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Probability and Nonlinear Systems

the 1930s and first published in this coun-try in 1957. See Excerpts from The Scottis Book. )Ulam's incredible feel for mathematicswas due to a rare combination of intu-itions, a common feature of almost allgreat mathem aticians. He had a verygood sense of combinatorics and ordersof magnitude, which included the abilityto make quick, crude, but in-the-ballparkestimates. Those talents, comb ined withthe more ordinary abilities to analyze aproblem by means of logic, geometry, orprobability th eory, already mad e him veryunusual. Besides, he had a good intuitionfor physical phenomena, which m otivatedmany of his ideas.Ulam's intuition, as exhibited in nu-merous problems formulated over a spanof more than fifty years, cov ered an enor-mous range of subjects. Th e problems oncomputing, physical systems, evolution,and biology were stimulated by new de-velopments in those fields. Many othersseemed to spring from his head. He usu-ally had some prime examples in mindthat motivated his choice of mathemati-cal model or method. In this regard oneof his favorite quotes, from Shakesp eare'sHenry VIII was

Things done without examplein their issueAre to be feared.

In approaching a complicated problemStan first searched for simplicity. Hehad no patience for complicated theoriesabout simple objects, much less complexobjects. That philosophical dictum hap-pened to match his personality. He couldnot hold still for the time it would taketo learn, let's say, modern abstract alge-braic geometry, nor could he put up withthe generalities of category theory. Also,he was familiar with, and early in his ca-reer obtained fundamental results in, mea-sure and probability theories. That back-ground led him to approach many prob-lems by placing them in a probabilistic

framework. Instead of considering justone possible outcome of a process, onecan consider an infinite number of possi-ble outcomes at once by randomizing theprocess. T hen one can apply the power-ful tools of probability, such as the lawsof large numbers, to determine the like-lihood of a given outcome. The famousMonte Carlo method is a perfect exam-ple of that approach. In fact, one ofthe favorite sayings of Erdos and Ulam,both of whom worked in combinatorics(in which the number of outcomes is fi-nite) and probability, was

The infinite we do right away;the finite takes a little longer.Stan's interest in probability dates backto the early 1 930s, when he and Lom nicki

proved several theorems concerning itsfoundatio ns. In particular, they sho wedhow to construct consistent probabilitymeasures for systems involving infinite(as opposed to finite) sequences of inde-pendent random variables and, more gen-erally, for Markov processes. (In Markovprocesses probabilities governing the fu-ture depend only on the present and areindependent of the past.) At about thesame time Kolmogorov, independently,proved his consistency theorem, whichincludes the Ulam and Lomnicki resultsas well as many more. Those resultsguarantee the existence of a probabilitymeasure on classes of objects generatedby various random processes. The objectsmight be infinite sequences of numbersor more general geometrical or topologi-cal objects, such as the homeomorphisms(one-to-one, onto maps) discussed in de-tail later in this article. Stan's interest inprobability continued after World War 11,when he and Everett wrote fundamentalpapers on multiplicative processes (bet-ter known as branching processes). Thosepapers were stimulated by the need tocalculate neutron multiplication in fissionand fusion devices. (David Hawkin s, in'T he Sp irit of Play, discusses some of

the earliest work that Stan and h e did onbranching processes.)Stan's background in probability m adehim a leader among the outstanding grou pof intellects who, during the late 1940sand early 1950s, recognized the potentialvalue of the computer for doing experi-mental mathem atics. They realized thatthe computer was an ideal tool for an-alyzing stochastic, or random, processes.While formal theorems gave rules on howto determine a probability measure on aspace of objects, the computer open ed upthe possibility of generating those objectsat random . Simply stated construction sthat yield complicated objects could beimplemented on the computer, and if onewas lucky, demonstrable guesses couldbe made about their asymptotic, or long-term, behavior. That was the approachStan took in studying deterministic aswell as random recursions. In additionhe invented cellular automata (lattices ofcells and rules for evolution at each cell)and used them to simulate growth patternson the computer.The experimental approach to mathe-matics has since become very popular andhas tremendously enhanced our vision ofcomplex physical, chemical, and biologi-cal systems. Without the fortuitous con-junction of the computer and probabil-ity theory, it is very unlikely that wewould have reached today's understand-ing of those nonlinear systems. Such sys-tems present a challenge analogous to thatNewton would have faced if the earthwere part of a close binary or tertiarystar system. (One can speculate whetherNewton could have ever unraveled thelaw of gravitation from the complicatedmotions of such a system.) At presentresearchers are trying to formulate limit-ing laws governing the long-term dynam-ics of nonlinear systems that are analo-gous to the major limiting theorems inclassical probability theory. The attemptto construct appropriate p robability m ea-sures for such systems is o ne of the topicsI will discuss in more depth.

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Other interests that Ulam maintainedthroughout his life were logic and set the-ory I remember a conference' on largecardinal numbers in New York a fewyears ago. Stan was the honored partic-ipant. More than fifty years earlier hehad shown that if a nontrivial probabilitymeasure can be defined on all subsets ofthe real numbers, then the cardinal num-ber, or size, of the set of all the subsetsexceeded the wildest dreams of the time.(See Learning from Ulam: MeasurableCardinals, Ergodicity, and Biomathemat-ics. ) But that large cardinal of his isminuscule compared with the cardinals oftoday. After listening to some of the con-ference talks, Stan said that he felt likeWoody Allen in Sleeper when he woke upafter a nap of many years and was con-fronted with an unbelievably large num-ber on a McDonald's hamburger sign.

There is a serious aspect to that re-mark. Stan felt that a split between math-ematics and physics had developed duringthis century. One factor was the traumathat shook the foundations of mathemat-ics when Cantor's set theory was foundto lead to paradoxes. That caused mathe-matics to enter a very introspective phase,which continues to this day. A tremen-dous effort was devoted to axiomatiz-ing mathematics and raising the level ofrigor. Physics, on the other hand, expe-rienced an outward expansion and devel-opment. (The situation is somewhat re-versed today, as internal issues concern-ing the foundations of physics receive at-tention.) As a result, university instruc-tion of mathematicians has become so rig-orous and demanding that the mathemat-ical training of scientists has been takenover by other departments. Consequently,instruction in applied mathematics, ormathematical methods, is often at a fairlylow level of rigor, and, even worse, someof the important mathematical techniquesdeveloped during this century have notmade their way into the bag of tools ofmany physical scientists. Stan was veryinterested in remedying the situation and

believed the Center for Nonlinear Studiesat Los Alamos could play a significantrole.

