Random Walks and Their Applications: Widely used as mathematical models, random walks play an important role in several areas of physics, chemistry, and biology

Sigma Xi, The Scientific Research Society

Random Walks and Their Applications: Widely used as mathematical models, random walksplay an important role in several areas of physics, chemistry, and biologyAuthor(s): George H. WeissSource: American Scientist, Vol. 71, No. 1 (January-February 1983), pp. 65-71Published by: Sigma Xi, The Scientific Research SocietyStable URL: http://www.jstor.org/stable/27851819 .

Accessed: 22/10/2014 07:10

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Sigma Xi, The Scientific Research Society is collaborating with JSTOR to digitize, preserve and extend accessto American Scientist.

http://www.jstor.org

This content downloaded from 67.210.62.178 on Wed, 22 Oct 2014 07:10:51 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=sigmaxi

http://www.jstor.org/stable/27851819?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Random Walks and Their

Applications

George H. Weiss

Widely used as mathematical models, random walks play an important role in several areas of physics, chemistry, and

biology

The simplest example of a random walk is that of a drunkard moving on a one-dimensional array of regu larly spaced points. At regular intervals he flips a coin and makes one step to the right or left, depending on whether the coin comes up heads or tails. At the end of n steps the drunkard can be at any one of 2n +1 sites and the probability that he is at any site can be calculated.

Quite generally, one can define a random walk as a sum of random variables. In the example of the drunkard's walk each of the random variables can take on the values ?1 with probability V2. Obviously, more sophisticated examples can be devised, some of which will be de scribed in this article.

The notion of the random walk in the simple form

suggested above is implicit in the studies of gambling done by Ferm?t and Pascal in the seventeenth century (2). Since that time the mathematical problems suggested by random walks, which now make up an important part of probability theory (2), have been analyzed at great length: a recent bibliography of articles related to ran dom walks contains over 800 references and is far from

complete (3). The random walk, although a simple concept, has come to play an important role in several areas of scientific investigation. In this article I will re view some of the ideas and problems considered to be of interest in the analysis of random walks, and discuss several of the many applications of the theory to the

physical and biological sciences. The first, and rather informal, description of a

random walk was posed as a problem by the English statistician Karl Pearson in a letter to Nature in 1905 (4).

He solicited the solution to a problem phrased as follows: "A man starts from a point 0 and walks / yards in a

straight line; he then turns through any angle whatever and walks another / yards in a second straight line. He

repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + dr from his starting point 0." Shortly thereafter Lord

Rayleigh pointed out that he had considered a similar

problem much earlier (5). In retrospect we can now

recognize that much of the research on central limit theorems (theorems related to the distribution of the

George H. Weiss received a Ph.D. in applied mathematics from the

University of Marylatid. He is now Chief of the Physical Sciences

Laboratory, Division of Computer Research and Technology, at the

National Institutes of Health. His interests can be summarized as the

application of probabilistic methods to problems in the physical sciences.

Address: Division of Computer Research and Technology, National

Institutes of Health, Bethesda, MD 20205.

sum of large numbers of random variables), which was undertaken from the seventeenth century onward, can be rephrased in terms of random walks and leads to

important results in their study. Few mathematical models have found applications

in such a wide range of scientific disciplines as have random walks. They play a key role in describing con

figurational properties of polymers, and form the basis of much of the current thinking on energy transfer in

amorphous solids. In addition, random walks are used to describe the motion of microorganisms on surfaces and to analyze data from such experiments. Many of the

applications in polymer physics are described by Ya makawa (6), and other applications are reviewed by Rubin and myself (7). Clearly, this article can mention

only a few of these applications and can provide a su

perficial description at best. One of the significant roles played by random walks

is that of a microscopic model for the phenomenon of diffusion or Brownian motion. In our formulation of the drunkard's walk, let the distance between points on the lattice be AL, and let the time between successive steps be AT. Then, if AL and AT are allowed to shrink to zero in such a way that (AL)2/AT remains constant and equal to the diffusion constant D, the equation governing the distribution of the displacement of the random walker from his starting point can be shown to be

?P d2P ?t dx2

which is a partial differential equation that describes diffusion or heat flow. The function P(x,t) can be inter

preted either as a probability density or as a concentra tion of diffusing matter at position x at time t. The lim

iting process that start? from a discrete random walk and

passes to a continuum picture forms the basis of Ein stein's studies of Brownian motion (8).

