Statistical-thermodynamical foundations of anomalous diffusion

Abstract

It is shown that Tsallis's generalized statistics provides a natural frame for the statisticalthermodynamical description of anomalous di usion. Within this generalized theory, a maximumentropy formalism makes it possible to derive a mathematical formulation for the mechanisms that underly L evy-like superdi usion, and for solving the nonlinear Fokker-Planck equation. I Introduction: Di usion processesAmong the elementary processes that underly natural phenomena, di usion is certainly one of the most ubiquitous. In an ensemble of moving elements atoms, molecules, chemicals, cells, or animals each element usually performs, at a mesoscopic description level, a random path with sudden changes of direction and velocity. As a result of this highly irregular individual motion, which is microscopically driven by the interaction of the elements with the medium and of the elements with each other, the ensemble spreads out. At a macroscopic level, this collective behavior is in contrast with the individual microscopic motion extremely regular, and follows very well de ned, deterministic dynamical laws. It is precisely this smooth macroscopic spreading of an ensemble of randomly moving elements that we associate with di usion. One of the rst systematic observations of di usion was made by the botanist Robert Brown in 1828. He noticed that pollen particles dispersed in water exhibit a v ery irregular, swarming motion. In 1905, Einstein conjectured that this Brownian motion" is due to the interaction of pollen with the water molecules and, in fact, proved that microscopic particles suspended in a liquid perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat" 1 . Since then, Brownian motion is used as a synonym of di usion.A v ery suitable and very useful mathematical model for Brownian motion is provided by random walks 2 . In its simplest version, a random walker is a point particle that moves on a line at discrete time steps t. At each step, the walker chooses to jump to the left or to the right with equal probability, and then moves a xed distance x. This stochastic process can be readily generalized, rstly, b y allowing the walker to move in a many-dimensional space. In addition, time can be made continuous by associating a random duration with each jump or by i n troducing random waiting times between jumps. Finally, the length of each jump can be also chosen at random from a continuous set with a prescribed probability distribution.Being a stochastic process, a random walk admits a probabilistic description in terms of probability distributions for the relevant quantities 3 . In particular, one is interested at studying the probability of nding the walker in a certain neighborhood dr of point r in general, in a d-dimensional space at time t, P r; t dr. Note that, besides its interpretation as a probability distribution, P r; t can be related to the density in an ensemble of noninteracting identical random walkers. In fact, if the ensemble contai...

Gradient pattern analysis of extended convection–diffusion

“…Therefore, the focus is on the case 1 < q < 2. For the two dimensional case one finds the following sub-regimes: (a) the interval 1 < q < 1.5 corresponds to a transition to a superdiffusive process, and (b) the interval 1.5 ≤ q < 2 an anomalous superdiffusion is obtained [6].…”

“…The generic scenario we assume in this paper is a hybrid diffusion process (HDP) composed by three sequential, and time dependent, phenomenological regimes (R 1 [6]. Thus, in this first regime of the process the generation of 2D-cumulative energy distributions occurs, and we can use the following notation to characterize these distributions U q (x, y).…”

“…For physical purposes (mainly, long-range correlations), from [6], the generalized distribution is given by:…”

“…The parameter β is a Lagrange parameter associated with the temperature [6]. Many distributions, U(S i,j ) β 1/2 versus S i,j β 1/2 , for typical values of q, can be obtained from Equation 1.…”

“…The GPA formalism performs numerical investigation of spatio-temporal complex patterns by using computational operators such as AAF (Asymmetric Amplitude Fragmentation) [4] and CEF (Complex Entropic Form) [6]. Usually, the classical measures of complex variability doesn't take into account the directional information contained in a vector field.…”