Brownian motion continuous time random walk

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... 

As far as the physical mechanism underlying the Cole-Cole equation is concerned, we first remark that Eq. (9) arises from the diffusion limit of a continuous-time random walk (CTRW) [17] (see Section II.A). In this context one should recall that the Einstein theory of the Brownian motion relies on the diffusion limit of a discrete time random walk. Here the random walker makes a jump of a fixed mean-square length in a fixed time, so that the only random... 

We can view Brownian motion as a continuous time random walk, visualised as a walk along a line, beginning at Xo = 0 and moving at each incremental time interval At either up or down by an amount If we denote the position of... 

One of the important issues is the possibility to reveal the specific mechanisms of subdiflFusion. The nonlinear time dependence of mean square displacements appears in different mathematical models, for example, in continuous-time random walk models, fractional Brownian motion, and diffusion on fractals. Sometimes, subdiffusion is a combination of different mechanisms. The more thorough investigation of subdiffusion mechanisms, subdiffusion-diffiision crossover times, diffusion coefficients, and activation energies is the subject of future works. 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

Brownian motion of particles is the governing phenomenon associated with transitions between states in the above examples as well as in the mathematical derivations in the following [4, p.203]. If we consider a particle as system and the states are various locations in the fluid which the particle occupies versus time, then the transition from one state to the other is treated by the well-known random walk model. In the latter, the particle is moving one step up or down (or, alternatively, right and left) in each time interval. Such an approach gives considerable insight into the continuous process and in many cases we can obtain a complete probabilistic description of the continuous process. 

This is best understood intially by considering the process of diffusion. Ghromatographic peaks represent chemical species that have been concentrated in space and time and the process of diffusion will immediately disperse them in space as a function of time. The conceptual basis of diffusion lies in the concept of the random walk model, wherein particles/molecules in suspension or solution are being jostled continuously by collisions with other particles or molecules. This is also referred to as Brownian motion, and is readily apparent when observing small particles with a microscope, such as some pollen grains, that seem to be in constant and random motion as they gradually spread out from any center of concentration. 

Particles suspended in a fluid are continuously bombarded by the surrounding fluid molecules. This constant bombardment results in a random motion of the particles known as Brownian motion. A satisfactory description of this irregular motion ( random walk ) can be obtained ignoring the detailed structure of the particle-fluid molecule interaction if we assume that what happens to the aerosol fluid system at a given time t depends only on the system state at time t. Stochastic processes with this property are known as Markov processes. 

Models of Brownian motion and of the geometry of ideal polymer chains are based on the concept of a random walk. Let us start at time t = 0 and flip a coin. If heads, we take a step to the right of length /. If tails, we take a step to the left. We continue to do this, taking n steps, and measure the net displacements (our final position) (Figure 7.7). 

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