The theory of diffusion is intimately related to, and sometimes indistinguishable from, the theory of random walks; many properties are similar in both models. For

example, the solution to the diffusion equation (Eq. 1) is the familiar bell-shaped Gaussian curve shown in

Figure 1, if all the matter is initially concentrated at a

single point. It is easy to show that the distribution of the end-to-end distance of a discrete /z-step random walk

approaches the Gaussian form under a wide variety of circumstances. This is, indeed, the main point of several central limit theorems. Further, unless we build in a mechanism for a net drift (for instance, if the drunkard

flips a biased coin), the average displacement after //

1983 January-February 65



Steps is zero and the standard deviation of the dis

placement is proportional to n1/2 in the discrete walk and to t1/2 for the diffusion process.

Many interesting problems specifically concern random walks on lattices, which have important appli cations in solid state physics. George P?lya, a distin

guished emeritus professor of mathematics at Stanford

University, was the first to discuss a problem involving random walks in which the lattice feature is essential (9).

He studied the drunkard's walk described earlier, and asked for the probability that the drunkard, whose

peregrinations are assumed to start at the origin, will return to that point at some time during the random

walk. P?lya proved the remarkable result that in one and two dimensions the probability of return is 1?i.e., the drunkard is certain to return at some time during the course of the walk. In three or more dimensions the

probability of return is less than 1. Although return to the origin is a certain event in one and two dimensions, it can also be shown that the average time to return to that point is infinite.

Certain properties of random walks on lattices have no natural analogs in the diffusion case. For example, models of trapping in both physical and biological contexts lead to the consideration of how many distinct sites are visited during an /?-step random walk. This

problem arises because a random walker as described so far can occupy a fixed site more than once in the course of his walk. The average number of distinct sites visited is readily calculated if n is very large (10). In one di

mension the expected number of sites is proportional to

n112, in two dimensions to n/ln n, and in three or more dimensions to n. These results are valid provided that the average distance traveled in a single step is finite, as is the case in the drunkard's walk, and provided there is no net drift. A calculation of the standard deviation of the number of distinct sites requires a much more intricate analysis than that of the expected number, ex

cept in one dimension. The case of three dimensions is

perhaps of greatest interest. There the standard deviation is proportional to (n X In n)112 and the distribution function is Gaussian, if n is large (11). The standard de viation, but not the distribution function, is known in two dimensions, again if n is large. Exactly how large n

must be for these results to be valid is still an open question, and poses a challenging mathematical

problem. Other properties suggest themselves as a means of

characterizing the configurations of random walks. For

example, what is the distribution of the maximum dis

placement of a random walker after n steps? This prob lem was first studied in depth by Erd?s and Kac, who found distribution functions for the maximum dis

placement under very weak restrictions on the detailed structure of the random walk, and who showed that the

maximum displacement of an tt-step walk is of the order of n1/2 if n is large (12). This problem also arises in the

study of random walks used as models of polymer con

figurations. It was tackled in that context by Hollings worth, who found that, as with diffusion processes, the maximum displacement in time t is proportional to f1/2 if t is large (13).

A similar, but somewhat simpler, property of ran dom walks is that of occupancy, or the number of times the random walker has visited a single point or a set of

Position

Figure 1. Random walks can be used to model such processes as

diffusion. The curves represent the probability that a random

walker who starts from the origin at time zero is between points x and x + dx at time f, or the concentration of a substance at

point x at time f. As time increases, the random walker's position becomes more uncertain, as is evidenced by the flattening

profiles of the curves. These bell-shaped curves are referred to in

the physical sciences as Gaussian distributions, and in statistics as

normal distributions.

points on a lattice after having taken n steps. This

property does have an analog for diffusion processes: the amount of time spent in a given region by a particle that diffuses for a time t. In a one-dimensional lattice for an

h-step random walk, the average number of times a point is visited is proportional to n1/2 if n is large, while in a three-dimensional lattice the average number tends to a calculable constant (24). For diffusion processes, the

analogous problem was analyzed by Darling and Kac, who gave results for the distribution of the occupancy that are valid if the time of diffusion is long (15).