Stan was associated, either directly orthrough inspiration, with the three re-search problems described in Part I11 ofthis article. Each is an example of how aprobabilistic approach and computer sim-ulation can be combined to illuminate fea-tures of nonlinear systems. Since somebackground in modem probability theoryis needed to follow the solutions to theproblems, Part I1 provides a tutorial onthat subject, which starts with a bit of his-tory and concludes with several profoundand useful theorems. Fortunately MarkKac and Stan Ulam gave a very insightfulsummary of the development of probabil-ity theory in their book Mathematics andLogic: Retrospect and Prospects haveadapted and extended their discussion tomeet the needs of this presentation buthave retained their broad perspective onthe history .of mathematics and, in somecases, their actual words.

t rom th

ionof Ulam's own translation into En

nd the work stimulated by Problem 4has played a major role in understandinthe consequences of th

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PartPROBABILITY and NONLINEAR SYSTEMS

*The material quoted in this tutorial from Mathem tics and Logic has been reprinted with permis-sion from Encyclopedia Britannica, Inc.

A TUTORIALon PROBABILITY.MEASURE, and th laws ofLARGE NUMBERSA mentionedfoundationsoutcomes is in the introduction, Stan Ulam contributed to the me asure-theoreticthat allow one to define a probability when the num ber of possibleinfinite rather than finite. Here I will explain why this extensionis so necessary and so powerful and then use it to introduce the laws of large numbersThose laws are used routinely in probability and its applications (several times, foexample, during solution of the prob lems discussed in Part 111). Following the logic oKac and Ulam I begin at the beginning.*

arly Probability TheoryProbability theory has its logical and historical origins in simple problems of

counting. Consider games of chance, such as tossing a coin, rolling a die, or drawinga card from a well-shuffled deck. No specific outcome is predictable with certaintybut all possible outcome s can usually be listed or described. In many instances thenumber of possible outcom es is finite (though perhaps exceedingly large). Suppose weare interested in some subset of the outcomes (say, drawing an ace from a deck ofcards) and wish to assign a number to the likelihood that a given outcome belongs tothat subset. Our intuitive notion of probability suggests that that number should equathe ratio of the number of outcomes yielding the event 4, in the case of drawing anace) to the number of all possible events 52, for a full deck of cards).

This is exactly the notion that La place used to forma lize the definition of probabilityin the early nineteenth century. Let A be a subset of the set of all possible outcomesand let P(A) be the probability that a given outcome is in A For situations such tha

is a /rote set and all outcomes in are equally probable, Laplace defined P(A ) athe ratio of the number v(A) of elements in A to the total number v(f2) of elements o

; that is,

However, the second condition makes the definition circular, for the concept of probability then is depende nt upon the c oncept of equiprobab ility. A s will be described laterthe more mod em definition of probability does not have this difficulty.

For now let us illustrate how L aplace s definition reduces the calculation oprobabilities to counting. Suppose w e toss a fair coin (one for which hea ds and tail

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are equally probable) n times and want to know the probability that we will obtainexactly m heads, where 1 < m < n. Each outcome of n tosses c-an be representedas a sequence, of length n, of H 's and T's (HTHH . THH, for example), where Hstands for heads and T for tails. The set of all possible outcomes of n tosses isthen the set of all possible sequences of length n containing only H ' s and T's. Thetotal number of such sequences, v f l ) , s 2". How many of these contain H exactlym times? This is a relatively simple problem in counting. The first H can occur in npositions, the second in n 1 positions, . and the mth in (n m 1) positions. Soif the H 's were an ordered sample (HI,Hz,. . ,Hm), he number of sequences with mH 's would equal n(n 1)(n - 2) . . n - m 1 . But since all the H 's are the same,we have overcounted by a factor of m (the number of ways of ordering the H 's). Sothe number of sequences of length n containing m H 's is

n(n 1). . . (n - i n + 1) n-m m (n - m)(The number n /m (n - m) , often written 3,s the familiar binomial coefficient,that is, the coefficient of x m y n m n the expansion of x y)"). Since the number ofsequences with exactly m H ' s is 3nd the total number of sequences is 2", we haveby Laplace's definition that the probability P(m, n ) of obtaining m heads in n tossesof a fair coin is

Consider now a coin that is "loaded" so that the probability of a head in a singletoss is 116 (and the probability of a tail in a single toss is 516). Suppose again we tossthis coin n times and ask for the probability of obtaining exactly m heads. To describethe equiprobable outcomes in this case, one can resort to the artifice of thinking of thecoin as a six-faced die with an H on one face and T's on all the others. Using thisartifice to do the counting, one finds that the probability of m heads in n tosses of theloaded coin is

nP m ,n) = m\{n - m) ( A ) i ) n -m .Suppose further that the coin is loaded to make the probability of H irrational

\ / 2 / 2 , for example). In such a case one is forced into considering a many-faced dieand passing to an appropriate limit as the number of faces becomes infinitely large.Despite this awkwardness the general result is quite simple: If the probability of ahead in one toss is p , 0 < p < 1, and the probability of a tail is 1 - = q then theprobability of m heads in n tosses is

nm (n - m)

Building on earlier work of deMoivre, Laplace went further to consider whathappens as the number of tosses gets larger and larger. Their result, that the numberof heads tossed obeys the so-called standard normal distribution of probabilities, was amajor triumph of early probability theory. (The standard normal distribution function,Los Alamos ScienceSpecial ssue 1987

STANDARD NORMALDISTRIBUTION FUNCTION

b) STANDARD NORMAL DENSITYFUNCTION

Fig. 1. Almost two centuries ago Laplaceshowed that the number Nu of heads ob-tained in a large number n of tosses of acoin fair or loaded) follows the standardnormal distribution of probabilities. Moreprecisely, he showed that the probabilityof Nu being equal to or less than npt\/np 1where p is the probability ofa head in a single toss and is some num-ber) can be approximated, for large n by thestandard normal distribution function F t )shown in a). The derivative of a distribu-tion function when it exists) is called a fre-quency, or density, function. Shown in b)is the density function f t) for the standardnormal distribution function. Note that thevalue of the distribution fuction at some par-ticular value of t, say 0 s equal to the areaunder the density function from 00 o 3.

