Correlation functions time independent

Remark. For fixed t the correlation function in Eq. (21) exhibits a (t )a decay. A (f )a 1 decay of an intensity correlation function was reported in experiments of Orrit s group [4] for uncapped NCs (for that case ot = 0.65 0.2). However, the measured correlation function is a time-averaged correlation function [Eq. (12)] obtained from a single trajectory. Here the correlation function is independent of t thus no comparison between theory and experiment can be made yet. 

The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

This last point suggests an alternative interpretation of the transport coefficient as the one corresponding to the correlation function evaluated at the point of maximum flux. The second entropy is maximized to find the optimum flux at each x. Since the maximum value of the second entropy is the first entropy sM(x), which is independent of x, one has no further variational principle to invoke based on the second entropy. However, one may assert that the optimal time interval is the one that maximizes the rate of production of the otherwise unconstrained first entropy, 5(x (x, x), x) = x (x,x) Xs(x), since the latter is a function of the optimized fluxes that depend on x. 

For the initial time t = 0, the above formula (A2.7) is identical with the result of averaging over random orientations in a surface plane. In the course of time, the "memory" of the initial orientation fades, the condition t w 1 (w 1 is the average period between reorientations) permitting an independent averaging over ea(t) and e (0), and the correlation function (A2.7) tends to zero. 

The autocorrelation function G(t) corresponds to the correlation of a time-shifted replica of itself at various time-shifts (t) (Equation (7)).58,65 This autocorrelation defines the probability of the detection of a photon from the same molecule at time zero and at time x. Loss of this correlation indicates that this one molecule is not available for excitation, either because it diffused out of the detection volume or it is in a dark state different from its ground state. Two photons originating from uncorrelated background emission, such as Raman scattering, or emission from two different molecules do not have a time correlation and for this reason appear as a time-independent constant offset for G(r).58... 

This time-independent expression obviously has to be identified with the equilibrium distribution. We thus obtain the following functional relation between the equilibrium correlations (k 0) and the velocity distribution ... 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

Aging. If we assume independent exponential relaxations for the CRRs, we obtain the following expression for the two-times correlation function ... 

The following problem is in a certain sense the inverse of the one treated in the two preceding sections. Consider a photoconductor in which the electrons are excited into the conduction band by a beam of incoming photons. The arrival times of the incident photons constitute a set of random events, described by distribution functions/ or correlation functions gm. If they are independent (Poisson process or shot noise) they merely give rise to a constant probability per unit time for an electron to be excited, and (VI.9.1) applies. For any other stochastic distribution of the arrival events, however, successive excitations are no longer independent and therefore the number of excited electrons is not a Markov process and does not obey an M-equation. The problem is then to find how the statistics of the number of charge carriers is affected by the statistics of the incident photon beam. Their statistical properties are supposed to be known and furthermore it is supposed that they have the cluster property, i.e., their correlation functions gm obey (II.5.8). The problem was solved by Ubbink ) in the form of a... 

However, Waite s approach has several shortcomings (first discussed by Kotomin and Kuzovkov [14, 15]). First of all, it contradicts a universal principle of statistical description itself the particle distribution functions (in particular, many-particle densities) have to be defined independently of the kinetic process, but it is only the physical process which determines the actual form of kinetic equations which are aimed to describe the system s time development. This means that when considering the diffusion-controlled particle recombination (there is no source), the actual mechanism of how particles were created - whether or not correlated in geminate pairs - is not important these are concentrations and joint densities which uniquely determine the decay kinetics. Moreover, even the knowledge of the coordinates of all the particles involved in the reaction (which permits us to find an infinite hierarchy of correlation functions = 2,...,oo, and thus is... 

Deviation from standard chemical kinetics described in (Section 2.1.1) can happen only if the reaction rate K (t) reveals its own non-monotonous time dependence. Since K(t) is a functional of the correlation functions, it means that these functions have to possess their own motion, practically independent on the time development of concentrations. The correlation functions characterize the intermediate order in the particle distribution in a spatially-homogeneous system. Change of such an intermediate order could be interpreted as a series of structural transitions. 

In dynamic or quasi-elastic light scattering, a time dependent correlation function (i (0) i (t)) = G2 (t) is measured, where i (0) is the scattering intensity at the beginning of the experiment, and i (t) that at a certain time later. Under the conditions of dilute solution (independent fluctuation of different small volume elements), the intensity correlation function can be expressed in terms of the electric field correlation function gi (t)... 

These correlation functions are stationary that is, they are independent of the time t. Furthermore, since the vector U is normalized,... 

As mentioned previously the above relations follow from the fact that at long times the value of the random variable in each correlation function becomes statistically independent of its initial value—provided of course interactions are present. 

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