Position

Figure 2. A modification of the random walk to allow for

momentum produces the curves shown that represent the

concentration of diffusing matter. The parameters for these

curves are c = 1 and T = V2. The fact that the concentration is

equal to zero outside of the vertical lines is interpreted to mean

that there is a finite speed of diffusion, which is equal to the

parameter c in Eq. 2. The values of x such that x2 < c2t2 represent the region in which diffusion occurs.

66 American Scientist, Volume 71



Some applications have required that new kinds of random walks be analyzed, and these new walks in turn have led to new insights in different subject areas. I will

emphasize, in what follows, some modifications of random walks that have been suggested in this way by problems in chemistry and physics. These modifications have occasionally set profoundly difficult mathematical

problems; they may also be of interest in subject areas outside their original framework.

One such modification of the simple random walk was proposed by the British hydrodynamicist G. I. Tay lor, in a study of turbulent diffusion (26). In our discus sion of the drunkard's walk there was no property that

might be regarded as an analog of momentum. That is to say, the drunkard has the option of reversing his di rection at every step, regardless of the direction of his last step. A more realistic model of the motion of a

physical object needs to take into account some form of

memory of where the object has come from and the likelihood that it will continue to move in the same di rection. The simplest such model is Taylor's modification of the random walk, in which the coin toss decides not the direction of the drunkard's next step but the persis tence of motion. In other words, with probability equal to p the drunkard makes his next step in the same di rection as his last one, and with probability 1 ? p he re verses direction. The ordinary drunkard's walk occurs when p

= V2 The interesting feature of Taylor's modified walk

is that as the distance between points and the time be tween steps decrease, one no longer obtains the diffusion

equation (Eq. 1) but rather

?^P 1?P= , d2P ?t2 T ?t C

?x2 (2)

in one dimension where T and c2 are constants. In con trast to the Gaussian profile shown in Figure 1, the so lution to Eq. 2, the so-called telegrapher's equation, be haves as shown in Figure 2. The most striking result is that P(x,t) is zero outside a boundary that moves with a

Figure 3. This schematic diagram of a carbon chain illustrates two

angular degrees of freedom of polymeric substances, or how a

chain can be flexible. The bond angle is denoted by 6, and is

generally fixed for a given type of polymer. If one considers the

bonds Ci ?

C2 and C2 ?

C3 to define a plane, then C4 lies on the

surface of a cone that has a vertex at C3 and that is generally not in the plane. The angle defining the amount of rotation that gives the exact location of C4 is known as the bond rotation angle.

These two angles represent two degrees of freedom.

speed equal to ?c. This is interpreted to mean that, in contrast to simple diffusion as described in Eq. 1, the

speed of propagation is finite when momentum is taken into consideration. That is to say, Eq. 1 implies that P(x,t) > 0 for all x, whenever t > 0. This is not the case for the solution to the telegrapher's equation (Eq. 2). However, as time increases, the concentration at a fixed point will

increasingly be approximated by that obtained from the

simple diffusion equation.

Polymer configurations A polymeric substance is one made up of chains of re

peating molecules; the units of a chain, or the monomers, are usually identical. Familiar examples of polymers include rubber, plastics, and synthetic fibers. The chains are not totally rigid but may be flexible in various ways, one of which is shown in Figure 3. Because of this flex

ibility within a chain, and the motion of the entire chain in one direction or its rotation due to Brownian motion, a polymer in solution can be in any one of a large num ber of configurations. These configurations have been modeled in terms of random walks with a considerable amount of success, and much current research has been

performed at the intersection of the chemical and mathematical disciplines. In the models of polymers the number of bonds is analogous to the length of a random walk; thus, a polymer with n bonds is modeled as an

tt-step random walk.

All the physical techniques used to describe the features of polymers at the molecular level, or to relate

macroscopic properties to microscopic structure, are af fected by the nature of polymeric configurations. In the case of elasticity, for example, the number of configu rations possible in a randomly coiled chain exceeds that in an extended chain. Therefore, randomly coiled chains" -

have a higher entropy than extended ones, which means that when a polymer chain is stretched, a force is gen erated that can be regarded as purely entropie. Methods of physical chemistry that are widely used to determine the structure and properties of polymers can be roughly categorized either as hydrodynamic, those related to

viscosity and sedimentation, or as optical, those related to x-ray and light scattering. Some of the fundamental

papers in these areas have been reprinted by Hermans

(17). In large part these investigations, most of which calculate quantities measurable on a macroscopic scale, start with random walks.