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BERTRAND S PARADOXWhat is the probability P that arandomly chosen chord of circle islonger than the side of theequilateral triangle inscribed withinthe circleThis question cannot be answered by us-ing Laplace's definition of probability, sincethe set of all possible chords is infinite, asis the set of desired chords (those longerthan the side of the inscribed equilateral tri-angle). However, the question might be ap-proached in the two ways depicted here anddescribed in the text. Although both ap-proaches seem reasonable, each leads to adifferent answer

Fig 2.

call it F(t), is given byF (t) - x 2 /2 .

-m9

the function dF dt ( l / v ^ ) e - t 2 / 2 is called the standard normal density functioThe deMoivre-Laplace result can be stated as follows. As n gets larger

larger, the probability that NH he number of heads tossed, will be less than or equanp +t^/pqn (where t is some number) is approximated better and better by the standnormal distribution function. Symbolically,

lim P(Nn < izp + t J n p q ) e-x2 /2 .n cn

In other words, P(NH < np + tJnpq) is approximated by the area under the standnormal density function from 00 to t, as shown in Fig. 1. (In modem terminology is called a random variable; this term and the terms distribution function and denfunction will be defined in general later.)

The de Moivre-Laplace theorem was originally thought to be just a special propof binomial coefficients. However, many chance phenomena were found empiricto follow the normal distribution function, and it thus assumed an aura of universaat least in the realm of independent trials and events. The extent to which the nordistribution is universal was determined during the 1920s and 1930s by LindebFeller, and others after the measure-theoretic foundations of probability had blaid. Today the de Moivre-Laplace theorem (which applies to independent trials, egoverned by the same probabilities) and its extension to Poisson schemes (in wheach independent trial is governed by different probabilities) are regarded simplyspecial cases of the very general central limit theorem. Nevertheless they were seeds from which most of modem probability theory grew.

Bertrand s ParadoxThe awkwardness and logical inadequacy of Laplace's definition of probabi

made mathematicians suspicious of the whole subject. To make matters worse, attemto extend Laplace's definition to situations in which the number of possible outcomeinfinite resulted in seemingly even greater difficulties. That was dramatized by Bertrawho considered the problem of finding the probability that a chord of a circle choat random be longer than the side of an equilateral triangle inscribed in the circl

If we fix one end of the chord at a vertex of the equilateral triangle (Fig. 2a), can think of the circumference of the circle as being the set l of all possible outcomand the arc between the other two vertices as the set A of favorable outcomes (tis, those resulting in chords longer than the side of the triangle). It thus seems proto take 113, the ratio of the length of the arc to the length of the circumference, as desired probability.

On the other hand we can think of the chord as determined by its midpoint athus consider the interior of the circle as being the set of all possible outcomes. Tset of favorable outcomes is now the shaded circle in Fig. 2b, whose radius is ohalf that of the original. It now seems equally proper to take 114 for our probabil

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the ratio of the area of the smaller circle to that of the original circle. XIOM OF DDITIVITYThat two seemingly appropriate ways of solving the problem led to differentanswers was so striking that the example became known as Bertrand's paradox.It is not, of course, a logical paradox but simply a warning against uncritical use ofthe expression at random. One must specify exactly how something is to be done atrandom.Coming as it did on top of other ambiguities and uncertainties, Bertrand's paradox Samplegreatly strengthened the negative attitude toward anything having to do with chanceand probability. As a result, probability theory all but disappeared as a mathematical Disjoint Eventsdiscipline until its spectacular successes in physics (in statistical mechanics, for ex- ad pample) revived interest in it early in the twentieth century. In retrospect, the logicaldifficulties of Laplace's theory proved to be minor, but clarification of the foundations probability of E , or ,==of probability theory had a distinctly beneficial effect on the subject. Probability of El + Probability of E2Axioms of Modern Probability Theory

The contemporary approach to probability is quite simple. From the set f of all XIOM OF COMPLEMENT RITYpossible outcomes (called the sample space), a collection of subsets (called elementaryevents) is chosen whose probabilities are assumed to be given on e and or all Onethen tries to calculate the probabilities of more complicated events by the use of twoaxioms.Axiom of additivity If El and are events, then El or 2' is an event. Moreover,if l and E2 are disjoint events, (that is, the subsets corresponding to l and Ei haveno elements in common), then the probability of the event E1 or E2 is the sum ofthe probabilities of El and E2, provided, of course, that El and can be assignedprobabilities. Symbolically,

P(El U E2)= P( Ei ) P(E2) provided El nE2 = 0

Sample Space

Probability of not E) = Probability of 0 E =1 Probability of E

Axiom of complementarity If an event E can be assigned a probability, then theevent not E also can be assigned a probability. Moreover, since the whole samplespace fl is assigned a probability of 1

P(not E) = P(^t E) = 1 (E).

Why these axioms? What is usually required of axioms is that they should

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through approximating forms. Finally, at the heart of the subject is the selectiof elementary events and the decision on what probabilities to assign them. Henonmathematical considerations come into play, and we must rely upon the empiricworld to guide us toward promising areas of exploration. These considerations alead to a central idea in modem probability theory-independence.

The efinition of IndependenceLet us return to the experiment of tossing a coin times. In attempting to constru

any realistic and useful theory of coin tossing, we must first consider two entirdifferent questions: (1) What kind of coin is being tossed? (2) What is the tossimechanism? The simplest assumptions are that the coin is fair and the tosses independent. Since the notion of independence is central to probability theory,

must discuss it in some detail.Events E and F are independent in the ordinary sense of the word if the occurren

of one has no influence on the occurrence of the other. Technically, the two eve(or, for that matter, any finite number of events) are said to be independent if the rof multiplication of probabilit ies is applicable; that is, if the probability of the jooccurrence of E and F is equal to the product of their individual probabilities,

Kac and Ulam justified this definition of independence as follows:In other words, whenever E and F are independent, there should be a rule

that would make it possible to calculate Prob. {E and F} provided only that oneknows Prob. {E} and Prob. { F } . Moreover, this rule should be universal; it shouldbe applicable to every pair of independent events.

