Random walk one-dimensional

This is an important problem first described by Lord Rayleigh in 1919. It is important because it finds applications in numerous scientific fields. We present a simple illustration of the problem. 

Consider a drunkard walking on a narrow path, so that he can take either a step to the left or a step to the right. The path can be assumed one-dimensional and the size of each step always the same. Let p be the probability of a step to the right and q the probability of a step to the left, so that p -I- = 1. If the drunkard is completely drunk, we can assume that p = q = /2. This is a one-dimensional random walk (Fig. 2.6). 

Consider bJ steps, to the right and ni to the left, so that hr +ni = N. If m is the final location, so that nR — ni =m, what is the probability of any position m after N steps Since any step outcome (right or left) is independent of the rest, the probability of a particular sequence of nR and ni independent steps is equal to the product of probabilities of each of the steps, i.e., 7 . Because the specific sequence of steps is 

Consider the following question what is the width of the distribution with respect to N for large A In other words, where is the drunkard likely to be after N steps Assuming A oo, we can use Stirling s approximation and write 

Therefore the binomial distribution becomes a Gaussian distribution for a large number of steps. This distribution peaks at w = 0. 

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As a consequence of these various possible conformations, the polymer chains exist as coils with spherical symmetry. Our eventual goal is to describe these three-dimensional structures, although some preliminary considerations must be taken up first. Accordingly, we begin by discussing a statistical exercise called a one-dimensional random walk. 

For a one-dimensional random walk, the probability of n j heads after n moves is supplied by application of the bionomial distribution formula ... 

This result enables us to calculate the probability of any specified outcome for the one-dimensional random walk. We shall continue to develop this one-dimensional relationship somewhat further, since doing so will produce some useful results. 

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... 

Consider a one-dimensional random walk, with a probability p of moving to the right and probability q = 1 — p of moving to the left. If p = g = 1/2, the distribution has mean p = 0 and spreads in time with a standard deviation a = sJijA. In general, though, p = (p — g)t and a = y pgt. In particular, as p moves away from the center value 1/2, the center of mass of the system Itself moves with velocity P = p — q. 

Parameter setup for Example 2.1A. One-dimensional random walk... 

Figure S5.4 Three one-dimensional random walks each individual walk, (a)-(c), gives the position of the diffusing atom, x after the Mh jump. The final positions reached are (a) x = 0, (b), x = —2a, and (c) x = —8a.

Fick s (continuum) laws of diffusion can be related to the discrete atomic processes of the random walk, and the diffusion coefficient defined in terms of Fick s law can be equated to the random-walk displacement of the atoms. Again it is easiest to use a one-dimensional random walk in which an atom is constrained to jump from one... 

The 67-fold amplification obtained for polymer 3 is restricted by an inherent limitation of the wired in series design. As the exciton travels in a one-dimensional random walk process down the polymer chain, it has equal opportunity to visit a preceding or an ensuing receptor. This represents 134 random stepwise movements for 134 phenylene ethynylene units, and so much of the receptor sampling by the exciton is redundant. Increasing the efficiency of receptor sampling requires maximization of the number of different receptors that an exciton can visit throughout its lifetime. To achieve this end we extended the polymer sensor into two dimensions by use of a thin film and thereby increased the sensitivity. 

A one-dimensional random walk is not necessarily symmetric with respect to jumps toward the right and toward the left. If the chemical potential gradient is sufficiently weak we may still approximate the jump length distribution by an exponentially decaying function, but distinguish that toward the right from that toward the left. 

In a one-dimensional random walk with excursion from the origin, the mth moment of the jump length distribution can be derived from... 

This corresponds to a one-dimensional random walk with an absorbing barrier, with the transition probabilities calculated from quantum mechanics. The time dependent distribution of the reactant molecules among... 

Consider a one-dimensional random walk on a lattice of N sites j = 0,1,2,... N — 1 with absorbing barriers at j = — 1 and j = N. Let fj t denote the probability that the walker is at site j after / steps, fjfi being the initial probability distribution. Suppose that P(j i) = Pt-j is the probability of making a step from site j to site i (ignoring the barriers). Then, taking account of the absorption, the distribution after / steps is easily seen to be given by... 

Exercise. Show for the one-dimensional random walk with persistence that the distribution approaches a Gaussian. 

Equation 7.46 demonstrates that if each jump of a walk occurs randomly (i.e., is uncorrelated), the average displacement is zero and the center of mass of a large number of individual random jumpers is not displaced. Equation 7.47 gives the mean-square displacement of a random walk, NT(r2). Although Eqs. 7.46 and 7.47 were derived here for one-dimensional random walks, both are valid for two- and three-dimensional random walks. 

Figure 2.1 A one-dimensional random walk of particles placed at z = 0 at t = 0. The particles occupy only the positions 0, , 28, 3, FAS.

The flat distribution implies that a free one-dimensional random walk in the potential energy space is realized in this ensemble. This allows the simulation to escape from any local minimum-energy states and to sample the configurational space much more widely than the conventional canonical MC or MD methods. 

We can alternately exchange pairs of neighboring temperature values and pairs of neighboring pressure values during the replica-exchange simulation. Moreover, if we fix the temperature, we can have only the pressure-exchange process as a special case, which yields a one-dimensional random walk in the volume space. 

Note that in Eq. 6.34 the mean-square displacement is used, rather than the root-mean-square displacement. For a one-dimensional random walk, the mean-square displacement is given by 2Dt, and for a two-dimensional random walk, 4Dt. Since the jump distance (a vector) is A, if the jump frequency is now defined as F = n/t (the average number of jumps per unit time), then on combining Eq. 6.33 and 6.34 gives ... 

For the sake of simplicity, the special case of a one-dimensional random walk will be considered. The sailor starts off from X = 0 onthexaxis.Hetossesacoimheads—he moves forward in the positive x direction, tails—he moves backward. Since, for an honest coin, heads are as likely as tails, the sailor is equally as likely to take a forward step as a backward step. Of course, each step is decided on a fresh toss and is uninfluenced by the results of the previous tosses. After allowing him N steps, the distance x from the origin is noted (Fig. 4.13). Then the sailor is brought back to the bar (x = 0) and started off on another try of N steps. 

It has further been shown (Section 4.2.6) that in the case of a one-dimensional random walk, depends on time according to the Einstein-Smoluchowski equation... 

The numerical coefficient 5 has entered here only because the Einstein-Smoluchowski equation = 2Dt for a one-dimensional random walk was considered. In general, it is related to the probability of the ion s jumping in various directions, notjust forward and backward. For convenience, therefore, the coefficient will be taken to be unity, in which case... 

The microscopic view of diffusion starts with the movements of individual ions. Ions dart about haphazardly, executing a random walk. By an analysis of one-dimensional random walk, a simple law can be derived (see Section 4.2.6) for the mean square distance-cc traversedby an ionin atimef. This is theEinstein-Smoluchowski equation... 

In the one-dimensional random-walk problem, the expression for is found by mathematical induction as follows. Consider that after N - 1 steps, the sailor has progressed a distance If he takes one more step, the distance from the origin will be either... 

This equation has been derived for a one-dimensional random walk, but it can be shown to be valid for three-dimensional random flights, too. 

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