Physical considerations, at this point, suggest that random walks similar to the drunkard's walk are inad

equate to describe polymer configurations because, al

though the drunkard can return to a given point, no two atoms can occupy the same space. In other words, the random walk model of a polymer must not intersect it self; it must be self-avoiding. A walk on a one-dimen sional lattice, therefore, must move in one direction

only, since a reversal leads to intersection. From the mathematical point of view a self-avoiding random walk is profoundly different from, say, the drunkard's walk. In the latter, all the information required for the statis tical prediction of step n + 1 is contained in the knowl

edge of the location of the drunkard after n steps. This limitation on the amount of memory needed is known

technically as the Markov property. The self-avoiding walk does not have this property since one needs the




whole history of the walk to predict the next step. In deed, the mathematical techniques required to find

properties of self-avoiding walks are much more difficult than those needed for comparable properties of Mar kovian walks (18). Much of what is known about the

properties of self-avoiding walks has been discovered

by simulation, supplemented by approximate theoretical

analyses. Both simulation and a number of approximate

theories have now clearly established that the mean

squared displacement of an tt-step self-avoiding walk is

proportional to n3/1 in two dimensions and to n6/5 in three dimensions, in contrast to the displacement of the drunkard's walk, which is proportional to n in all di mensions. Furthermore, the probability density for the location of the self-avoiding walker after n steps changes from the Gaussian form shown in Figure 1 to the den sities plotted in Figure 4. The curve for a two-dimen sional self-avoiding walk is markedly different from the Gaussian curve, while for a three-dimensional walk the difference is much less apparent. As the number of di mensions increases the effects of self-avoidance dimin ish, which means that in walks of four or more dimen sions the Gaussian curve, or diffusion approximation, is very accurate. Even though much effort and ingenuity have been expended on the study of self-avoiding walks, it is possible to obtain considerable information about

polymer configurations from simpler random walks that do not take self-avoidance into account. The relevance of these simpler models for the interpretation of data on

light scattering, for example, depends on minimizing

Position

Figure 4. If a random walk cannot intersect itself, mathematical calculations of its properties are difficult. The function PGauss(xvf)

represents the probability that a random walker is at a certain

point, if the walk is unrestricted. However, if the walker cannot cross his own path, P2(x,t) is proportional to the probability that he is at a certain point in a two-dimensional walk; P$(x,t) is

proportional to the same probability in a three-dimensional walk. The last two of these densities were conjectured by Domb and his colleagues (19) on the basis of extensive simulations that

essentially enumerated all possible self-avoiding walks for up to 18 steps.

long-range interactions, which means that experiments must be conducted in sufficiently dilute solutions and at an appropriate temperature (20).

Solid state physics Diffusion processes are most appropriately set in the context of a continuum in which there is no micro structure. Much of the analysis of a long polymer chain in solution can be modeled in terms of diffusion because of the large number of bonds in the chain. There are, however, problems related to short polymer chains, those with about ten bonds each, that require a more detailed modeling of structure. Considerable work on this has been done in the field of polymer physics (21 ).

However, a lattice structure is fundamental to the notion of a solid, and therefore much recent research in solid state physics has been phrased in terms of random walks on lattices. I will sketch some developments in the

theory of random walks suggested by those investiga tions.

One of the variations on the drunkard's walk that is simple to describe, but that has far-reaching conse

quences in different applied areas, is the continuous time random walk, or CTRW (14). In my description of the drunkard's walk in the first paragraph of this article, the element that made the walk random (i.e., the toss of a

coin) was invoked at regular intervals. In order to model more realistically the time during which the walk occurs, one assumes that the intervals between successive tosses of the coin are randomly distributed. This mathematical

prescription corresponds to a picture of a particle mov

ing along a lattice and remaining at any given site for a random amount of time. Thus, instead of steps being taken at times 0, t, It, 3t,.. ., the steps are taken at times hr tir hf / with the intervals between the times gen erally assumed to be identically distributed random variables. When the average values of the random in tervals are finite, say, equal to At, then, at sufficiently long times (specifically, for times t such that t / At ? 1), the probability density of the walker's position after n

steps tends toward the Gaussian form just as when the intervals are regular. There are, of course, differences to be expected at shorter times.