Such a rule takes on the form of a function f (x ,y ) of two variables x y, andwe can summarize by saying that whenever E and F are independent we have

Prob. {E and F } f (Prob. {E},Prob. {F}Let us now consider the following experiment. Imagine a coin that can be

'loaded' in any way we wish (i.e., we can make the probability p of H any numberbetween 0 and 1) and a four-faced die that can be 'loaded' to suit our purposes alsoThe faces of the die will be marked 1,2,3,4 and their respective probabilities will bedenoted p,,p2, p3, p4; each pi is nonnegative and pl p p3 p4 = 1. We must noassume that whatever independence means, it should be possible to toss the coin anthe die independently. If this is done and we consider (e.g.) the event H and (1 or2)' then on the one hand

Prob. {Hand (1 or 2)} = f@,p i + p 2while on the other hand, since the event 'H and (1 or 2)' is equivalent to the event( H and 1) or (H and 2),' we also have

Prob. {H and(1 or 2)} Prob. {H andl} Prob. {H and 2} =f @,p i ) f (p,pz)Note that we have used the axiom of additivity repeatedly. Thus

f @ ~ \~ 2 )f(P,p1)+f(P,p2)for all p ,pl ,p2 restricted only by the inequalities~ .o.Tlamos Science Special Issue 1

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0 5 7 5 11 O< Pl , 0

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What is thefirst to toss

probability that will e the to toss a head? This can happen either on the first toss, or on the third (the first ta head being tails), or on the fifth (the first four being tails), and so on. The event that A wtoss the first head is thus decomposed into an infinite number of disjoint events. If

coin is fair and the tosses independent (so that the rule of multiplication applies), tthe probabilities of these events are

1 1 12 23 1and the probability that will toss the first head is simply the sum of a geomeseries:

This result hinges on one very crucial proviso: that we can extend the axiomadditivity to an infinite number of disjoint events. This proviso is the third axiommodem probability theory.

Axiom of countable additivity If E l ,E 2 ,E 3 , is an infinite sequence of disjevents, then U r i is an event and

Note that in solving the last problem we not only needed the axiom of countaadditivity but also assumed that the probabilities used for finite sequences of trare well defined on events in the space of infinite sequences of trials. Whether sprobabilities could be defined that satisfy the axioms of additivity, complementarand countable additivity was one of the central problems of early twentieth-centmathem atics. That problem is really the problem of defining a measu re because,we will see below, the axioms of probability are essentially identical with the requiproperties of a measure.Measure Theory The most familiar examples of measures are areas in a planevolumes in three-dimensional Euclidean space. These measures were first develoby the Greeks and greatly extended by the calculus of Newton and Leibnitz. mathem atics continued to develop, a need arose to assign measu res to sets less tamthan smooth curves, areas, and volume s. Studies of converg ence and divergenceFourier series focused attention on the sizes of various sets. For exam ple, gia trigonometric series a cos n t b sin n t , can one assign a measure to the set s for which the series converges? (Cantor's set theory, which ultimately became cornerstone of all of modem mathematics, originated in his interest in trigonomeseries and their sets of convergence.) For another example, how do es one assigmeasure to an uncoun table set, such as Cantor's midd le-third set? (See CantoMiddle-Third Set .) Answ ers to such questions led to the developm ent of meastheory.

The concept of measure can be formulated quite simply. O ne wants to be ablLos lanzos Science Special Issue

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assign to a set A a nonnegative number p(A), which will be called the measure of A,with the following properties.

roperty : If Al,A2,. are disjoint sets that are measurable, that is, if each Ai canbe assigned a measure /^(A,), hen their union Al U A2 U (that is, the set consistingof the elements of A,, A2, . is also measurable. Moreover,

/^(Ai UA2U = / < A i ) + p O + * -roperty 2: If A and B are measurable and A is contained in B A B), then B A

(the set composed of elements that are in B but not in A is also measurable. Byproperty 1 then, p. B A) = p(B) ^(A).Two additional properties are assumed for measures on sets in a Euclidean space.roperty 3: A certain set E, the unit set, is assumed to have measure 1: p E ) = 1.roperty 4: If two measurable sets are congruent (that is, a rigid motion maps one

onto the other), their measures are equal.

When dealing with sets of points on a line, in a plane, or in space, one chooses E to bean interval, a square, and a cube, respectively. These choices are dictated by a desire tohave the measures assigned to tame sets agree with those assigned to them previouslyin geometry or calculus.

Can one significantly enlarge the class of sets to which measures can be assignedin accordance with the above properties? The answer is a resounding yes, provided(and it is a crucial proviso) that in property 1 we allow infinitely man y A s. When wedo, the class of measurable sets includes all (well, almost all-perhaps there may besome exceptions the sets considered in both classical and modem mathematics.

Although the concept of countable additivity had been used previously by Poincare,the explicit introduction and development of countably additive measures early in thiscentury by mile Bore1 and Henri Lebesgue originated a most vigorous and fruitfulline of inquiry in mathematics. The Lebesgue measure is defined on sets that areclosed under countably infinite unions, intersections, and complementations. (Such acollection of sets is called a cr-field.) Lebesgue s measure satisfies all four propertieslisted above. Lebesgue s measure on the real line is equivalent to our ordinary notionof length.But how general is the Lebesgue measure? Can one assign it to every set on theline? Vitali first showed that even the Lebesgue measure has its limitations, that there aresets on the line for which it cannot be defined. The construction of such nonmeasurablesets involves the use of the celebrated axiom of choice. Given a collection of disjointsets, one can choose a single element from each and combine the selected elementsto form a new set. This innocent-sounding axiom has many consequences that mayseem strange or paradoxical. Indeed, in the landmark paper on measurable cardinalsmentioned at the beginning of this article, Ulam showed (with the aid of the axiom ofchoice) that if a nontrivial measure satisfying properties 1 through can be defined onall subsets of the real line, then the cardinality of the real numbers is larger than anyoneLos lamos Science pecial Issue 1987

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Probability and Nonlinear Systems-- . --

CANTOR'S D ring the last quarter of the nineteenth century, Georg Cantor introduced aseries of concepts that now form the cornerstone of all modem mathematics-set theory. Those concepts arose from Cantor's attempt to depict the setsMIDDLE of convergence or divergence of, say, trigonometric series. Many such sets havepathological properties that are illustrated by his famous construction, the middle-third set. This set is described by the following recursion. Consider the closed uniTHIRD SET interval [o, 11. First remove the middle-third open interval, obtaining two intervals[O, 1/31 and [2 /3, 11. Next remove from each of these intervals its middle-thirdinterval. We now have four closed subintervals each of length 119. Continue theprocess. After n steps we will have 2 closed subintervals of [0,1] each of lengthI 1/3 . From each of these we will remove the middle-third interval of length 1 /3n+Continue the process indefinitely. Cantor's middle-third set, K, consists of all numbersin [0,1] that are never removed.

