Three-dimensional random walk

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

The results of the three-dimensional random walk, based on the freely-jointed chain, has permitted the derivation of the equilibrium statistical distribution function of the end-to-end vector of the chain (the underscript eq denotes the equilibrium configuration) [24] ... 

Figure S5.3 Typical three-dimensional random walk, redrawn from a Mathematica calculation (see Further Reading, Chapter 5, S. Wolfram).

Polymer molecules in a solution undergo random thermal motions, which give rise to space and time fluctuations of the polymer concentration. If the concentration of the polymer solution is dilute enough, the interaction between individual polymer molecules is negligible. Then the random motions of the polymer can be described as a three dimensional random walk, which is characterized by the diffusion coefficient D. Light is scattered by the density fluctuations of the polymer solution. The propagation of phonons is overdamped in water and becomes a simple diffusion process. In the case of polymer networks, however, such a situation can never be attained because the interaction between chains (in... 

Equation 7.46 demonstrates that if each jump of a walk occurs randomly (i.e., is uncorrelated), the average displacement is zero and the center of mass of a large number of individual random jumpers is not displaced. Equation 7.47 gives the mean-square displacement of a random walk, NT(r2). Although Eqs. 7.46 and 7.47 were derived here for one-dimensional random walks, both are valid for two- and three-dimensional random walks. 

The absorption of energy by the grains produces conduction electrons and either free or trapped holes. The conduction electrons and the holes diffuse initially by a three-dimensional random walk. In chemically sensitized crystals, the holes are trapped by products of chemical sensitization which thus undergo photo-oxidation. Rapid recombination between a trapped hole and an electron is avoided by the delocalization as an interstitial Ag ion of the nonequilibrium excess positive charge created at the trapping site. Latent pre- and sub-image specks are formed by the successive combination of an interstitial Agi ion and a conduction electron at a shallow positive potential well. 

Treating Brownian motion as a three-dimensional random walk , the mean Brownian displacement x of a particle from its original... 

Further calculation shows that for a three-dimensional random walk the same result is obtained. 

Consider a particular random walk on a lattice with each step having independent Cartesian coordinates of either -fl or —1. The projection of this three-dimensional random walk onto each of the Cartesian coordinate axes is an independent one-dimensional random walk of unit step length (see Fig. 2,8 for an example of a two-dimensional projection). The fact that the one-dimensional components are independent of e ch other is an important property of any random walk (as well as any ideal polymer chain). 

This probability distribution function for the displacement of a one-"dimensional random walk can be easily generalized to three-dimensional random walks. The probability of a walk, starting at the origin of the coordinate system, to end after N steps, each of size b, within a volume dRxdRydR of the point with displacement vector R is P3ti N,R) dRjcdRydRz (see Fig. 2.11). Since the three components of a three-dimensional random walk along the three Cartesian coordinates are independent of each other, the three-dimensional probability distribution function is a product of the three one-dimensional distribution functions ... 

A similar treatment can be applied to two- and three-dimensional random walks, where the root-mean-square displacements are and respectively (19, 21). 

This analysis can be extended to two- and three-dimensional random walks by assuming that particle motion in each dimension is independent. The mean square displacement and r.m.s. displacement for higher dimension random walks become ... 

Note that Equations (5.3)-(5.6) have been derived for the case of the onedimensional random walk. This is because these equations will be used later in analysis of sedimentation under gravity where motion only in one direction (one dimension) is of interest. Only motion of particles in the direction of the applied gravitational force field is of interest in sedimentation lateral motion in the other two orthogonal directions is not. In the case of the three-dimensional random walk, the analogy to Equation (5.3) would be L = V6af. 

Figure 1.17. Step motion in a three-dimensional random walk on a cubic lattice.

Just as in P r) for the three-dimensional random walk on a discrete lattice (Eq. 1.18), G(ri, Y2, n) consists of three independent factors ... 

Baur JE, Motsegood PN (2004) Diffusional interactions at dual disk microelectrodes comparison of experiment with three-dimensional random walk simulations. J Electroanal Chem 572 29 0... 

Nagy G, Sugimoto Y, Denuault G (1997) Three-dimensional random walk simulation of diffusion controlled electrode processes (I) a hemisphere, disc and growing hemisphere. J Electroanal Chem 433 167-173... 

Fig. 11.36 Afreelyjointedchainis like a three-dimensional random walk, each step being in an arbitrary direction but of the same length.

Our treatment has implicitly assumed that the line diffusion will approximate a three-dimensional random walk. We might intuititively expect this to obtain if Dj) > 2a, so that a diffusing atom will assay a three-dimentional network of boundaries. This condition is well met for the shortest times in 0 2 each temperature. 

ABSTRACT. Excimer fluorescence is developed as a quantitative probe of isolated chain statistics and intermolecular segment density for miscible and immiscible blends of polystyrene (PS) with poly(vinyl methyl ether) (PVME). Rotational isomeric state calculations combined with a one-dimensional random walk model are used to explain the dependence of the excimer to monomer intensity ratio on PS molecular weight for 5% PS/PVME blends. A model for a three-dimensional random walk on a spatially periodic lattice is presented to explain the fluorescence of miscible PS/PVME blends at high concentrations. Finally, a simple two-phase morphological model is employed to analyze the early stages of phase separation kinetics. 

The objective of this section is to present a quantitative analysis of the photostationary state fluorescence of miscible and immiscible PS/PVME blends. In Section 4.1 we develop the onedimensional random walk model that is used in conjunction with rotational isomeric state calculations to analyze low concentration miscible blends. In Section 4.2 we treat miscible blends having high PS concentration using a spatially periodic three-dimensional random walk model. Finally, in Section 4.3 we present a simple two phase morphological model and demonstrate how it may be used to monitor phase separation kinetics. 

The familiar, flexible polymer chains are simply modeled as isotropic, three dimensional random walks. The monomer size, a, is the step length and the polymerization degree, A, is identified with the number of steps. The probability density... 

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