It is somewhat surprising that models in which the

average values of the random intervals are infinite play an important role in describing transport in amorphous media (22, 23). For instance, a basic experimental setup for studying how an electrical charge is transported in certain amorphous solids involves placing a sample between two semitransparent planar contacts and

flashing a light at it. This causes carriers of one electrical

sign to move across the sample, inducing a time-de

pendent conduction current to flow in an external cir cuit. The measured current can be used to estimate the time that it takes for a carrier to traverse the sample.

If that time is a constant, then the current should have the simple form shown at the left of Figure 5. If the mechanism of transport across the sample is by simple diffusion of carriers, the current is modified as shown in the middle of Figure 5. However, experiments on

amorphous arsenic triselenide show that holes, pro duced by the light pulse, propagate through the sample in a manner inconsistent with the model of a simple random walk. Typical results obtained in this type of




experiment are illustrated at the right of Figure 5. Spe cifically, if the charge moves by diffusion, then a con

sequence of the theory is that the average displacement s of a single hole is proportional to the time, t, and the standard deviation of that displacement is proportional to f1/2, which means that the ratio of the average dis

placement to the standard deviation is also t1/2. How ever, analysis of data resembling those shown at the

right of Figure 5 leads to a constant ratio of these quan tities, thereby ruling out a picture of simple diffusion.

Scher and Montroll showed that the observed re sults could be mimicked by a CTRW in which the times between the steps have an infinite average value (22). This finding implies that some carriers are held at some

points for relatively long periods of time as they move across the sample. The mathematical assumption needed to reproduce the experimental data is that the probability density function for At is proportional to (Af)"(1

+ a)

when 0 < a < 1 and At is sufficiently large. This model

predicts that at first the current behaves like C\t~(1 ~

a\ where C\ is a constant, and later like C2^"(1

+ a). The observed behavior is in very good agreement with data on amorphous arsenic triselenide when a is close to 0.45

(24). Similar qualitative behavior has been found for the

transport of holes in other organic polymers.

Inhomogeneous media

One of the most active research areas at the present time

involving random walks concerns their properties in media that are only statistically homogeneous; that is to

say, the averaged properties of the media are uniform in space but the properties are inhomogeneous on a

microscopic scale. A good example in this category is the so-called trapping problem, which relates to a random walk on a uniform lattice in which each site can be a trap, with probability c, or an ordinary point, with probability 1 ? c. When a random walker lands on a trap he disap pears. This model has been used to describe the lumi nescence of organic solids (25) and the photosynthetic process (26, 27), and to model the kinetics of annealing defects in metallurgy (28) and other phenomena in solid state physics (29). As an example, during photosynthesis the conversion of a photon and the consequent pro duction of oxygen involve the absorption of the photon by a chlorophyll molecule and the creation of an exciton. The exciton moves by a random walk to neighboring chlorophyll molecules until it reaches a trap, where a chemical reaction occurs that leads to the production of

oxygen and the disappearance of the exciton (26).

The questions related to trapping are simply stated.

Suppose that a random walker is set down on an arbi

trary point of a lattice containing traps. How long, on the

average, will it take before the walk is terminated by trapping? How far will the random walker wander be fore trapping occurs? No complete answer is known to either of the two questions, although it is easy to produce a formal expression for the probability that the random walker survives until at least step n. If Sn denotes the number of distinct sites visited in an M-step walk on a lattice without traps, then the desired probability can be

written

Pn = (1 -

since survival requires that each point visited by the walker must not have been a trap. The parameter S? is a random variable whose properties have not been

completely elucidated, as I have already mentioned. In

fact, its properties are known only if the lattice is mul tidimensional and if n is large. In essence this restricts

any analysis of trapping to lattices with a low concen tration of traps. A recent calculation using Jain and Pruitt's theory (11) suggests that there is a satisfactory method of predicting how long it will take for a random walker to be trapped for concentrations of traps below about 5% (30). Little is known about the statistical

properties of how far the walker can go before encoun

tering a trap. There is clearly much still to be learned about trapping problems, both from analysis and from