This set possesses a myriad of wonderful properties. For example, K is uncount-able and yet has Lebesgue measure zero. To see that K has measure zero, consider

I the set {[O 11 K}, which consists of the open intervals that were removed at someI stage. At the nth stage 2 ' open intervals of length 113 were removed from the

remainder. So, by the countable additivity of measure,+ +2 -'/3* + ...= (1 /3)(1+ 2/23 + (2/3)2 + . = 1.omplementarity (K) = 0 hich is what we wanted to

I

on Cantons middle-third set is discussednvariant Measures, mConsider the closed unit interval [O. 11I0 1

I

I I3 1

I 7 89 9 1

$ j 1*

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imagined. (See "Learning from Ulam: Measurable Cardinals, Ergodicity, and Biomath- BANACH TARSKI PARADOXematics.") Another example is the Banach-Tarski paradox.Banach and Tarski proved that each of two solid spheres Sl and Sz of different

radii can be decomposed into the same finite number of sets, say SI = A \ UAz U . . U A nand s2= B UB2 U U n such that all the A 's and all the B 's are among themselvespairwise disjoint and yet A is congruent to B for all i . It is therefore impossible todefine meaures for these sets, since their union in one fashion yields a certain sphereand their union in a different fashion yields a sphere of different size That sucha construction is possible rests on the complicated structure, earlier pointed out byHausdorff, of the group of rigid motions of three-dimensional Euclidean space.

We close this section on measure theory with a few comments from Kac and Ulam."Attempts to generalize the notion of measure were made from necessity. For

example, one could formulate theorems that were valid for all real numbers exceptfor those belonging to a specific set. One wanted to state in a rigorously defined How can it be thatway that the set of these exceptional points is in some sense small or negligible. One s = A u As u u Ancould 'neglect' merely countable sets as small in the noncountable continuum of all Sg = 61 8points but in most cases the exceptional sets turned out to be noncountable, thoughstill of Lebesgue measure 0 In the theory of probability one has many statementsthat are valid 'with probability one' (or 'almost surely'). This simply means that they and Ai is congruent to Bi for 1 i nhold for 'almost all' points of an appropriate set; i.e., for all points except for a set ofmeasure 0. In statistical mechanics one has important theorems that assert propertiesof dynamic systems that are valid only for almost all initial conditions.

One final remark:The notion or concept of measure is surely close to the most primitive intuition.

The axiom of choice, that simply permits one to consider a new set Z obtained byputting together an element from each set of a family of disjoint sets, sounds soobvious as to be nearly trivial. And yet it leads to the Banach-Tarski paradox

One can see why a critical examination of the logical foundation of set theorywas absolutely necessary and why the question of existence of mathematical con-structs became a serious problem.

If to exist is to be merely free from contradiction as Poincari decreed, we haveno choice but to learn to live with unpleasant things like nonmeasurable sets orBanach-Tarksi decompositions."

Consistency Theorems for Probability MeasuresNow let us return to probability theory and consider the construction of countably

additive probability measures. To see that a finitely additive measure cannot always beextended to a countably additive measure, consider the set of integers and take aselementary events the subsets A of l such that either the set A is finite or the set Q Ais finite. Set 0 if A is finite

1 if A is finite.So, p f2) = 1 and p satisfies the axioms of finite additivity and complementarity.Los Alamo.7 Science Special ssue 987

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M PPING ELEMENT RY EVENTSONTO THE UNIT INTERV LLet each elementary event be one of the setsof all infinite binary sequences with the firsttwo digits fixed Then there are four e lementaryevents.

E i ) Length of the unit interval;=I

However, if were countably additive, then we would hav e the contradiction

Now consider the problem of defining a countably additive probability meason the sample space Cl of all infinite two-letter sequences (each of which represethe outcome of an infinite numb er of independent tosses of a fair coin). Take aselementary event a set E consisting of all sequences whose first r letters are speci(m = 1,2 , . . . Since there are 2rn such eleme ntary events, w e use the a xiom of fiadditivity to assign a probability P of 112 ' to each such event. Can this funcP which has been defined on - he elementary events, b e extended to a countaadditive measure defined on the u-field generated by the elementary events? Uand Lomnicki proved such an extension exists for any infinite sequence of independtrials. Kolmogorov obtained the ultimate consistency results by giving necessary sufficient conditions under which an extension can be made from a finitely addito a countably additive measure, includ ing the case of non-indepe ndent trials. Thextensions put the famous limiting laws of probability theory, such as the laws of lanumbers, on solid ground.

In the case of coin tossing we have chosen our elementary events to be sof infinite sequences whose first rn digits are fixed and have assigned them eacprobability of 112' in agreement with the finitely additive measu re. Now we wshow that the measure defined by these choices is equivalent to Lebesgue's measurethe unit interval [0,1] and is therefore a well-defined countably additive measure. Fassociate the digit 1 with a head and the digit 0 with a tail and encode each outcoof an infinite number of tosses as an infinite sequence of 1's and 0's (101 1 0 .. ,exam ple), which in turn can be looked upon as the binary representation of a real num

(0 < t < 1). In this way we establish a correspondence between real numbers[0,1] and infinite two-letter sequences; the correspondence can be made one-to-by agreeing once and for all on which of the two infinite binary expansions to twhen the choice presents itself. (For instance, we m ust decide between .0100 0. . .OO 111 . as the binary representation of 1 4.

The use of the binary system is dictated not only by considerations of simplicAs one can easily check, the crucial feature is that each elementary event maps intointerval whose length is equal to the corresponding probability of the event. In ffixing the first r letters of a sequence corresponds to fixing the first m binary digitsa number, and the set of real numbers whose first r binary digits are fixed covers interval between i lm nd + 1)/2' , where s 0 ,1 ,2 , . . , or 2 ' 1, dep end inghow the first r digits are fixed. Clearly the length of such an interval, 1/l' , s equathe probability of the corresponding elementary event. Thus the probability measurethe sample space fl of all infinite two-letter sequences maps into the ordinary Lebesmeasure on the interval [0,1] and is therefore equivalent to it.

