Random walks disordered systems

To characterize the dynamic movement of particles on a fractal object, one needs two additional parameters the spectral or fracton dimension ds and the random-walk dimension dw. Both terms are quite important when diffusion phenomena are studied in disordered systems. This is so since the path of a particle or a molecule undergoing Brownian motion is a random fractal. A typical example of a random fractal is the percolation cluster shown in Figure 1.5. 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

The general solution (15.22) allows for very different phase profiles (see Fig. 15.15). The regularity of the wave pattern depends on the relative influence of the mean slope c > compared to the fluctuations Q and can be measured by the quality factor = < /var(10j ) and the autocorrelation r of the 4>i. As demonstrated in Fig.15.16, for sufficiently large systems, both and r only depend on the product 7 0 we find r 1, and the solution is essentially a random walk (see Fig.2a,b). With increasing values of ya the correlations r are reduced and eventually become negative. Furthermore increases with the product ycr, and for 7 > 1 can rise drastically (see Fig. 15.16c). Thus, with increasing disorder of the system we obtain more regular patterns until synchronization is lost. 

Statistical fractals are generated by disordered (random) processes. An element of disorder is typical of most real physical phenomena and objects. The fact that disorder, i.e., the absence of any spatial correlation, is a sufficient condition for the formation of fractals was first noted by Mandelbrot [1]. A typical example of this type of fractal is the random-walk path. However, real physical systems are often inadequately described by purely statistical models. Among other reasons, this is due to the effect of excluded volume. The essence of this effect lies in the geometric restriction that forbids two different elements of a system to occupy the same volume in space. This restriction is to be taken into account in the corresponding modelling [10, 11]. The best-known examples of this type of models are self-avoiding random walk, lattice animals and statistical percolation. 

As we have learned from the present review, the interplay between the self-avoiding condition of a linear chain and the self-similar nature of the substrate on which the chain is located displays a very rich behavior. The resulting compound random process attains its maximum complexity in the case of a random substrate modelled by percolation, responsible for the occurrence of multifractality in the statistical behavior of SAWs on such disordered structures. It is important to note that some aspects of possible multifractality of SAWs on percolation cluster have not been studied yet. Already in the much simpler system of random walks on percolation cluster, that is walks that can intersect themselves [90], these quantities show a rich multifractal behavior. ITie extension of such a study to S.W s on percolation is only at a preliminar sta, and further numerical investigations are required before general conclusions can be drawn. 

Conductivity is of course closely related to diffusion in a concentration gradient, and impedance spectroscopy has been used to determine diffusion coefficients in a variety of electrochemical systems, including membranes, thin oxide films, and alloys. In materials exhibiting a degree of disorder, perhaps in the hopping distance or in the depths of the potential wells, simple random walk treatments of the statistics are no longer adequate some modem approaches to such problems are introduced in Section 2.1.2.7. 

The above qualitative picture is supported by the data of frequency-dependent transport properties. The frequency dependence of conductivity provides another, independent way of determining the effective dimensionality d. In disordered systems the conductivity as a function of frequency usually follows a power law o- o>. Considering that the basic process of conduction is an anomalous diffusion, i.e., a random walk of the charge carriers on a network of effective dimensionality d < d, where d is the space dimension ( = 3 in the present case), the exponent s can be expressed as = 1 - did. This expression with data of conductivity versus frequency given in the literature [106] leads to values for d that agree satisfactorily with those obtained from spin dynamics. 

Dense polymer systems, such as melts, glasses, and crosslinked networks or rubber are extremely complex materials. Besides the local chemical interactions and density correlations which are common to all disordered hquids and solids, the chain conformations also play an important role. Their influence is twofold. First the intrachain entropy dominates over the positional entropy of the center of mass of the chains. This leads to the well known effect that a weak effective repulsion between different types of chains is sufficient to drive phase segregation. The static and dynamic properties of mixtures of two types of chains is an important and challenging problem, which is reviewed by Binder in Chapter 7 of this volume. Here we consider dense melts of ehains of the same chemical composition. In this case the entropy is at its maximum when the chains have a random walk structure. Since the average end-to-end distance Nfor a random... 

Fig. 9.2 The disordered pinning model, based on simple random walk, with binary charges. In the two cases / = 0.1, h instead takes value 0.0045 and 0.0048. In both cases to suppress fluctuations we plot only a point every 10 . A slope is visible in both cases and it suggests a free energy of about 10 in the first case and of about 1.5 -10 in the second (note that the slope has to be divided by 2 since the real length of the system is 2N). Note also that in the second case the partition function is still well below 1 for a system of length 200 millions. For h = 0.0049 (data not plotted) the slope is less clear. However, since logcosh(O.l) = 0.004991..., these graphs suggest that the quenched critical point is very close to the annealed one (see Section 5.5). With reference to (9.3), we have chosen A = B = 6 and No = 1000.

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