically indistinguishable to the exact one, independently of the energy of the state in the Q2 space. The application of coarse-graining works equally well for the cross-correlation functions as for the autocorrelation functions. For further details on the application of coarse-graining for efficiently computing the correlation functions, we refer to the discussion in Ref. [41]. 

As for the properties themselves, there are many chemically useful autocorrelation functions. For instance, particle position or velocity autocorrelation functions can be used to determine diffusion coefficients (Ernst, Hauge, and van Leeuwen 1971), stress autocorrelation functions can be used to determine shear viscosities (Haile 1992), and dipole autocorrelation functions are related to vibrational (infrared) spectra as their reverse Fourier transforms (Berens and Wilson 1981). There are also many useful correlation functions between two different variables (Zwanzig 1965). A more detailed discussion, however, is beyond the scope of this text. 

The dynamic RIS model is used to calculate the conformational and first and second orientational autocorrelation functions for PE. Various sequence lengths and directions in the chain are considered. 

Fig. 50. Velocity autocorrelation function for argon at 90 K. O, From the molecular dynamics calculation of Rahman [451] ——, from the Langevin approximation, exp — mDtfkuT D is taken as 2.72 X 10 9 m2 s"1.

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...

Configurational relaxation in the absence of flow is governed by the normal modes and their corresponding relaxation times. The autocorrelation function for the end-to-end vector is a measure of the configurational memory ... 

The reptation model (225) also appears to produce a Rouse spectrum at long times. In order to renew its configurations a chain must diffuse out of the tunnel defined by the fixed obstacles along its length. De Gennes calculates the autocorrelation function for the end-separation vector, obtaining... 

Fig. 4. Spectral density for the velocity autocorrelation function for vibrations of abody-centered cubic lattice, extrapolated from 7 even moments, by the method of Section IV.

It has recently been pointed out by Gordon1 that the root-mean-square fluctuations in the sampled values of the autocorrelation function of a dynamical variable do not necessarily relax to their equilibrium values at the same rate as the autocorrelation function itself relaxes. It is the purpose of this paper to investigate the relative rates of relaxation of autocorrelation functions and their fluctuations in certain systems that can be described by Smoluchowski equations,2 i.e., Fokker-Planck equations in coordinate space. We exhibit the fluctuation and autocorrelation functions for several simple systems, and show that they usually relax at different rates. 

The intermolecular potential consists of the sum of Eqs. (176), (177), (178), and (179). This simulation was done for 216 and 512 molecules but again only the autocorrelation functions for 512 molecules are discussed here. This potential is the strongest angular dependent potential we considered. The results from this potential indicate that it is slightly stronger than that in real liquid carbon monoxide. For example the mean square torque/TV2), for this simulation is 36 x 10-28 (dyne-cm)2 51 and the experimental value is 21 x 10-28 (dyne-cm)2. If this potential is taken seriously, then it should be pointed out that this small discrepancy in torques could be easily removed by using a smaller quadrupole moment. This would be a well justified step since experimental quadrupole moments for carbon monoxide range from 0.5 x 10-26 to 2.43 x 10-26 esu.49... 

Fig. 6.12. A Typical CARS signal trajectory revealing the particle number fluctuations of 110-nm polystyrene spheres undergoing free Brownian diffusion in water. The epi-detected CARS contrast arises from the breathing vibration of the benzene rings at 1003cm 1. B Measured CARS intensity autocorrelation function for an aqueous suspension of 200-nm polystyrene spheres at a Raman shift of 3050 cm-1 where aromatic C-H stretch vibrations reside. The corresponding translational diffusion time, td, of 20 ms is indicated. (Panel B courtesy of Andreas Zumbusch, adapted from [162])...

The magnitudes of < (j>(t)> versus time are shown in Fig. 4. The autocorrelation function for the positive displacement along Qx in the coupled potential (lowest curve in Fig. 4), starts out at 1 and drops to 0 over a shorter period of time than in the uncoupled potential (middle line). The Fourier transforms of these < (t) > give the spectra shown in Fig. 3. This reasoning explains why a positive displacement results in a broader progression and a negative displacement results in a narrower progression. 

Fig. 6-21. Autocorrelation and partial autocorrelation functions for ARIMA(1,0,0) and ARIMA(0,0,1)...

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