The space of all infinite sequences of 0's and 1's is infinite-dimensional in sense that it takes infinitely many coordinates to describe eac h point of the spWhat we did was to construct a certain countably additive measure in the space was natural from the point of view of independent tosses of a fair coin.

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This approach immediately suggests extensions to m ore general infinite-dimensia1 spaces in which each coordinate, instead of just being 0 or 1, can be an elemena more general set and need not even be a number. Such extensions, called prodmeasures, were introduced by Lomnicki and Ulam in 1934. (Stan's idea of writinbook on measure theory emphasizing the probabilistic interpretation of measure is subject of the accompanying letter from von Neumann to Ulam.) Mea sures for of curves have also been developed. Th e best known an d most interesting of thwas introduced by Norbert Wiener in the early 1920s and motivated by the theoryBrownian motion. Mathematicians have since found new and unexp ected applicatiof the Wiener measure in seemingly unrelated parts of mathematics. Fo r exampleturns out that the Wiener measure of the set of curves emanating from a point pspace and hitting a three-dimensional region R is equal to the electrostatic potentiap generated by a charge distribution that make s the boundary of the conductor Requipotential surfac e on which the potential is equal to unity. Since the ca lculationsuch a potential can be reduced by classical methods to solving a specific differenequation, we establish in this way a significant link between classical analysis measure theory.Random Variables and istribution Functions

Having introduced the measure-theoretic foundations of probability, we now tto a conven ient formalism for analyzing problems in probability. In many p roblemspossible outcomes can be described by numerical quantities called random variabFor example, let X be the random variable describing the outcome of a single tossa fair coin. Thus, set X e qual to 1 if the toss yields a head and to 0 if the toss yia tail. This is an exam ple of an elementary random variab le; that is, X is a funcwith a constant value on some elementary event and another constant value on comp lementary event. In general a random variable is a real-valued fun ction defion the sample space that can be constructed from elementary random variablesforming algebraic combina tions and taking limits. For example, NH the numberheads obtained in tosses of a coin, is a random variable defined on the sam ple spconsisting of all sequences of T's and H 's of length n ; its value is equal to Ey=where X, 1 if the ith toss is a head and Xi 0 otherwise.

In evaluating the outcomes of a set of mea sureme nts subject to random fluctuatiowe are often interested in the mean, or expected, value of the random variable bemeasured. The expected value E(X) (or m ) is defined as

where X (w) is the value of X at a point w in the sample space a nd P w ) is the probabimeasure defined on the sample space. In the case of a fair coin, P( X 1 112 P ( X 0 112, so the expected value of X is a simple sum:

The expected value of a random variable X is most easily determined by knowosAlamos Science pecial Issue

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its distribution function F This function, which contains all the information we needto know about a random variable, is defined as follows:

where the set X < is the set of all points in l such that X(w) < t. The form of thisfunction is particularly convenient. It allows us to rewrite E(X), which is a Lebesgueintegral over an abstract space, as a familiar classical integral over the real line:

Furthermore, if X has a density function (t ) = dF(t)/dt, then

The expected value is one of the two commonly occurring averages in probability andstatistics; the other is the variance of X , denoted by ^(x) or var(X). The variance isdefined as the expected value of the square of the deviation of X from its mean:

The standard, or root-mean-square, deviation of X is defined as a X) VvarCX .Figures 3 and 4 illustrate two distribution functions, the binomial distributionfunction for the number of heads obtained in five tosses of a fair coin and a normal

distribution function with a positive mean.

The Laws o Large NumbersA historically important problem in probability theory and statistics asks for esti-

mates on how a random variable deviates from its mean, or expected, value. A simplerough estimate is, of course, its root-mean-square deviation. An estimate of a differentnature was obtained by the nineteenth-century mathematician Chebyshev. This esti-mate, known as Chebyshev s inequality, gives an upper limit on the probability that arandom variable deviates from its mean E(Y) by an amount equal to or greater thana a > 0):Chebyshev s inequality: P(\Y E(Y)\ :> a) var(y)/a2.This fundamental inequality will lead us to the famous laws of large numbers, whichtell us about average values for infinite sequences of random variables. We begin byreturning again to the coin loaded in such a way that p is the probability of a head ina single toss. If this coin is tossed a large number of times n , shouldn t the frequencyof heads, Nn In, be approximately equal to p , at least in some sense?

This question can be answered on several levels. Let Xi be the random variabledescribing the outcome of the ith toss. Set Xi 1 if the ith toss is a head and i 0 if

BINOMIAL DISTRIBUTION FUNCTIONFig. 3. The distribution function F t ) or thenumber of heads obtained in n independenttosses of a fair coin is a binomial distribu-tion so called because the probability of ob-taining k heads in n tosses of the coin isgiven by a formula involving binomial coef-ficients namely i ) . Shown here is thebinomial distribution function for the num-ber of heads obtained in five tosses of thecoin. The value of F t ) equals the probabil-ity that the number of heads is equal to orless than t.

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a) NORM L DISTRIBUTIONFUNCTION

m - a G m + ~

b) NORM L DENSITY FUNCTIONdF t )f t )= t

Fig. 4. So many random variables can bedescribed, at least approximately, by thedistribution func tion shown in a) that it isknown as the normal distribution function.Examples of such random variables includethe number of heads obtained i n very manytosses of a coin and, as a general exper-imental fact, accidental errors of observa-tion. The value of Fi t) equals the probabil-ity that the value of the random variable isequal to or less than t m)/u , where mis the mean, or expected, value of the ran-dom variable and is its standard deviation.The mean here is assumed to be positive.)

Shown in b) is the normal density functionf t) = dF t )/d t, which gives the probabil-ity that the value of the random variable ist m)/u.

the ith toss is a tail. Then NH = XI . . .+Xn. Also, the distribution function for eXi is the same, namely, t < o

1 < t(Random variables that have the same distribution function are said to be identicadistributed.) Now the expected value of NH n is easy to compute:

Thus, on the simplest level our guess is right: The frequency of heads, N H / ~approximately equal to p in the sense that the expected value of NH /n is p . But sureeven in a very long series of tosses, it would be foolish to expect NH/ n to exacequal p (and NT/n to exactly equal 1 -p). What one is looking for is a statement tholds only in the limit as the number of tosses becomes infinite.

