Random walk correlated

Dunn et al. (D7) measured axial dispersion in the gas phase in the system referred to in Section V,A,4, using helium as tracer. The data were correlated reasonably well by the random-walk model, and reproducibility was good, characterized by a mean deviation of 10%. The degree of axial mixing increases with both gas flow rate (from 300 to 1100 lb/ft2-hr) and liquid flow rate (from 0 to 11,000 lb/ft2-hr), the following empirical correlations being proposed ... 

Hollander, 1972) using basically the Noyes diffusion model (Noyes, 1954, 1955, 1956, 1961), treats the radicals as if they go on a random walk . It must be remembered, however, that the two radicals must maintain the correlation of their electron spins (Jee f ) or tl distinction between T and S manifolds for a given radical pair would be lost. 

An alternative way to describe the phenomenon is to consider that the ground state of a chain is already divided into domains at any temperatures. In order for the system to follow a small variation of the magnetic field some domains have to reverse their spin orientation. This occurs through a random walk of the DWs, that is, equal probability for the DW to move backward or forward, which implies that the DW needs a time proportional to d2 to reach the other end of a domain of length d. Given that d scales as the two spins correlation length, ., which, for the Ising model, is proportional to exp(2///rB7 ), for unitary spins, the same exponential relaxation is found... 

A second modification to the random-walk model of diffusion is required if motion is not random but correlated in some way with preceding passage through the crystal... 

Figure 5.17 Correlated motion during vacancy diffusion (a) vacancy can jump to any surrounding position and its motion follows a random walk (b, c) the motion of a tracer atom is correlated, as a jump into a vacancy (b) is most likely to be followed by a jump back again (c).

The vacancy will follow a random-walk diffusion route, while the diffusion of the tracer by a vacancy diffusion mechanism will be constrained. When these processes are considered over many jumps, the mean square displacement of the tracer will be less than that of the vacancy, even though both have taken the same number of jumps. Therefore, it is expected that the observed diffusion coefficient of the tracer will be less than that of the vacancy. In these circumstances, the random-walk diffusion equations need to be modified for the tracer. This is done by ascribing a different probability to each of the various jumps that the tracer may make. The result is that the random-walk diffusion expression must be multiplied by a correlation factor, / which takes the diffusion mechanism into account. 

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

The correlation factor, for any mechanism, is given by the ratio of the values of the mean square displacement of the atom (often the tracer) moving in a correlated motion to that of the atom (or vacancy) moving by a random-walk process. If the number of jumps considered is large, the correlation factor/can be written as... 

The correlation factor,/, is defined by the ratio of the tracer diffusion coefficient to the random-walk diffusion coefficient (Section 5.6) ... 

To obtain a more complete description, we need to find an analytic expression for the pre-exponential factor Dq of the diffusion coefficient by considering the microscopic mechanism of diffusion. The most straightforward approach, which neglects correlated motion between the ions, is given by the random-walk theory. In this model, an individual ion of charge q reacts to a uniform electric field along the x-axis supplied, in this case, by reversible nonblocking electrodes such that dCj(x)/dx = 0. Since two... 

In the random-walk model, the individual ions are assumed to move independently of one another. However, long-range electrostatic interactions between the mobile ions make such an assumption unrealistic unless n is quite small. Although corrections to account for correlated motions of the mobile ions at higher values of n may be expected to alter only the factor y of the pre-exponential factor Aj., there are at least two situations where correlated ionic motions must be considered explicitly. The first occurs in stoichiometric compounds having an = 1. but a low AH for a cluster rotation the second occurs for the situation illustrated in Fig. 3.6(c). 

The amount of decrease of the resonance width may be simply estimated in the following way 50). Let the motion of the spins be characterized by a time tc, that is t is the average time a spin stays in a definite environment or the correlation time for the motion. This environment will cause a difference 5w in the precessional frequency of the spin which may be positive or negative from some average value to. During the time Tc the spin acquires a phase angle 60 = TcSu in addition to that acquired by the uniform precession at to. If we consider the motion to be a random walk process (51), after n such intervals during a time t the mean square phase acquired will be... 

Diffusion can be modeled as a random walk in three dimensions, and the value of the diffusion coefficient can be computed by the correlation formula... 

The parameter ( r(f) - r(O)p) is the ensemble average of the square of the distances between the initial and the later positions for each particle in the system, and the factor of 3 takes care of the three-dimensional nature of the random walk. Given an ensemble of M particles, the correlation function of properties r t) and r(0) is given by... 

Ramsey theory, 22 201-204 Random-fragmentation model, Szilard-Chalmers reaction and, 1 270 Random-walk process, correlated pair recombination, post-recoil annealing effects and, 1 288-290 Rare-earth carbides, neutron diffraction studies on, 8 234-236 Rare-earth ions energy transfer, 35 383 hydration shell, 34 212-213 Rare gases... 

A point that has not been investigated is the possibility of considering u(k) a coloured noise instead of white noise, and therefore a non diagonal E. For example, the choice of a tridiagonal Ey would imply the assumption of u(k) a random walk process. On the one hand, by imposing a correlation among successive values of u(k), the flexibility of the output is reduced, and for example a delta function could not be recuperated. On the other hand, smoother outputs and better solutions could be obtained if good "a priori" estimations of the real autocorrelations of u(k) could be provided. 

Atomic jumps in random walk diffusion of closely bound atomic clusters on the W (110) surface cannot be seen. A diatomic cluster always lines up in either one of the two (111) surface channel directions. But even in such cases, theoretical models of the atomic jumps can be proposed and can be compared with experimental results. For diffusion of diatomic clusters on the W (110) surface, a two-jump mechanism has been proposed by Bassett151 and by Cowan.152 Experimental studies are reported by Bassett and by Tsong Casanova.153 Bassett measured the probability of cluster orientation changes as a function of the mean square displacement, and compared the data with those derived with a Monte Carlo simulation based on the two-jump mechanism. The two results agree well only for very small displacements. Tsong Casanova, on the other hand, measured two-dimensional displacement distributions. They also introduced a correlation factor for these two atomic jumps, which resulted in an excellent agreement between their experimental and simulated results. We now discuss briefly this latter study. 

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]. 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

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