computer simulations. Another variant of the problem of statistically ho

mogeneous but locally inhomogeneous random walks is that of transport in disordered media. A disordered

medium is one whose properties at any point are only known in a statistical sense. In this type of problem, the random walker may be trapped temporarily and then -

released. An early version of this random walk was de

veloped by Giddings and Eyring as a microscopic model of the Chromatographie process (31). In their model a molecule traversing a chromatography column is as sumed at any given time to be in one of two states: mo bile or stationary. The molecule travels at a constant

speed down the column when it is in the mobile phase, but trapping into the stationary phase and release back into the mobile phase occur randomly in time. While in the column the molecule's position and phase are pre dictable only in a probabilistic sense. It is assumed that when the molecule reaches the end of the column it is withdrawn; the problem of interest is to calculate the distribution of withdrawal times. When the rates at

TmeTime Time

Figure 5. The speed at which an electrical charge moves through a

sample can be measured by a conduction current. At the left the

charge is assumed to move at a constant speed until it leaves the

sample at time T. In the middle, the speed of the charge is

modified by adding a diffusion component. However,

experimental results produce a curve like that shown at the right, which can be modeled using a continuous-time random walk.

Two noteworthy features of the curve are a plateau that appears

relatively early and a tail that is much more pronounced than

would be expected on the basis of diffusion alone.




Figure 6. Experiments suggest that the motion of some bacteria on

a surface consists of a hunting phase and a traveling phase.

During the hunting phase the bacterium rotates around an axis

that remains nearly stationary, and in the traveling phase it

moves in an almost straight line. The angle between successive

lines has a random distribution.

which transitions are made between the mobile and

stationary phases are assumed to be equal at all positions and times, and dependent only on which is the initial and which the final phase, an exact solution for this model is available (31 ). With more complicated schemes for transitions between the two states, the distribution of withdrawal times will be approximately Gaussian when there are many events of trapping and release and when both the trapping and release rates are finite

(32). A more complex random walk, the multistate

hopping model, involves a single mobile state and some

number, greater than one and possibly infinite, of

trapping states. This model has been analyzed in con siderable detail in recent years because it serves as a

model for the displacement of charge in a dielectric due to an applied electrical field (23, 33). A somewhat dif ferent approach to the problem of transporting charge in a disordered medium, in which the walker moves between randomly placed trapping centers, was origi nally suggested by Scher and Lax (34). Their theory re

places the disordered medium by a regular lattice but retains the effect of disorder by replacing the diffusion

process by a ctrw. Although this procedure appears to be an ad hoc one, it does not conflict with the more

readily visualized hopping model (23, 35) and leads to a useful interpretation of a considerable body of data.

A biological model

My final example of an application of random walks concerns the motion of microorganisms on surfaces or in suitable media. It is not generally known that Pear son's (4) original query on random walks was related to his theory of the migration of populations (36). One reason for studying the motion of microorganisms is to

develop an assay to measure Chemotaxis, the response of organisms to chemical stimuli. Many investigators have remarked that the motion of bacteria or other

microorganisms appears to be random. Gail and Boone were the first to confirm experimentally that aspects of the motion of mouse fibroblasts in tissue culture are describable in terms of a two-dimensional random walk

(37) . Experiments on bacteria suggest that their motion can be approximately represented as a sequence of

nearly linear segments separated by periods during which the bacteria move only within a restricted area as

they appear to hunt for a new direction of linear motion

(38, 39). A schematic illustration of this type of motion is shown in Figure 6. Notice that both the length of the linear segments and the angles between them are random.

If we suppose that a chemical stimulus is located somewhere on the surface, then at least two kinds of

response by the organism have been identified experi mentally. In the first response, the time spent traveling along a given line varies according to whether the bac terium is moving toward the stimulus or away from it

(38) . In the second response, the angles between the linear segments are not completely random because the

organism tends to turn toward the stimulus (40). No doubt other biasing mechanisms also exist in nature. The

theory of these types of random walks was first devel

oped by Patlak (41) and was later presented in a more

immediately applicable form by Nossal and myself (42). Nossal has recently published an excellent review of

quantitative aspects of the motion of microorganisms (43). Reading the literature makes clear that what is

lacking at the present time is a good set of statistical tests to sort out the different mechanisms of chemotactic re

sponse in the experimental data.