Bernoulli proved such a theorem: As n gets larger and larger, the probability tNHIn differs from its expected value p by more than a positive amount e tends to

limn>owhere P is the probability measure on fin, the space of all sequences of H s and of length n. No matter what positive e is chosen, the probability that the differebetween the frequency of heads and p , the probability of a head in a single trial, excee can be made arbitrarily small by tossing the coin a sufficiently large number of tim

Let us see how Bernoulli s theorem follows from Chebyshev s inequality. Finotice that var(Xi) p(1 for all i . Second, the random variables Xi , . . ,X,, independent (the outcome of the ith toss has no influence on the outcome of jth). Nofrom the fact that E (XY ) = E (X)E (Y ) for independent random variables, we get

nvar(X1 + +Xn)= ar(xi) = np(1 ).

;=1

So, by Chebyshev s inequality( l N ~ / npi > e < np(1 -p)/n2e2 = p ( l -p)/e^n.

Thus, for each e > 0

Notice that the measure-theoretic background of Bernoulli s theorem is trivial least as far as coin-tossing is concerned), since the events of interest correspondfinite sets. That is, for each n we need only estimate how many trials of length n theare such that the number of heads differs from np by more than en. Nevertheless, tsimple argument just given can be generalized to prove the famous weak law of larnumbers.

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Probability and Nonlinear System

Weak law of large numbers: Let Xi,X 7,X3, be independent, identically distributedrandom variables such that var(Xl) < oo Then for each e > 0

lim P ([( XI Xn)/n E(Xl)l e = 0.n o oIn other words, for any positive e the probability that the deviation between thefrequency in trials and the expected value in a single trial exceeds e can be madearbitrarily small by considering a sufficiently large number of trials.

For our coin-tossing example NH n approximately equals in another sense also.Suppose one asks for the probability that the frequency of heads (in the limit as thenumber of tosses becomes infinite) is actually equal to p. The answer was obtained byBore1 in 1909:

Notice the complexity of the question. In order to deal with it, the sample space i isnow by necessity the set of all infinite two-letter sequences w and the subset of interestis the set A of those sequences for which

M w)lim p .11 oo n

where N,,(w) is the number of H s among the first n letters of the infinite sequencew It takes some work just to show that A is an event in the sample space 0 Unlikethe question that led to the weak law of large numbers, this question required the fullapparatus of modem probability theory. An extension of Borel s result by Kolmogorovis known as the strong law of large numbers.Strong law of large numbers: Let XI , 2 X 3 be independent, identically distributedrandom variables such that E(\Xi\) < oo Then

An A pplication of the Strong Law of Large N umbers Let us illustrate the powerof the strong law of large numbers by using it to answer the following question: Whatis the probability that, in an infinite sequence of tosses of a fair coin, two heads occurin succession?We will first answer this question using only the rules governing the probabilitiesof independent events. In particular, we will use the axioms of countable additivity andcomplementarity and the rule of multiplication of probabilities. Let A

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Since the eventsAi,A2,A ,... are independent, the events 0-Ai,O-A2,Q-A3,...are also independent, and we can apply the rule of multiplication of probabilities:

Finally, by the axiom of complementarity, P U A ~ ) ; that is, there exists, wprobability 1 some k such that the (2k 1)th and (2k)th tosses are heads.

Now we will answer the same question by using the strong law of large numbeLet Xi be the random variable such that

1 i fw ?A ;Xi (w) = 0 i fw $A ;Then XI ,Xz,X3,. is a sequence of independent random variables. Also, they all hathe same distribution: (P(Xi 1) 114,P(X; = 0) = 314, and E (Xi) = 114. Therefoaccording to the strong law of large numbers,

lim (XI +Xll) /n = 4 with probabilityn- m

This result is stronger than that obtained above. It guarantees, with probability 1existence of infinitely many k's such that heads occur on the (2k 1)th and (2ktosses; further, the set of all such k's has an arithmetic density of 114.

Borel's theorem marked the beginning of the development of modem probabity theory, and Kolmogorov's extension to the strong law of large numbers greaexpanded its applicability. To quote Kac and Ulam:

Like all great discoveries in mathematics the strong law of large numbers hasbeen greatly generalized and extended; in the process it gave rise to new problems,and it stimulated the search for new methods. It was the first serious venture out-side the circle of problems inherited from Laplace, a venture made possible only bydevelopments in measure theory. These in turn were made possible only because opolarization of mathematical thinking along the lines of set theory.

The polarization Kac and Ulam were referring to concerns the great debate at tturn of the century about whether the infinite in mathematics should be based upCantor's set theory and its concomitant logical difficulties. The logical problems habeen met, and today we use Cantor's theory with ease.he Monte Carlo Method One of Stan Ulam's great ideas, which was first develop

and implemented by von Neumann and Metropolis, was the famous Monte Camethod. It can be illustrated with Chebyshev's inequality. Suppose that we neto quickly get a rough estimate of w sinx)/x3 dx. Setting t = \/x the problem this to estimate t sin(1 I t ) dt . Let y , , . yn be independent random variables eauniformly distributed on [O, I ] . That is, for all i P (a < yi < b) = b , where ais a subinterval of [0,1]. Now set f (t) t sin (l/ t) and for each let Xi = f y ; ) .Th

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XI, ,X n is a sequence of independent identically distributed random variables. Also,

and

By Chebyshev's inequality we have

< var (1 /n )EX a2 var(x1)/na2-) :Thus if is large, (1 /n )xk,Xi is, with high probability, a good estimate of the valueof the integral. For example, if a = 0.005 and n = 134,000, then

In other words, if we chose 134,000 numbers y i , . . y134,000 independently and atrandom from [O, 11 then we are 90 percent certain that (1/ 134,000)~ , ~ ~ 0 0 0i sin(1 y; )differs from the integral by no more than 0.005. So, if we can statistically sample theunit interval with numbers y , , n, then

(The reader may well wonder why such a large number of sample points is requiredto be only 90 percent certain of the value of the integral to within only two decimalplaces. The answer lies in the use of Chebyshev's inequality. By using instead thestronger central limit theorem, which will be introduced below, many fewer samplepoints are needed to yield a similar estimate.)

