Simple random walks

One of the simplest idealizations of a flexible polymer chain consists in replacing it by a random walk on a periodic lattice, as shown in Fig. I.l. The walk is a sucession of N steps, starting from one end (a) and reaching an arbitrary end point ( ). At each step, the next jump may proceed toward any of the nearest-neighbor sites, and the statistical weight for all these possibilities is the same. The length of one step will be called a. 

This description was apparently initiated by Orr in 1947. It is convenient from a pedagogical point of view all chain properties are easy to visualize. For instance, the entropy. S(r) associated with all chain conformations starting from an origin (r = 0) and ending at a lattice point r, is simply related to the number of distinct walks 92Ar(r) going from (0) to (r) in N steps  

The main features of the number 9 w(r) are discussed now. First, the total number of walks is simple to compute if each lattice site has z neighbors, the number of distinct possibilities at each step is z, and the total number is 

The factors N arise ftom normalization conditions. We purposely do not write the complete numerical value of the constant in front of eq. (1.6) these constants would obscure our ruguments. They can be found in standard textbooks on statistics.  

6) gives a formula for the entropy of the chain at fixed elongation 

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In this relatively simple random walk model an ion (e.g., a cation) can move freely between two adjacent active centres on an electrode (e.g., cathode) with an equal probability A. The centres are separated by L characteristic length units. When the ion arrives at one of the centres, it will react (e.g., undergoes a cathodic reaction) and the random walk is terminated. The centres are, therefore absorbing states. For the sake of illustration, L = 4 is postulated, i.e., Si and s5 are the absorbing states, if 1 and 5 denote the positions of the active centres on the surface, and s2, s3, and s4 are intermediate states, or ion positions, LIA characteristic units apart. The transitional probabilities (n) = Pr[i-, —>, Sj in n steps] must add up to unity, but their individual values can be any number on the [0, 1] domain. 

Figure 2.7 Simple random walk in 3-D showing equal magnitude steps r, to //. The end-to-end vector is R...

To show the relationship between pn(m) expressing the probabilities of numbers and p x) describing a continuous spatial distribution of a quantity like concentration, we make use of the analogy between the integers n and m, which describe the simple random walk model shown in Fig. 18.1, and the time and space coordinates t and x, that is t = n At and x = m Ax. The incremental quantities, At and Ax, are characteristic for random motions the latter is the mean free path which is commonly denoted as X = Ax, the former is associated with the mean velocity ux= Ax/At = XIAt. Thus, we get the following substitution rules ... 

Individual steps of restricted walks are no longer independent, since the restriction introduces a correlation. However, this correlation is short range in character and falls off exponentially with increasing separation of steps. The short range correlation is insufficient to change the characteristic features of the walk from those of a simple random walk. 

In order to estimate the end-to-end distance r0 we assume, as a first approximation, that the chain segments can move freely with respect to each other in all directions. The chain contains n segments, each with a length bo. The next simplification is, that we consider the chain in two dimensions. We now have a simple random walk ( drunk man s walk ) problem. We situate one chain end in the origin of an x-y coordinate system and we build the chain step by step with randomly chosen angles tp (see Figure 2.7). The position of the other end is than given by... 

The same result is obtained if one considers a simple random walk in two dimensions, i.e., the walk is performed on a 2-dimensional lattice. Here, the walker (particle) moves either vertically or horizontally at each time step (t0 units of time) with equal probabilities. Two configurations for eight-time-step... 

In a second model II, more emphasis is given to the motion toward the output and less to the other directions. The probabilities for motion in the different directions are now defined differently. While in the simple random walk the probability for motion in a specific direction is 1/z, here the probability for motion in the output direction is (1/z) + e, while the probability in any of the other five directions is... 

Enumeration of Random Walks. Counting simple random walks was reported by Klein et al.216 In parallel to the generation of walks from the powers of the adjacency matrix (see, for example, our Report in ref. 2) that may be viewed as an identification of the distribution for equipoise random walks, Klein et al.216 generated the distribution for simple random walks by powers of a Markov matrix M with elements that are probabilities for associated... 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

The continuous limit of a simple random walk model leads to a stochastic dynamic equation, first discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. 

We define the variable of interest as Xj, where j = 0,1,2,... indexes the time step, and in the simplest model a step is taken in each increment of time, which for convenience we set to one. The operator B lowers the index by one unit such that BXj = Xj j so that a simple random walk can be written... 

In the simple random walk the steps are statistically independent of one another. The simplest generalization of this model is to make each step dependent on the preceding step in such a way that the second moment is... 

In the science of complexity the system response X(t) is expected to depart from the totally random condition of the simple random walk model, since such fluctuations are expected to have memory and correlation. In the physics literature, anomalous diffusion has been associated with phenomena with longtime memory such that the autocorrelation function is... 

The time interval between steps were assumed to be a constant finite value in the simple random walk model. If, however, we explicitly take the limit where the time interval vanishes, then the discrete walk is replaced with a continuous rate. We begin our discussion of the scaling of statistical processes by considering one of the simplest stochastic rate equations and follow the development of Allegrini etal. [49]. 

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