Random Walk in One Dimension

The use of distribution functions is well illustrated by an important and classical problem, the random walk/ Let us assume that a marker is placed at the origin x = 0) of the x axis. We shall now allow it to move one step in the +x direction with a probability p and one step in the negative direction of the x axis with a probability q, such that p + g = 1. We are now concerned with the position of the marker with respect to its starting position after n moves have been made. If p = g, Ve would intuitively guess that the marker would be near its starting position, there being equal probability of plus and minus moves. 

We define P x,n) as the probability of finding the marker at the position X after n moves have been made. P(x,n) is a bounded function, since [a ] cannot exceed +n. (Note that x and n are both integers.) 

Now the chance that the marker will be at the position x after n moves is exactly the chance that the total number of positive moves exceeds the total number of negative moves by the quantity x  

But the chance that there will be positive moves and n moves is given by the well-known binomial formula  

P XjU) is the famous binomial or Bernouilli distribution, and it has a very useful asymptotic form when n is a very large number such that, as w — 00, O. We can then write 

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That volume bounded by the distance away from the electrode over which a redox-active species can diffuse to the electrode surface, within the timescale of the experiment being undertaken. From considerations of random walk in one dimension it can be shown that the distance / which a species moves in a time / is given by ... 

A very important application of the Markov dynamics is random walk. In the special case of random walk/(x) = 0 and g(x) = 1, then the diffusion equation for a random walk in one dimension is... 

Let us take a random walk in one dimension, stepping a distance / every t time interval. After n such steps, the elapsed time is t = n(5r). The mean-squared displacement during the random walk is... 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

In general, a dispersed particle is free to move in all three dimensions. For the present, however, we restrict our consideration to the motion of a particle undergoing random displacements in one dimension only. The model used to describe this motion is called a onedimensional random walk. Its generalization to three dimensions is straightforward. 

According to the model of random walk in three dimensions, the diffusion coefficient of a molecule i, can be expressed as one-third of the product of its mean free path A, and its mean three-dimensional velocity u, (Eq. 18-7a). In the framework of the molecular theory of gases, u, is (e.g., Cussler, 1984) ... 

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... 

Quantification of polymer entropies is done through the statistics of random walk (Flory, 1953). The model is based on a drunk trying to walk in one dimension Because of his state, the next step the drunk takes could be to the right or to the left with equal probability, but his stride rranains of idmtical length and at every step he waits for the same length of time. The key probability density function is that of end-to-end displacement x, that is, the distance betweai the beginning and the end, which for a linear polymer in one dimraision is Ganssian ... 

Random Walk A linear flexible polymer chain can be modeled as a random walk. The concept of the random walk gives a fundamental frame for the conformation of a polymer chain. If visiting the same site is allowed, the trajectory of the random walker is a model for an ideal chain. If not allowed, the trajectory resembles a real chain. In this section, we learn about the ideal chains in three dimensions. To familiarize ourselves with the concept, we first look at an ideal random walker in one dimension. 

Let s start with a walk in one dimension. Each step has unit length in either the +x or -x direction, chosen randomly with equal probability. Steps are independent of each other. In a given walk of N total steps, m steps are in... 

The distribution for a random chain in one dimension (random walk) may be expressed in the form of Bernoulli s equation ... 

Fig. 8.11 One possible path of a random walk in three dimensions. In this general case, the step length is also a random variable. (Available at http // www.ki.inf.tu-dresden. de/ fritzke/research/TS/ exampleLhtml.)...

These sets of equations also describe the classic case of Brownian motion or random walk. The initial condition is that all M particles were at the central point, and then spread in one dimension (along a line), two dimensions (along a... 

