Anomalous diffusion motion

The connection between anomalous conductivity and anomalous diffusion has been also established(Li and Wang, 2003 Li et al, 2005), which implies in particular that a subdiffusive system is an insulator in the thermodynamic limit and a ballistic system is a perfect thermal conductor, the Fourier law being therefore valid only when phonons undergo a normal diffusive motion. More profoundly, it has been clarified that exponential dynamical instability is a sufRcient(Casati et al, 2005 Alonso et al, 2005) but not a necessary condition for the validity of Fourier law (Li et al, 2005 Alonso et al, 2002 Li et al, 2003 Li et al, 2004). These basic studies not only enrich our knowledge of the fundamental transport laws in statistical mechanics, but also open the way for applications such as designing novel thermal materials and/or... 

This finding implies a sublinear increase of (r (t)) with time. Thus, the incoherent neutron scattering studies qualify the motion of the hydrogens as an anomalous diffusion involving a sublinearly increasing (r (t)). 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

After internalization in phase I, most polyplexes show anomalous or confined diffusion (phase II) followed by active transport (phase III) [37]. The anomalous diffusion and confinement displayed by the MSD analysis represent the local microenvironment of the particles in the cytoplasm where the cytoskeleton, organelles and large macromolecules are local obstacles for free diffusion. Suh et al. [41] tracked internalized polyplexes with a temporal resolution of 33 ms and found diffusive motion of polyplexes where the corresponding trajectories showed hop diffusion (Fig. 4). These hop diffusion patterns can be interpreted as the cages of the... 

In this section, we extend the previous study to the case of non-Ohmic dissipation, in the presence of which the particle damped motion is described by a truly retarded equation even in the classical limit, and either localization or anomalous diffusion phenomena are taking place. Such situations are encountered in various problems of condensed matter physics [28]. 

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... 

Equation (5) is a partial differential equation of infinite order, and cannot be solved in general. Since all properties of glass vary slowly in space and time, the left-hand side of Eq. (5) can be truncated, the motion of holes in response to molecular fluctuations is then treated as an anomalous diffusion process [12]... 

In this chapter we discuss a novel fast diffusion process that was experimentally discovered by Yasuda, Mori, and co-workers (YM) [7] in nano-sized binary metal clusters, because it is supposed to be a typical manifestation of an anomalous diffusion process peculiar to microclusters. The authors have reported some numerical results of MD simulation on rapid alloying (RA) [8]. Special attention is paid to the diffusion of atoms in cluster [9]. The aim of the present work is to realize RA by elucidating what kind of diffusive motion is relevant to RA. 

Investigating the diffusive properties of single-particle motion allows us to predict the characteristics of the macroscopic motion of concentration fields [cf. Eq. (4)]. In this framework it is important to identify the conditions that may lead to anomalous diffusion that implies as a consequence the failure of the Fickian description of transport that is, Eq. (4) does not hold anymore. From Eq. (3) it is easy to obtain the following relation [19] ... 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

This leads us to one of the standard, but often inappropriate, explanations of anomalous diffusion using fractional Brownian motion with the probability density... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

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