Time-correlation function fluctuation

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

For liquids, few simple and widely accepted theories have been developed. The shear viscosity can be related to the way in which spontaneous fluctuations relax in an equilibrium system, leading to the time correlation function expression " " ... 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

The time dependent friction coefficient, per solute mass p, is related to the fluctuating forces exerted by the solvent on the solute coordinate x through their time correlation function ... 

FIG. 21. The influence of potential on step fluctuations, x(t), may be described by means of a time correlation function F(t) = ((x(t) - x(0) ). At negative potentials, fluctuations are due solely to mass transport along the steps, while at more positive potentials the magnitude of the fluctuations increases rapidly. This is attributed to the onset of adatom exchange with terraces as well as the electrolyte, which occurs even at the potential well below the reversible value for Ag/Ag+. (From Refs. 207, 208.)... 

Ensemble average fluctuation with time, CORRELATION FUNCTION ENTATIC STATE ENTERING GROUP ENTEROPEPTIDASE ENTEROTOXIN ENTHALPY... 

From the above linear theory the time-correlation function for the thermal fluctuation < ,(1) decays exponentially with the decay rate rth(q) given by... 

The power dissipation is linearly related to <7BB(k, co) which is called, for obvious reasons, the power spectrum of the random process Bk. It should be noted that the energy dissipated by a system when it is exposed to an external field is related to a time-correlation function CBB(k, t) which describes the detailed way in which spontaneous fluctuations regress in the equilibrium state. This result, embodied in Eq. (51), is called the fluctuation... 

Time-correlation functions are of central importance in understanding how systems respond to weak probes in the linear approximation. According to the fluctuation dissipation theorem of the preceding section, spectro-... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

The quantities in (4.40) are single time quantities. According to Eq. (4.38) we need the special correlation function (8pay(PP t)8 Uya( p pt)) this function is closely connected to the correlation function (SpSp). According to the discussions presented in Refs. 12 and 28 for the determination of such correlation functions, one has to start from a differential equation for the corresponding two-time correlation functions and use the relevant single-time correlation function as initial condition. From the equation for the fluctuations 8p, which reads... 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

The sequence of the observed frequencies, resolved on the time scale, may be regrouped in a form giving a quantity S(t) which may be related to a time correlation function CAE(t) which represents the ensemble average of solvent fluctuations. 

If AE can be considered to be a small perturbation in system properties, the solvation response can be estimated using the linear response approximation (LRA), which relates S(t) to the time correlation function (TCF) C0(t) of fluctuations SAE = AE - (AE) of AE in the unperturbed system [23],... 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

For the dynamical distribution it will in general be necessary to consider both the auto and cross time correlation functions of the 0-1 and the 1-2 frequencies (117). For example, if the fluctuations, <5A(t), in the anhar-monicity are statistically independent of the fluctuations in the fundamental frequency, the oscillating term (1 — elAt3) in Equation (18) would be damped. In a Bloch model the fluctuations in anharmonicity translate into different dephasing rates for the 0-1 and 1-2 transitions that were discussed previously for two pulse echoes of harmonic oscillators. Thus we see that even if A vanishes, the third-order response can be finite (94). 

It is well known that the fluctuational behavior of the solvent molecules is characterized by the normalized time correlation function of the fluctuation, p (r), which is expressed as a following equation using observed time resolved fluorescence spectra. 

Fig. 7 shows the time correlation functions of solvent fluctuation in accordance with eq (7) using the values plotted in Fig. 6. In this calculation we used a(0)=30cm as the spectral width of the exciting laser pulse and 1430cm 1360cm and 1250cm for a(o>) in acetonitrile, methanol, and ethanol, respectively. In Fig. 7 the reported correlation functions obtained by the dynamic Stokes shift measurements of LDS7S0 in acetonitrile and 1-aminonaphthalene in methanol in accordance with eq (6) are also plotted by dashed lines. According to the literatures -, the correlation function decayed more than 80% in acetonitrile in the first 200fs and about 60% in methanol in the first 500fe. The peak shift of the absorption spectra in the present work was completed in the first Ips in methanol and ethanol solutions as indicated in Fig. 5 (2). However, the peak shift in acetonitrile solution could not be observed in the time resolution of our system. It is obvious fiom above results that the major part of the energy relaxation due to the fast response of the solvent dynamics is taken place in a few...

The fluctuation relations for the various viscosity coefficients involve time correlation functions. Some of them are ensemble dependent and others are ensemble independent. Only correlation functions the components of which couple with the angular velocity of the director or its conjugate torque density... 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

An insight into dynamical processes can be obtained from examining the equilibrium fluctuations SA = A A). Both the probability distribution P[5(A)] and the time correlation function... 

S. Kielich. Time-correlation functions for new cross-multipole field fluctuations in binary light scattering by unlike polar molecules. J. Phys. (Paris), 45 L.389-L.394 (1982). 

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