Problems with Random Walks

The dimensions of a polymer chain in solution are important to the rheological properties of the system. More specific to the question of colloidal stability, however, such dimensions play a vital role in the ability of an adsorbed polymer to stabilize (or destabilize) a lyophobic colloid as discussed below and in Chapter 10. 

Macromolecular species have played an indispensable role in the stabilization of colloidal systems since the first prelife protein complexes came into existence. We (humans) have consciously (although usually without knowing why) been making use of their properties in that context for several thousand years. Today macromolecules play a vital role in many important industrial processes and products, including as dispersants, stabilizers, and flocculants as surface coatings for protection, lubrication, and adhesion for the modification of rheological properties and, of course, for their obvious importance to biological processes. 

In order to understand the role of polymers in their various surface and colloidal applications, it is necessary to understand when, where, why, and how they adsorb at interfaces. While the interactions that control polymer adsorption at the monomer level are the same as those for any monomolecular species, the size of the polymer molecule introduces many complications of analysis that must be treated in a statistical manner, which means that we seldom really know what the situation is but must make educated guesses on the basis of the best available evidence. 

FIGURE 14.2. When polymers adsorb at a surface, only a fraction of the monomer units need be involved for strong binding to occur. As a result, the configurations of the adsorbed chains can vary from the least likely end-group attachment (a) to the more random (and more probable) attachments as loops, trains, and tails (b). 

FIGURE 14.3. Because of the nature of polymer adsorption, monolayer coverage is almost assured and high-affinity isotherms are the norm. 

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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. 

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. 

Such a chain is also called a chain of random (free) walk, since Kquation 89 matches the distribution calculation of random walk probability of a structural element in space with 71 jumps (steps) from one position to another vvith the length of each jump (step) defined by the probability distribution t Rj) (Flory, 1953 Yamakawa, 1971). Hence. Kquation 89 relates the statistics of polymer chains to the problems of random walk and diffusion. As the diffusion equation is mathematically similar to Schrddinger s one, the common ideology and common mathematical solutions unite the conformational tasks of a polymer chain, the state of quantum-mechanical systems, and the field theory. 

The one-dimensional random walk of the last section is readily adapted to this problem once we recognize the following connection. As before, we imagine that one end of the chain is anchored at the origin of a three-dimensional coordinate system. Our interest is in knowing, on the average, what will be the distance of the other end of the chain from this origin. A moment s reflection will convince us that the x, y, and z directions are all equally probable as far as the perfectly flexible chain is concerned. Therefore one-third of the repeat units will be associated with each of the three perpendicular directions... 

Equation (1) can be used in a general way to determine the variance resulting from the different dispersion processes that occur in an LC column. However, although the application of equation (1) to physical chemical processes may be simple, there is often a problem in identifying the average step and, sometimes, the total number of steps associated with the particular process being considered. To illustrate the use of the Random Walk model, equation (1) will be first applied to the problem of radial dispersion that occurs when a sample is placed on a packed LC column in the manner of Horne et al. [3]. 

A problem with this model is that very early values and recent values have an equal contribution to the precision of the forecast. The random walk model provides a good forecast of trend but is less efficient with cyclical and seasonal variations. 

An approach that does not suffer from such problems is the ABF method. This method is based on computing the mean force on and then removing this force in order to improve sampling. This leads to uniform sampling along . The dynamics of corresponds to a random walk with zero mean force. Only the fluctuating part of the instantaneous force on remains. This method is quite simple to implement and leads to a very small statistical error and excellent convergence. 

A second problem with the random walk model concerns the interaction between segments far apart along the contour of the chain but which are close together in space. This is the so-called "excluded volume" effect. The inclusion of this effect gives rise to an expansion of the chain, and in three-dimensions, 2 a, r3/5 (9), rather than the r dependence given in equation (I). 

The mathematical translation of the plane-source problem is as follows. Initially, there is a finite amount of mass M but very high concentration at a = 0, i.e., the density or concentration at a = 0 is defined to be infinite (which is unrealistic but merely an abstraction for the case in which initially the mass is concentrated in a very small region around a = 0). The initial condition is not consistent with that required for Boltzmann transformation. Hence, other methods must be used to solve the case of plane-source diffusion. Because this is the classical random walk problem, the solution can be found by statistical treatment as the following Gaussian distribution ... 

An ancient but still instructive example is the discrete-time random walk. A drunkard moves along a line by making each second a step to the right or to the left with equal probability. Thus his possible positions are the integers — oo < n < oo, and one asks for the probability pn(r) for him to be at n after r steps, starting from n = 0. While we shall treat this example in IV.5 as a stochastic process, we shall here regard it as a problem of adding variables. 

Exercise. Formulate the random walk problem in 1.4 as a Markov chain. There is no normalized ps owing to the infinite range. Remedy this flaw by considering a random walk on a circular array of N points, such that position JV + 1 is identical with 1. Find ps for this finite Markov chain. Is it true that every solution tends to ps ... 

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. 

However, in this case it is no longer possible to prove that J = 0. One cannot exclude the possibility of a constant flow from — oo to +00, as in the asymmetric random walk. Such solutions would describe, for instance, diffusion in an open system, such as diffusion through a medium between two reservoirs with different densities. The stationary solution is no longer unique, but depends on the current J, which depends on additional information concerning the physical problem one is dealing with. See Exercise. 

Exercise. For the infinite symmetric random walk an explicit solution of an absorbing boundary problem can be obtained by the reflection principle. Solve the M-equation with initial condition p (0) = 5wm — 5w m. The solution pn(t) for n> 0 obeys... 

Exercise. Write the corresponding equations for the case of a left exit point L < m. Exercise. Solve the first-passage problem for the simple symmetric random walk. Show that any site R is reached with probability nR m= 1, but that the mean first-passage time is infinite. 

From a formal point of view, (2.2.53) describes random walks on a onedimensional lattice of enumerated sites. Unlike standard problems with constant transition probabilities between sites, in (2.2.53) these probabilities depend on a site number and are essentially non-linear. Figure 2.11 shows possible transitions in the model under consideration and the relevant transition rates. 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

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... 

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