In order to get a better understanding of the slow dynamics in complex systems, the currently available theoretical tools for treating jump processes in rugged energy land-scapes have to be extended and supplemented. Our work focused on two issues.
In the first part, a new Effective Medium Approximation (EMA) scheme was for-mulated that allows us to treat asymmetric transition rates. We have shown that the new EMA scheme describes well the diffusion properties of the examined hopping model when smooth Densities of States (DOS) are used. The value of the long–time diffusion coefficient, however, is over–estimated by the EMA (as it is for other EMA schemes, too). This problem has been resolved by a proper rescaling of the tempera-ture in the effective medium. We applied the new scheme to ionic motions in glasses (including concentration dependence in the dilute limit) and showed that this approx-imation provides a scaling behavior of conductivity spectra comparable to that ob-served in experiment.
Since in real disordered materials fluctuations in the energy barriers will be present, too, it is important to note that the EMA approach can be extended straightforwardly to this case. In addition to the averaging over site energies in the EMA self–consistency equation, one would have to average over the barrier distribution as well.
An interesting modification that could be applied to the EMA scheme presented in chapters 3 and 4 is the “Percolation Path Approximation” (PPA), first proposed by DYRE [Dyr93] for barrier disorder. The PPA assumes that the dc– and the ac–
diffusion for all relevant frequencies follows the percolation paths in the disordered system. In the simplest case these paths are assumed to be one–dimensional, leading to an one–dimensional EMA. Since only paths with energiesE.Ec=E0lnqc (where
qc is the percolation threshold) contribute to the dynamics on the percolation paths, the DOS would have to be cut at E = Ec. Probably it would be necessary to take into account in the PPA that different ranges of energies contribute significantly to the averaged diffusion coefficient for the initial and for the the final state which leads to an asymmetric cut–off of the DOS in the averaging procedure of the transport quantities.
In the second part of this work we have studied phyical aging phenomena on the base of a new hopping model, the Quenched Jump Model (QJM). Arguments have been presented based on the statistical theory of extreme values that the DOS of the deepest minima in the complex energy landscape is expected to have an exponential tail under generic conditions. The exponential tail in the energy distribution density transforms into a PARETO(algebraic) tail of the distribution density for the effective hopping times. A method has been applied for treating sums of random variables
drawn from singular PARETO–type distribution functions, i. e. that have a diverging first moment.
We have developed a Partial Equilibrium Concept (PEC) and applied it to the QJM.
The PEC equation of the QJM has been analyzed by an exact mathematical treatment and has been shown to yield the correct aging behavior. We note that the partial equi-librium approximation proposed here should be applicable to a wide class of hopping models of non–equilibrium systems. It has been shown, both by Monte–Carlo simu-lations and by evaluating the PEC formula, that quite general scaling laws can occur within the QJM, including normal aging, subaging, ultraslow aging and generalized scaling. To our knowledge, the hopping model presented here for the first time allows one to understand analytically this rich variety of aging phenomena that are caused by thermally activated processes.
We have demonstrated that the appearance of a specific time scale is rather sen-sitive to the form of the (effective) transition rates between low–lying states. If the dynamics in the space of states is mapped onto a jump motion by means of some quantitative analysis, one would expect that the form of the jump rates is not unique but should vary with the position in configuration space. In this situation many char-acteristic time scales can occur that give rise to multiple scaling regimes in thet-tw plane. To observe these phenomena in experiments is a challenging task, since the separation of the various time scales can be difficult, while neglecting them can lead to wrong scaling laws.
In the literature the view has been held [Mon96] that in a high–dimensional state space a mean–field treatment of the energy landscape is appropriate since states are rarely revisited when the connectivity is large. However, we have shown that a mean–
field approximation (by means of the Annealed Jump Model) exhibits different decor-relation properties both when approaching the aging regime (!1+) and in the aging regime itself. The success of the PEC also for large connectivities1of the states demonstrates that the QJM in general does not show mean–field like behavior.
The results obtained for the QJM suggest that this model is a good base for fur-ther investigations of non–equilibrium dynamics in hopping models. For example it would be interesting to study how the aging dynamics is modified by an additional energy barrier disorder and by correlated energy disorder. New insight in the dynam-ical structure could be gained also by examining localization properties within the Quenched Jump Model. A possible measure for “incoherent localization” are partic-ipation ratiosyq(t) as defined in [Com98], i. e. the disorder averaged probability for
qparticles being on the same site at timet. Further topics of interest would be the applicability of the concept of a “time–dependent temperature” (as introduced e. g. in [Cug97]), the validity of the generalized fluctuation–dissipation theorem (as shortly outlined in section 1.3), temperature– and (external) field–cycling “experiments” and the study of memory and rejuvenation effects (see e. g. [Bou00]).