Although I have focused on physical and chemical models that lead to an analysis of random walks and diffusion in disordered media, there has been a parallel development of ideas in the realm of biology to furnish models for population dynamics in random environ ments (44). Pure mathematicians and probabilists are also considerably interested in the subject of random walks in random environments (45). One of the richest areas for research on random walks at present is their occurrence in disordered media. A considerable amount of analysis has been done for one-dimensional systems because of their relative simplicity. Similar analysis for multidimensional systems tends to pose more difficult mathematical problems and to provide approximate results of less certain validity than the exact results available for one-dimensional systems.

Random walks have played a significant role in several other areas, but lack of space prevents a detailed

description of any of these. They include the theory of

sequential analysis used in the design and analysis of statistical experiments (46) and the theory of the stock market (47). What I have tried to emphasize in this article is the interplay over the years between theory and ap plication. The interaction has been fruitful both in

shedding light on phenomena in which random ele ments are important and in leading to more intensive

investigations of the mathematical tools needed to de scribe these phenomena.

References 1. I. Todhunter. 1865. A History of the Mathematical Theory of Probability.

Reprinted 1965. New York: Chelsea.

2. F. Spitzer. 1976. Principles of Random Walk. Springer-Verlag. 3. L. H. Liyanage, C. M. Gulati, and J. M. Hill. 1980. A Bibliography on

Random Walks. Mathematics Dept., Univ. of Wollongong, Aus tralia.




4. K. Pearson. 1905. The problem of the random walk. Nature 72: 294.

5. Lord Rayleigh. 1905. The problem of the random walk. Nature 72:318.

_ 1880. On the resultant of a large number of vibrations of the same pitch and arbitrary phase. Phil. Mag. 10:73-78.

6. H. Yamakawa. 1971. Modern Theory of Polymer Solutions. Harper and Row.

7. G. H. Weiss and R. J. Rubin. 1982. Random walks: Theory and se lected applications. Adv. in Chem. Phys. 52:363-505.

8. A. Einstein. 1905. Uber die von der Molekular-kinetischen Theorie der W?rme gefordete Bewegung von in ruhenden Fl?ssigkeiten suspendierten Teilchen. Ann. der Physik. 17:549-60.

_ 1906. Zur Theorie des Brownsche Bewegung. Ann. der

Physik. 19:371-81.

9. G. P?lya. 1921. Uber eine Aufgabe der Wahrscheeinlichkeits

rechnung betreffend die Irrfahrt in Strassennetz. Math. Ann. 84: 149-60.

10. A. Dvoretzky and P. Erd?s. 1951. Some problems on random walks in space. In Proceedings of the Second Berkeley Symposium on Statistics and Probability, ed. J. Neyman, pp. 353-67. Univ. of California Press.

11. N. C. Jain and 1W. E. Pruitt. 1971. The range of transient random walks. /. Analyse Math. 24:369-93.

12. P. Erd?s and M. Kac. 1946. On certain limit theorems of the theory of probability. Bull. Am. Math. Soc. 52:292-302.

13. C. A. Hollingsworth. 1948. The average boundaries of statistical chains. /. Chem. Phys. 16:544-47.

14. E. W. Montroll and G. H. Weiss. 1965. Random walks on lattices.

II./. Math. Phys. 6:167-81.

15. D. A. Darling and M. Kac. 1957. On occupation times for Markov

processes. Trans. Am. Math. Soc. 84:444-58.

16. G. I. Taylor. 1921-22. Diffusion by continuous movements. Proc. Lond. Math. Soc. 20:196-212.

17. J. J. Hermans, ed. 1978. Polymer Solution Properties, Part II. Hydro dynamics and Light Scattering. Stroudsburg, PA: Dowden, Hutch

inson, and Ross, Inc.

18. C. Domb and M. S. Green. 1973-77. Phase Transitions and Critical

Phenomena, vols. 1-6. Academic Press.

19. C. Domb, J. Gillis, and G. Wilmers. 1965. On the shape and con

figuration of polymer molecules. Proc. Phys. Soc. 85:625-45.

20. P. J. Flory. 1953. Principles of Polymer Chemistry. Cornell Univ. Press.

21._ 1969. Statistical Mechanics of Chain Molecules. Wiley. 22. H. Scher and E. W. Montroll. 1975. Anomalous transit-time dis

persion in amorphous solids. Phys. Rev. B12:2455-77.

