Random walk model, molecule

Diffusion occurs when there is a concentration gradient of one kind of molecule within a fluid. In terms of random walk model, the average distance, x, after an elapsed time, t, between molecule collisions in a diffusion movement is characterized by the Einstein-Smoluchowski relation,... 

The random-walk model consists of a series of steplike movements for each molecule which may be positive or negative the direction being completely random. After (p) steps, each step having a length (s) the average of the molecules will have moved some distance from the starting position and will form a Gaussian type distribution curve with a variance of o2. ... 

The random walk model is certainly less suitable for a liquid than for a gas. The rather large densities of fluids inhibit the Brownian motion of the molecules. In water, molecules move less in a go-hit-go mode but more by experiencing continuously varying forces acting upon them. From a macroscopic viewpoint, these forces are reflected in the viscosity of the liquid. Thus we expect to find a relationship between viscosity and diffusivity. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

While the random walk model employed here is widely applicable to F(+) methods, it fails if the molecules do not transfer rapidly between velocity states, equivalent to many random steps. Such a limitation applies to electrodecantation (noted below), where the distances are too great for rapid diffusional exchange. The random walk model is most meaningful for zonal separation methods such as chromatography and field-flow fractionation. 

The simplest mechanisms leading to the dispersion (spreading) of a zone s molecules can be described by the classical random-walk model [9], as noted in Section 5.3. However this model does not fully account for the complexities of migration. It gives, instead, a simple approximation which inherits the most essential and important properties (foremost of all the randomness) of the real migration process. The random-walk model has been used in a similar first-approximation role in many fields (chemical kinetics, diffusion, polymer chain configuration, etc.) and is thus important in its own right. 

The random-walk model assumes steps of equal length forward and backward. Prove that the desorbed molecule takes a step forward of length equal to the length Rvtd of the backward step discussed in the text. (Hint Assume that a desorbed molecule takes a step after each mean sorption time ta.)... 

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... 

Though the ideal gas assumption would cause some error in predicting result, the reasonableness of the above suggested models can be explained by HIO (Higashi, Ito, Oishi) model (Higashi et al., 1963) which is based on the random walk of molecules. The HIO model was same with the model 4 in this paper when the single layer adsorption was assumed. 

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. 

The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length 1. To analyze the model one end of the chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shown in Fig. A2.1(b), and the probability, P(x,y,z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates x,y,z) may be calculated. Such calculation leads to an equation of the form (Young and Lovell, 1990) ... 

Figure 3.17. Simulation of sample zone dispersion in the absence of a chemical reaction by the random walk model. 5 = 400 jlL, Q = 1 mL/min, / = 0.8 mm, Dm = 1 x 10" mm/ s, 1400 molecules, and 40 cycles per simulated second (simsec). (By courtesy of D. Betteridge with permission of Elsevier Scientific Publishing Co.)...

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