Temporal correlation functions

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

The auto-correlation functions for the pressure and temperature Rjj fluctuations are presented in Fig. 2.41. It is clear that the temporal behavior of the temperature fluctuations corresponds to that of the pressure fluctuations (Hetsroni et al. 2002b). 

Temporal behavior of the correlation function was studied in Ref. 91 using a particular example of the correlation function of sin x(f) in a periodic potential with periodic boundary conditions. In that case the use of single exponential approximation had also given a rather adequate description. The considered... 

Therefore, the joint correlation functions Xvjl (a t)> being at least potentially observable, are more a theoretical than an experimental tool for the description of interacting particles in condensed media. Both these joint functions and macroscopic concentrations nv t) determine the lowest level to characterize the spatio-temporal structure of a system. 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

This chapter relates to some recent developments concerning the physics of out-of-equilibrium, slowly relaxing systems. In many complex systems such as glasses, polymers, proteins, and so on, temporal evolutions differ from standard laws and are often much slower. Very slowly relaxing systems display aging effects [1]. This means in particular that the time scale of the response to an external perturbation, and/or of the associated correlation function, increases with the age of the system (i.e., the waiting time, which is the time elapsed since the preparation). In such situations, time-invariance properties are lost, and the fluctuation-dissipation theorem (FDT) does not hold. 

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

In mathematical terms, the frozen turbulence approximation, known as Taylor s hypothesis, refers to a simple mathematical relation between statistical temporal auto-covariances and spatial correlation functions approximately valid for quasi-steady homogeneous turbulent flow. 

In all the 2BSM calculations presented here, the diffusion coefficient Dj equals 1, which defines the unit of frequency (inverse time) whereas the diffusion coefficient for the solvent, Dj varied from 10 (very fast solvent relaxation) to 1, 0.1, 0.01 (very slow solvent relaxation). In the Dj = 10 case, one finds that the reorientation of the solute is virtually independent of the solvent a projection procedure could easily be adopted in this case to yield a one-body Smoluchowski equation for body 1 with perturbational corrections from body 2. The temporal decay of the first and second rank correlation functions is then typically monoexponential. When the solvent is relaxing slowly (i.e., Dj is in the range 1-0.01), the effect of the large cage of the rapid motion of the probe becomes... 

Table X contains numerical data concerning the temporal decay of momentum correlation functions (for body 1, i.e., L,). One realizes immediately that in this case the influence of the cage body is much weaker than it was for orientational observables. For Wj = 50 the relaxation of the momentum of body 1 is almost totally decoupled from reorientation of body 2, even for large potentials. For Wj = 5, a cluster of eigenvalues close in value to the collision frequency is present. This is also the case for Wj = 0.5, but librational modes are beginning to play a nonnegligible role.

Another early method used to monitor the laser pulse was the three photon fluorescence (3PF) technique. ( 8, 9) The advantage of 3PF over TPF is two-fold the contrast ratio is 10 1 for 3PF as opposed to 3 1 for TPF and in addition to temporal information about the pulses available by TPF, 3PF also provides pulse shape information.(10-13) This additional information is obtained because the third-order correlation function which relates the 3PF intensity to the pulse Intensity includes dependence upon the phase of the pulse frequency components.(11) Again, the resulting fluorescence is photographed, and a densitometer trace is made to determine fluorescence intensities. Azulene is an example of a molecule which has been studied quite extensively.(14-20) Typical data are shown in Figure 2. 

In conclusion, in this section we presented the formal expressions for the absorption lineshape [Eq. (70)] and for spontaneous Raman and fluorescence spectroscopy. For the latter, we derived Liouville space expressions in the time and the frequency domain [Eqs. (74) and (75)], an ordinary correlation function expression [Eq. (76)], and, finally, the factorization approximation resulted in Eqs. (77) and (78). The factorization approximation is expected to hold in many cases for steady-state experiments and for time-resolved experiments with low temporal resolution. It is possible to observe a time-dependent shift of spontaneous emission lineshapes using picosecond excitation and detection [66-68]. This shift arises from the reorganization process of the solvent and also from vibrational relaxation that occurs during the t2 time interval. A proper treatment of these effects requires going beyond the... 

When radiation interacts with a sample scattering or diffraction occurs due to spatial and temporal correlations in the sample. In this section, the basic quantities and correlation functions will be introduced. In elastic and quasi-elastic scattering experiments the most important quantity is the magnitude of the so-called scattering vector given... 

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