23. G. Pfister and H. Scher. 1978. Dispersive (non-Gaussian) transient

transport in disordered solids. Adv. in Phys. 27:747-98.

24. M. E. Sharfe. 1970. Transient photoconductivity in vitreous As2Se3.

Phys. Rev. B2:5025-34. G. Pfister and H. Scher. 1977. Time-dependent electrical transport in amorphous solids: As2Se3. Phys. Rev. B15:2062-83.

25. H. B. Rosenstock. 1969. Luminescent emission from an organic solid with traps. Phys. Rev. 187:1166-68.

26. E. W. Montroll. 1969. Random walks on lattices. III. Calculation of first-passage times with application to exciton trapping on

photosynthetic units. /. Math. Phys. 10:753-65.

27. P. Argyrakis and R. Kopelman. 1978. Exciton percolation III. Sto chastic and coherent migration in binary and ternary random lattices. /. Theoret. Biol. 73:205-36.

28. J. R. Beeler. 1964. Distribution functions for the number of distinct sites visited in a random walk on cubic lattices: Relation to defect

annealing. Phys. Rev. 134:A1396-401.

29. H. Scher, S. Alexander, and E. W. Montroll. 1980. Field-induced

trapping as a probe of dimensionality in molecular crvstals. PN AS 77:3758-62.

30. G. H. Weiss. 1980. Asymptotic form for random walk survival

probabilities on 3-D lattices with traps. PNAS 77:1273-74.

31. J. C. Giddings and H. Eyring. 1955. A molecular dynamics theory of chromatography. /. Phys. Chem. 59:416-20.

32. G. H. Weiss. 1970. Contributions to the stochastic theory of Chro

matographie kinetics. Separation Sei. 5:51-62.

33. F. W. Schmidlin. 1977. Theory of trap-controlled transient pho

toconduction. Phys. Rev. B16:2362-85.

34. H. Scher and M. Lax. 1973. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B7:4491-502.

35. J. Klafter and R. Silbey. 1980. Derivation of the continuous-time random-walk equation. Phys. Rev. Lett. 44:57-58.

36. K. Pearson. 1906. A Mathematical Theory of Random Migration. London: Draper's Company Research Memoirs.

37. M. H. Gail and C. W. Boone. 1970. The locomotion of mouse fi broblasts in tissue culture. Biophys. J. 10:980-93.

38. H. C. Berg and D. A. Brown. 1972. Chemotaxis in Escherichia coli

analyzed by three-dimensional tracking. Nature 239:500-504.

39. R. Macnab and D. E. Koshland, Jr. 1972. The gradient-sensing mechanism in bacterial Chemotaxis. PNAS 69:2509-12.

40. R. Nossal and S. H. Zigmond. 1976. Chemotropism indices for

polymorphonuclear leukocytes. Biophys. J. 16:1171-82.

41. C. S. Patlak. 1953. Random walk with persistence and external bias. Bull. Math. Biophys. 15:311-38.

42. R. Nossal and G. H. Weiss. 1974. A generalized Pearson random walk allowing for bias. /. St at. Phys. 10:245-53.

43. R. Nossal. 1980. Mathematical theories of topotaxis. In Biological Growth and Spread. Mathematical Theories and Applications, ed. W.

Jager, H. Rost, and P. Tautu, pp. 410-40. Springer-Verlag. 44. M. Turelli. 1977. Random environments and stochastic calculus.

Theoret. Pop. Biol. 12:140-78.

45. F. Solomon. 1975. Random walks in a random environment. Ann. Prob. 3:1-31.

46. A. Wald. 1947. Sequential Analysis. Wiley. 47. L. Bachelier. 1900. Th?orie de la Speculation. Paris: Gauthier-Vil

lars. M. F. M. Osborne. 1959. Brownian motion in the stock market. /.

Opns. Res. Soc. Am. 7:145-73. E. F. Fama. 1970. Efficient capital markets: A review of theory and

empirical work. /. Finance 25:383-420.

1*

"30% of everything is a placebo. The other 70% makes things worse."



Documents

Random Walks and Their Applications: Widely used as mathematical models, random walks play an important role in several areas of physics, chemistry, and biology