Random walk step, effective

Jackson, Shen and McQuarrie (95) use an alternative method for the calculation of the obstructional effect of the molecular volume. Their starting point is the assumption that the obstruction becomes anisotropic as soon as the network is stretched. Rather than using probabilities of 1/3 for the random walk steps in any of the three cartesian directions, they derive for unidirectional strain... 

If one can define fully extended length, or contour length, L, of the molecule, then one can define yet another effective random-walk step size, or Kuhn length bx, by... 

Several additional characteristic lengths will also be used in this book. One is the effective random-walk step, which is defined by use of Eq. 2.1, with the number of freely-jointed segments set equal to the actual number of backbone bonds, n ... 

Random walks are often called Markov random walks. A Markov chain is a sequence of random events described in terms of a probability that the event under scmtiny evolved from a defined predecessor. In effect there is no memory of any preceding step in a Markov chain. Hidden Markov processes involve some short-term memory of preceding steps. 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

The erratic motion of a chromatographically migrating molecule resembles a random-walk process. In order to apply random-walk ideas to chromatography, we must identify the effective step lengths and step numbers associated with the molecular migration. This is the main task to follow. 

Figure 10 A schematic illustration of the effect of the presence of a step on the diffusion of a surface vacancy, (a) Schematic topography, with a step in the middle, (b) The recombination probability depends logarithmically on the distance, (c) Random walks that bring the vacancy far from the step will result on average in a much larger number of encounters with a tracer atom on the terrace than shorter random walks.

If x —> 0, then p(t = 1 /2, and we have the usual random walk process (i.e., initial effects are ignored). If x 0, then we have a process with memory, i.e. the particle remembers its state at the previous step (the position and direction of motion). Stability embodies the fact that po differs but little from 1. Equations (552) and (553) are now substituted into the recurrence equations (551) yielding... 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

Calculations of the mean energy of atoms in a cascade show that most recoils are produced near the minimum energy necessary to displace atoms, Ed. Due to the low-energy stochastic nature of these displacement events, the initial momentum of the incident particle is soon lost, and the overall movement of the atoms in a collision cascade becomes isotropic. This isotropic motion gives rise to an atomic redistribution that can be modeled as a random-walk of step size defined by the mean range of an atom with energy near Ed. The effective diffusivity, Dcas, for a collision cascade-induced random-walk process is expressed in the diffusion equation as (Andersen 1979)... 

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