Anomalous diffusion particles

Actually, a (possibly anomalously) diffusing particle is, first, an interesting system per se. Roughly speaking, its velocity thermalizes, but not its displacement with respect to a given initial position, which is out of equilibrium... 

N. Pother, Aging properties of an anomalously diffusing particle. Physica A 317, 371 (2003). 

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... 

One can see that the investigated equations of dynamics even in linear approximation describe anomalous diffusion of the mass centre of macromolecule moving amongst the other macromolecules. The displacement of every particle of the chain is also anomalous in comparison with case of a macromolecule in a viscous liquid. Now we shall consider, following work by Kokorin and Pokrovskii (1990, 1993), the displacement of each internal particle of the chain... 

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]. 

In all that follows, the discussion will be purely classical. We now want to address the question whether the study of the (possibly anomalous) diffusion of a particle in an out-of-equilibrium medium is likely to provide information about the out-of-equilibrium properties of the latter. Generally speaking, the medium in which a diffusing particle evolves may be, or not, in a state of thermal equilibrium. For instance, when it is composed of an aging medium such as a glassy colloidal suspension of Laponite [8,12,55,56], the environment of a... 

To begin with, we summarize some results about the (possibly anomalous) diffusion of a particle in a thermal bath. 

In this chapter, we have showed that a particle undergoing normal or anomalous diffusion constitutes a system conveniently allowing one to illustrate and to discuss the concepts of FDT violation and effective temperature. Our study was carried out using the Caldeira-Leggett dissipation model. Actually this model, which is sufficiently versatile to give rise to various normal or anomalous diffusion behaviors, constitutes an appropriate framework for such a study, in quantum as well as in classical situations. 

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] ... 

Let us now discuss in more general terms the anomalous diffusion problem considering moments of arbitrary order of the particle s displacement. Two cases are possible [26] (a) weak anomalous diffusion when a unique exponent is involved,... 

We reviewed some aspect of passive transport in fluid flows. As far as inert substances are concerned, we described the problem of transport from a Lagrangian point of view focusing on single-particle properties. In particular, the conditions for having asymptotic standard or anomalous diffusion have been discussed in details. As for the problem of reacting substances, we study the... 

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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