Random walks on square lattices with two or more dimensions are somewhat more complicated than in one dimension, but not essentially more difficult. One easily finds, for instance, that the mean square distance after r steps is again proportional to r. However, in several dimensions it is also possible to formulate the excluded volume problem, which is the random walk with the additional stipulation that no lattice point can be occupied more than once. This model is used as a simplified description of a polymer each carbon atom can have any position in space, given only the fixed length of the links and the fact that no two carbon atoms can overlap. This problem has been the subject of extensive approximate, numerical, and asymptotic studies. They indicate that the mean square distance between the end points of a polymer of r links is proportional to r6/5 for large r. A fully satisfactory solution of the problem, however, has not been found. The difficulty is that the model is essentially non-Markovian the probability distribution of the position of the next carbon atom depends not only on the previous one or two, but on all previous positions. It can formally be treated as a Markov process by adding an infinity of variables to take the whole history into account, but that does not help in solving the problem. 

Many years ago Polya [20] formulated the key problem of random walks on lattices does a particle always return to the starting point after long enough time If not, how its probability to leave for infinity depends on a particular kind of lattice His answer was a particle returns for sure, if it walks in one or two dimensions non-zero survival probability arises only for the f/iree-dimensional case. Similar result is coming from the Smoluchowski theory particle A will be definitely trapped by B, irrespectively on their mutual distance, if A walks on lattices with d = 1 or d = 2 but it survives for d = 3 (that is, in three dimensions there exist some regions which are never visited by Brownian particles). This illustrates importance in chemical kinetics of a new parameter d which role will be discussed below in detail. 

Diffusion is the random migration of molecules or small particles arising from motion due to thermal energy. A very simple derivation of Fick s first law, based on the random walk problem, can be obtained in one dimension. In this case, Jx(x, t), that is, the number of particles, N, that move across unit area, A, in unit time, x, can be defined as... 

RNA polymerase holoenzyme has lower affinity for nonspecific DNA sequences than for promoter sequences. The nonspecific affinity, however, allows the enzyme to bind to random-sequence DNA and then slide along the molecule in a unidimensional random walk until it encounters a promoter sequence, for which its binding affinity is higher. Diffusion in one dimension is much faster than diffusion in three dimensions, thereby explaining the observed rapid rate constant for the binding of RNA polymerase holoenzyme to promoter sequences. If one measured the encounter of the polymerase with the nonspecific regions of the DNA rather than with promoter sequences, the value of the rate constant would be much lower and would fit our expectations for a three-dimensional, diffusion-limited reaction between macromolecules. 

We begin with a simple example of a particle performing a discrete-time random walk (DTRW) in one dimension. Assume that it is initially at point 0. The random walk can be defined by the stochastic difference equation for the particle position... 

Diffusion results from Brownian motion, the random battering of a molecule by the solvent. Let s apply the one-dimensional random walk model of Chapter 4 (called random flight, in three dimensions) to see how far a peirticle is moved by Brownian motion in a time t. A molecule starts at position x = 0 at time t = 0. At each time step, assume that the particle randomly steps either one unit in the +x direction or one unit in the -x direction. Equation (4.34) gives the distribution of probabilities (which we interchangeably express as a concentration) c(x, N) that the particle will be at position x after N steps,... 

An early attempt to describe effects of coupled motions in relaxing dipole orientations was the defect diffusion model of Glarum (58) in which it is assumed that a dipole relaxes to a randomorientation either by exponential decay or by arrival of a nearest defect which completely destroys the correlation existing just before it arrives assumed to be determined by a random walk or diffusion process in one dimension, but otherwise unspecified. With T as the rotational relaxation time and as the diffusional time use of the diffusion solution for the absorbing wall problem (59) gives the frequency domain relaxation function... 

Einstein interpreted diffusion as being a result of the random thermal motion of molecules. Such a random motion is caused by fluctuations in pressure in a liquid. Thus, diffusion is closely related to Brownian motion. The Brownian motion consists of zigzag motion in aU directions. It is a random walk, as discussed in Chapter 5, and is described by the parameter (x ), the square mean displacement. The equation of motion in one dimension for the Brownian motion of a particle in solution is given by... 

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