The Random Walk Problem

The other problem we will discuss is the most likely distribution of a fixed amount of energy between a large number of molecules (the Boltzmann distribution). This distribution leads directly to the ideal gas law, predicts the temperature dependence of reaction rates, and ultimately provides the connection between molecular structure and thermodynamics. In fact, the Boltzmann distribution will appear again in every later chapter of this book. 

One of the most basic problems in statistics is called the random walk problem. Suppose you take a total of N steps along a north-south street, but before each step you flip a coin. If the coin comes up heads, you step north if the coin comes up tails, you step south. What is the probability that you will end up M steps north of your starting point (in other words, the probability that you will get M more heads than tails)  

If you toss N coins, the total number of distinct possible outcomes is 2N since there are two ways each coin can fall. For example, if N = 3 the eight outcomes are 

The number of these outcomes which have exactly heads and n tails is given by the binomial distribution  

The probability P nu r) of getting exactly rtu heads and nT tails is given by dividing Equation 4.1 by the total number of possible outcomes 2N. 

Starting from Z = 0, we want to know the probability / ( , z) = P(n,N, Az) that the particle has made a net number n of steps to the right (a negative n implies that the particle has actually moved to the left) after time Z = AAz. In other words, for a particle that starts at position = 0 we seek the probability to find it at position n (i.e. at distance A/ from the origin) at time Z, after making at total of N steps. An equation for P(p, t) can be found by considering the propagation from time t to Z -j- AZ  

Note that in (7.3) time is a continuous variable, while the position n is discrete. We may go into a continuous representation also in position space by substituting 

Here P(x, t) may be understood as the probability to find the particle in an interval of length Ax aboutx. Introducing the density f (x, Z) so that P x, t) =f x, Z) Ax and expanding the right hand side of (7.4) up to second order in Ax we obtain 

Note that even though in (7.5) we use a continuous representation of position and time, the nature of our physical problem implies that Ax and AZ are finite, of the order of the mean free path and the mean free time, respectively. 

To get a feeling for the nature of the solution of Eq. (7.5) consider first the case D = Q (that is also obtained if we truncate the expansion that led to (7.5) after first order). The solutions of the equation df /dt = —vdf/dx have the form f(x,t = f(x — vZ), that is, any structure defined by /moves to the right with speed V (drift velocity. This behavior is expected under the influence of a constant force that makes kr and ki different. The first term of (7.5) reflects the effect of the systematic motion resulting from this force. 

2) the terms that add to Pin, t) on the right hand side result from the walk. Thus, for example, krAtPin—, /)is the increase in P(n, t) due to the possibility of a jump from position n-1 to position n during a time interval At, while —krAtP n, t) is the decrease in P n, t) because of transition from n to n+l in the same period. Rearranging Eq. (7.2) and dividing by At we get, when At 0, 

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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 ... 

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... 

The fluctuating motion of suspended PS spheres can be related to the random-walk problem. Einstein was the first to show the connection between this random-walk Brownian motion and Pick s laws of diffusion. The resulting Stokes-Einstein relation is °... 

Second, the distance traveled, l, by a particle diffusing in a medium in a time t goes as tllz. This is, of course, the random walk problem. When we applied this to a polymer chain we were concerned with the distance between the ends in a walk of N steps here we are concerned with the distance traveled after a time t. Before we had 112 <- N, here we have 1/2 - t, or, more formally (Equation 13-68) ... 

If we assume that each of the individual errors may be assigned an equal probability of being positive or negative and all act independently, then the question of finding the distribution of possible errors is given precisely by the answer to the random walk problem with varied step. That is, the chance of making an error x is given by... 

The aim now is to seek a quantitative description of ionic random-walk movements. There are many exotic ways of stating the random-walk problem. It is said, for example, that a drunken sailor emerges from a bar. He intends to get back to his ship, but he is in no state to control the direction in which he takes a step. In other words, the direction of each step is completely random, all directions being equally likely. The question is On the average, how far does the drunken sailor progress in a time r ... 

Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient in order to determine it at any later time. The random walk problem discussed in Section 7.3 is an example of a stochastic Markov process. 

Unlike in the random walk problem, the transition rate out of a given state n depends on n The probability per unit time to go from n+1 to n is A (/j+1), and the probability per unit time to go from n to n — 1 is kn. The process described by Eq. (8.83) is an example of a birth-and-death process. In this particular example there is no source feeding molecules into the system, so only death steps take place. 

As an example without rigorous mathematical justification consider the master equation for the random walk problem... 

The basic idea is that the channels and polyhedral cavities of a zeolite crystal can be placed in correspondence with the bonds and vertices of a d — 3 dimensional, finite Cartesian lattice. Hence, the problem of determining the trajectories of guest molecules through zeolite crystals of finite extent is modeled by studying the random walk problem on, here, finite lattices of two different topologies. 

Equation 2G.19 is the basis for the Gaussian distribution. Its application to the random walk problem illustrates its usefulness. Setting... 

If in analogous fashion we introduce a scaled size parameter, r = x/xmax for the random walk problem and normalize the squared radii distribution Pn B ) of Eq.(2) at its maximum, the result takes the same form as Eq.(9) for the hydrogenic radial probability distribution, except that D is replaced by y. With this change. Figure 2 applies as well to the random walk problem. This illustrates the generic character of dimensional scaling. 

The dimensions of a linear flexible molecule in solution can be readily calculated using an extension of the random walk concept first introduced to describe the movement of gas molecules. The influence of chain connections, bond lengths, bond angles and short- and long-range steric interactions can all be introduced into the calculation. The random walk problem readily lends itself to computer-based numerical methods and it is possible to generate pictorial representations which reflect the chemical constitution of the backbone polymer and the interactions of the polymer with itself and the solvent. 

In a previous section reference was made to the random walk problem (Montroll and Schlesinger [1984], Weiss and Rubin [1983]) and its application to diffusion in solids. Implicit in these methods are the assnmptions that particles hop with a fixed jump distance (for example between neighboring sites on a lattice) and, less obviously, that jumps take place at fixed equal intervals of time (discrete time random walks). In addition, the processes are Markovian, that is the particles are without memory the probability of a given jump is independent of the previous history of the particle. These assumptions force normal or Gaussian diffusion. Thus, the diffusion coefficient and conductivity are independent of time. 

In the absence of microscopic characterization, chain defects are more difficult to model. Many kinds of defects or imperfections may exist that disturb the spin motion. So far, three parameters have been introduced to describe the spin motion the intra- and interchain diffusion rates and and the chain length N. Accounting for the influence of possible defects leads to the introduction of additional parameters. Many theoretical papers have been produced on the random walk problem in the presence of impurities, traps, or random hopping rates. It is out of the scope of this chapter to present a detailed discussion of this topic. Let us just envisage briefly a few typical cases. 

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