Correlation fluctuations

When interactions Uy in a binary alloy are purely pairwise, i,e, the rhs of Eq, (4) includes only the first term, one can derive an exact relation between F ci and its derivatives over c . Firstly we note that according to Eqs, (2)-(6) the fluctuation correlator / y =< (n — Ci)(iij — cj) > is related to the free energy as... 

For an individual molecule, fluctuations of the instantaneous electronic charge density away from its quantum mechanical average are characterized by the fluctuation-dissipation theorem (3, 4). The molecule is assumed to be in equilibrium with a radiation bath at temperature T then in the final step of the derivation, the limit is taken as T — 0. The fluctuation correlations, which are defined by... 

While the T — 0 limit has been taken formally, it should be noted that the electronic fluctuation correlations at room temperature are essentially equivalent to those at T = 0 for normal, nondegenerate systems. (Thermal photons at room temperature are generally unable to excite electronic transitions )... 

As illustrated in Fig. 3, the proton relaxation in super-paramagnetic colloids occurs because of the fluctuations of the dipolar magnetic coupling between the nanocrystal magnetization and the proton spin. The relaxation rate increases with the fluctuation correlation time and with the magnitude of this fluctuation. Different processes cause the fluctuation of the magnetic interaction. 

A technique, also known as fluctuation correlation spectroscopy, that uses a correlation algorithm to detect similarity of a fluctuating signal occurring over a period of time. This property aids in the kinetic characterization of fluctuations in light scattering, or other electromagnetic... 

Principal component analysis (PCA) is another tool that has been used extensively to analyze molecular simulations. The technique, which attempts to extract the large-scale characteristic motions from a structural ensemble, was first applied to biomolecular simulations by Garcia [28], although an analogous technique was used by Levy et al. [30]. The first step is the construction of the 3N x 3N (for an N-atom system) fluctuation correlation matrix... 

A concentration-fluctuation correlation coefficient, similar to that proposed by Danckwerts (D3, D5), may be defined. 

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

To model the experimental data we used a global-fit procedure to simulate EPS, integrated TG, heterodyne-detected TG, and the linear absorption spectrum simultaneously. The pulse shape and phase were explicitly taken into account, which is of paramount importance for the adequate description of the experimental data. We applied a stochastic modulation model with a bi-exponential frequency fluctuation correlation function of the following form ... 

It is possible to express the density fluctuation correlation function by the correlation operator g,2 = Fu - F1F2, and F, may be obtained easily ... 

Fig. 6a-e. Fermi sea correlation effects in photoionization (a) One-electron level scheme, (b)—(d) Feynman diagrams describing monopole (K = 0), dipole (K = 1) and quadrupole (K = 2) fluctuation (correlation) processes and (e) schematic pictures of the resulting photoelectron spectrum... 

There is no additional double-layer screening of fluctuations correlated across the separation /. 

S.9.a. Induced-dipole-induced-dipole fluctuation correlation... 

There is the maximum coefficient of charge fluctuations because L and R are ionic solutions, but there is no additional double-layer screening of fluctuations correlated across the separation Z. 

The prevalence of structural compared with dynamical information arises from a scarcity of appropriate spectroscopic techniques. For example, the measurement process in NMR spectroscopy takes place on a millisecond timescale. The dominating part of the fluctuation correlation function responsible for dephasing processes, on the other hand, decays on the picosecond timescale, so that spin transitions are strongly in the motional... 

A global fit was performed by applying the formalism outlined in Section III. A [Equation (7)—(12), (14)—(17)], which connects the transition frequency fluctuation correlation function (< three-pulse photon echo signal S(T, r) and by varying the five parameters in the right-hand side of Equation (20). The fit and the resulting first... 

Nevertheless, the responses of the test molecules embedded to the proteins differ significantly from that in solution (see Fig. 11). The fluctuation correlation function amplitude is considerably larger in solution for small times, but it decays much more quickly, so that only a very small quasi-static inhomogeneity (0.1 ps 2) remains within the observation window available in this experiment (6 ps). A much larger inhomogeneity remains for even longer times in the case of a protein environment. This result is consistent with the interpretation of optical photon echoes in protein environment (22-25,27), where a quasi-static contribution of the energy gap correlation function has been observed up to 100 ps. 

Because of the small size of the utilized test molecules and the fact that the molecules remain in their electronic ground states, accurate calculations of the transition frequency fluctuation correlation function of these systems fluctuations would seem to be achievable with state-of-the-art quantum dynamics calculations. 

Figure 11 The initial decay of the transition frequency fluctuation correlation function ( oio(t)< io(0)) obtained from global fits of the stimulated photon echo data of azide dissolved in D2O (solid line, same data as in Fig. 6), CA-N3- (dashed line) I Ib-N3 (dashed-dotted line), and Hb-CO (dashed-double-dotted line).

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