Correlation function temporal evolution

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. 

Figure 3 Temporal evolution of the correlation length A (pm) as determined by SALS on BLG/AG dispersions at 0.1 wt% total biopolymer concentration, pH 4.2 and Pr.Ps weight ratio of 2 1. Bars are standard deviation based on duplicate experiments. Drawn line is a power-law function with an exponent of 0.5...

True in-situ experiments on practical systems must be conducted in such a way that the catalytic performance is measured simultaneously with the spectroscopic or structural property of the experiment. Reliable in-situ experiments are performed at multiple steady states and a quantitative correlation between catalytic function (activity, selectivity) and spectroscopic/structural property is established. The preferred way of doing this should be a modulation of the reaction conditions coupled with observation of the temporal evolution of the spectroscopic signal. Only then, it is proven that a direct and physically meaningful correlation exists between structure/spectral property and catalytic function. 

The temporal evolution of the domain structure can be monitored by studying the time dependence of the equal-time correlation function... 

Probe diffusion was determined using quasi-elastic light scattering spectroscopy. QELSS monitors the temporal evolution of concentration fluctuations by measuring the intensity I(q,t) of the light scattered at time t, and calculating the intensity-intensity correlation function... 

The temporal and spatial variations of the field variable describe time evolution process of inhomogeneous system and the local free eneigy density determines the kinetic path. Hence, the key to the PFM calculation is the definitions of both the field variables and the local free energy density. In the present study, the CVM free energy(Eqs.9 and 10) is adopted as the local free energy density and the field variable i in Eq.(18) is replaced by correlation functions defined in the previous section. Then, Eq.(18) is rewritten as... 

Dual Lanczos transformation theory is a projection operator approach to nonequilibrium processes that was developed by the author to handle very general spectral and temporal problems. Unlike Mori s memory function formalism, dual Lanczos transformation theory does not impose symmetry restrictions on the Liouville operator and thus applies to both reversible and irreversible systems. Moreover, it can be used to determine the time evolution of equilibrium autocorrelation functions and crosscorrelation functions (time correlation functions not describing self-correlations) and their spectral transforms for both classical and quantum systems. In addition, dual Lanczos transformation theory provides a number of tools for determining the temporal evolution of the averages of dynamical variables. Several years ago, it was demonstrated that the projection operator theories of Mori and Zwanzig represent special limiting cases of dual Lanczos transformation theory. 

Equation [61] is only valid in the asjmiptotic Q" tail of the scattering function. A full treatment yields the dynamic stmc-tme factor also in the low Q limit. At r=0, this leads to the well-known Debye function (eqn [20]). Figure 33 displays the temporal evolution of the quasielastic SANS. The uppermost curve corresponds to the Debye function, while the lower curves visualize the decay of the sHucture factor due to the Rouse relaxation. For t —> CX> all internal chain correlations are lost and the stmcture factor displays the Gaussian density profile within a polymer coil. We note that the polymer coil in addition to the Rouse modes is also subject of translational diffusion. 

---