Relaxation process temporal evolution

In the previous section it has been shown that the temporal evolution of the pair correlations at the interchain peak is governed by the structural relaxation. If we move now towards more local scales - i.e. higher Q-values - we see that the static correlations observed in Spair(Q) correspond to pair correlations along a given chain. It is then natural to think that their time dependence might relate to dynamic processes other than the structural relaxation. 

To illustrate the relaxation of the electrons (Wilhelm and Winkler, 1979), the temporal evolution of their velocity distribution under the action of a time-independent field and of the important electron collision processes has been calculated by solving system (44). The solution procedure started at / = 0 from a Gaussian distribution as the initial value of the isotropic distribution and fi-om a vanishing anisotropic distribution, i.e., from... 

Furthermore, Fig. 12 illustrates the temporal evolution of the mean energy u ,(t)/n and the reduced particle current density j, t)/n in the neon plasma for the same relaxation processes as are considered in Fig. 9. The mean energy shows a monotone decrease from its initial value m (0)/m U, determined by the... 

The figure clearly illustrates that the boundary value f i,U) for the anisotropic distribution initiates a weakly damped, spatially periodic relaxation of the density and energy current density of the electrons, and that the corresponding relaxation length becomes very large and takes about 100 cm at this field. This periodic relaxation behavior is in substantial contrast to the largely monotone evolution of all important electron kinetic quantities in the temporal relaxation process shown above. 

Figure 11. Temporal evolution of the injection efficiency at 1.0 X 10 V/cm of evaporated Au contacts on 40 wt% TPD/polycarbonate as a function of Au deposition conditions. All Au contacts are 220 A. Panel A Au is deposited in two steps, 50 A and 170 A, at 10 A/sec. Panel B Au is deposited in 11 layers of 20 A each at 10 A/sec. Panel C Au is deposited in 11 layers at 2 A/sec. Panel D Au is deposited in a single continuous step at 2 A/sec. In all cases the injection efficiency is initially blocking. The overall relaxation process occurs in a much shorter time when Au deposition is carried out in several stages.

Following Leffler and Grunwald (11), relaxation can be defined as "the return to equilibrium of a system that has been slightly perturbed by the imposition of a change in one of the variables of state." Relaxation corresponds to an irreversible process which is usually studied by following the temporal evolution of the system after a sudden disturbance. The disturbance consists of the change of one physical variable of the system (i.e., temperature, pressure, external electric or magnetic field, etc.)... 

ESIPT processes are usually faster than 1 ps and the subsequent relaxation processes occur typically in less than 1 ns. The observation of the dynamics calls for techniques with a time resolution higher than what can be achieved by electronic means. This is the domain of pump-probe techniques [1, 2]. An ultrashort laser pulse excites the sample and initiates the process of interest. A second ultrashort pulse probes the properties of the sample after a delay, which is the temporal separation between the two pulses. The measurement is repeated with systematically varied delay times to sample the complete evolution of an observable during the process under... 

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

Let us consider now the nonequilibrium state of a fractal-like medium assuming that this nonequilibrium state is characterized by many events such that a subsequent event is separated by a certain time interval r, from a previous event. In this case, some intervals will be eliminated from a continuous process of system evolution by a definite law. Assume that such a process is caused by a temporal fractal state of dimensionality df the corresponding relaxation equation can be written as... 

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