Anomalous diffusion response

VII. Dynamic Approach to Anomalous Diffusion Response to Perturbation... 

VII. DYNAMIC APPROACH TO ANOMALOUS DIFFUSION RESPONSE TO PERTURBATION... 

As a step toward the study of thermodynamic equilibrium in the case of anomalous statistical physics, in Section VII we study how the generators of anomalous diffusion respond to external perturbation. The ordinary linear response theory is violated and, in some conditions, is replaced by a different kind of linear response. In Section VIII we review the results of an ambitious attempt at deriving thermodynamics from dynamics for the main purpose of exploring a dynamic approach to the still unsettled issue of the thermodynamics of Levy statistics. The Levy walk perspective seems to be the only possible way to establish a satisfactory connection between dynamics and thermodynamics in... 

This review is devoted to illustrating the problems that the processes of anomalous diffusion, of renewal nature, are raising in the adaptation of the traditional prescriptions of nonequilibrium statistical physics. In this section we explore the delicate issue of the response to external perturbations. 

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 the science of complexity the system response X(t) is expected to depart from the totally random condition of the simple random walk model, since such fluctuations are expected to have memory and correlation. In the physics literature, anomalous diffusion has been associated with phenomena with longtime memory such that the autocorrelation function is... 

Now, we shall demonstrate that the characteristic times of the normal diffusion process, namely, the inverse of the smallest nonvanishing eigenvalue 1 //.], the integral and effective relaxation times xint and xef obtained in [8,62,63], also allow us to evaluate the dielectric response of the system for anomalous diffusion using the two-mode approximation just as normal diffusion (Ref. 8, Section 2.13). Here, we can use known equations for xint, x,f, and X for the normal diffusion in the potential Eq. (163) [8,62,63] these equations are... 

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]. 

Pyun et al. started " their exploration of the anomalous current response with the CTs obtained from Lii.sCoOj which is the cathode material of almost all commercially available rechargeable lithium batteries today. They reported that the CTs obtained from U1.SC0O2 composite and thin fitm " electrodes hardly exhibit a typical trend of diffusion controlled lithium transport, i. e. Cottrell behavior. Furthermore, they have found that the current-potential relation obeys Ohm s law during the CT experiments. They thus suggested that lithium transport at the interface of the electrode and the electrolyte is mainly limited by the internal cell resistance, and not by lithium diffusion in the bulk electrode. This concept is called the cell-impedance controlled lithium transport. 

Following the developments outlined in [8,9], we now stress the fact that anomalous diffusion in the scahng form of Eqs. 119 and 120 is closely connected to descriptions based on fractional derivatives, given that they allow us to invert, in a simple way, the integral expressions which follow from the theory of Hnear response, when the anomalous behavior has a power-law character going as Eq. 119, with y < 1. For technical reasons and because of an intimate relation to linear response we prefer, as in [8,9], to extend the lower integration Emit in Eq. 121 to - oo in this way we obtain the Weyl-form. 

Kinetic response of surfaces defined by finite fractals has been addressed in the context of interaction of finite time independent fractals with a time-dependent diffusion field by a novel approach of Cantor Transform that provides simple closed form solutions and smooth transitions to asymptotic limits (Nair Alam, 2010). In order to enable automatic simulation of electrochemical transient experiments performed under conditions of anomalous diffusion in the framework of the formalism of integral equations, the adaptive Huber method has been extended onto integral transformation kernel representing fractional diffusion (Bieniasz, 2011). 

In summary, it is possible to understand the Perseus molecular complex and the dark cloud LDN 1622 microwave anomalous emission in terms of electric dipole emission of fulleranes if these molecules follow a size distribution similar to that proposed in the study of the UV extinction bump. The dominant microwave emission would be associated in both cases to the smaller fulleranes. These molecules could also be responsible for the diffuse microwave dust correlated emission at high Galactic latitude detected by the COSMOSOMAS experiment and WMAP. 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

---