The Monte Carlo method is a wonderful idea and, of course, tailor-made forcomputers. Although it might be regarded simply as an aspect of the more ancientstatistical sampling technique, it had many exciting new aspects. Three of these are(1) a scope of application that includes large-scale processes, such as neutron chainreactions; 2) the capability of being completely implemented on a digital computer;and (3) the idea of generating random numbers and random variables. How do wemechanically produce numbers yl , , ,, in [O,11 such that the y; s are independentand identically distributed? The answer is we don't. Instead, so-called pseudo-randomnumbers are generated. Many fascinating problems surfaced with the advent of MonteCarlo. Dealing with them is one of the major accomplishments of the group of intellectsgathered at Los Alamos in the forties and the fifties. (See The Beginning of the MonteCarlo Method. )

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entral Limit heoremWe close this tutorial by returning to the deMoivre-Laplace theorem and int

preting it in the modem context. Let Xi be a random variable describing the outcoof the ith toss of a coin; set Xi = 1 if the ith toss is a head and Xi = 0 otherwise. LSn be the number of heads obtained in n tosses; that is Sn = XI X,, Then de Moivre-Laplace theorem can be stated as follows:

Now np = nE(Xl) E S n ) and \np = &(Xi) = cr(Sn). So if we renormize S,, by setting Yn = s,, E ( s ~ ) ) / ~ ~ ( s ~ ) ,ach Yn has a mean of 0 and a standdeviation of 1 Then the equation above tells us that the distribution function oftends to the standard normal distribution. The central limit theorem is a generalizatiof this result to any sequence of identically distributed random variables. We state central limit theorem formally.Central limit theorem Let Xi,X2,X3, be a sequence of independent, identicadistributed random variables with E(Xl) = m and var(Xi) u oo Set S,Xi - - - X , , . Then

Thus the normal distribution is the universal behavior in the domain of independtrials under renormalization. Its appearance in so many areas of science has ledmany debates as to whether it is a law of nature or a mathematical theorem.

Thanks to the developments in modem probability theory, we begin our invegations with many powerful tools at our disposal. Those tools were forged durinperiod of tremendous upheavals and turmoil, a time when very careful analysis carrthe day. At the heart of that analysis lay the concept of countable additivity. SUlam played a seminal role in developing these tools and presenting them to us.

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PROBABILISTICAPPROACHESto NONLINEARPROBLEMS

Problem 1 Energy Redistribution:An Exact Solution to a Nonlinear Many-P article SystemUlam's talent for seeing new approaches to familiar problems is evident in one

he posed conce rning the-distrib ution of energy in physical systems. Will the energydistribution of an isolated system of N interacting particles always evolve to somelimiting energy distribution? And, if so, what is the distribution? (Note that thisquestion differs from the one asked in statistical mechanics. The re one assumes thatat equilibrium the system will have the most probable distribution. One then derivesthat the most probable distribution is the Boltzmann distribution, the density of whichis l

Obviously, following the evolution of a system of N interacting particles inspace and time is a very complex task. It was S tan's idea to simplify the situationby neglecting the spatial setting and redistributing the energy in an abstract randommanner. Wh at insights can one gain from such a simplification? On e can hope fornew perspectives on the original problem as well as on the standard results of statisticalmech anics. Also, even if the simplification is unrealistic, one can hope to dev elop som etechniques of analysis that can be applied to mo re realistic models. In this case DavidBlackwell and were able to give an exact analysis of an abstract, highly nonlinearsystem by using a combination of the machinery of probability theory and higher orderrecursions (Blackwell and Mauldin 1985 . We hope that the technique will be usefulin other contexts.

Let us state the problem more clearly and define what we mean by redistributingenergy in an abstract random manner. Assum e we have a vast numbe r of indistin-guishable particles with some initial distribution of energy, and that the average energyper particle is normalized to unity. Further, let us assume the particles interact onlyin pairs as follows: At each step in the evolution of the system, pair all the particlesat random and let the total energy of each pair be redistributed between the membersof the pair according to some fixed law of redistribution that is independent of thepairs. Iterate this procedure. Does the system have a limiting energy distribution and,if so, how does it depend on the redistribution law?

artPROBABILITY and NONLINEAR SYSTEMS

The Simplest Redistribution Law To begin we will consider the simplest redistri-bution law: each particle in a random pair gets one-half the total energy of the pair. Ifthe num ber of p articles in the system is finite, it is intuitively clear that und er iterationthe total energy of the system will tend to become evenly distributed-all the particlesLos Alamos Science Special Issue 1987

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SIMPLEST LAW FOR ENERGYREDISTRI UTION

Random Pair Outcomefter Redistribution

Equal Sharingof Total Energy

LIMITING ENERGY DISTRI UTION

Fig. 5 Consider a system of N particleswith some arbitrary initial distribution of en-ergy. Assume that the initial mean energyis 1 and that the particles interact in pairs.Assume further that the total energy of aninteracting pair is redistributed so that eachmember of the pair acquires one-half the to-tal energy of the pair. Then with probability1 the system reaches a limiting energy dis-tribution described by a step function with astep height of 1 at t = 1. That is, the proba-bility that the energy per particle is less thant equals 0 for t < 1 and equals 1 the initialmean energy) for t > 1.

will tend to have the same energy. So, a system with only finitely many particles halimiting distribution of energy, namely, a step function with a jump of size 1 at t =and moreover, no matter what the initial distribution of energy is, the system tendthis distribution under iteration.

Even for a system with a continuum of particles, our observations for the fincase still hold. In order to see this, we formalize the problem in terms of probabitheory.

Let X be a random variable corresponding to the initial energy of the particThus, the distribution function Fl associated with X is the initial distribution of enerFl ( t ) = P(X < t) is the proportion of particles with energy less than t. Our argumeand analysis will be based only on the knowledge of the energy distribution function how it is transformed under iteration by the redistribution law. In terms of distributfunctions, our normalization condition, that the average energy per particle is unmeans that the expected value of X nm t dFAt), equals 1.

We seek a random variable T(X) corresponding to the energy per particle aapplying the redistribution law once. To say that the indistinguishable particles paired at random in the redistribution process means that, given one particle in pair, we know nothing about the energy of the second except that its distribufunction should be the initial distribution function F l . In other words, we can descrthe energy of the randomly paired particles by two independent random variablesand X2, each having the same distribution as X. Thus the simplest redistribution laccording to which paired particles share the total energy of the pair equally, canexpressed in terms of T(X), X1, and X2 as

The new distribution of energy, call it F2 , that describes the random variable Twill be a convolution of the distributions of XI12 and X2/2. Since XI and X2 both hthe distribution F l , he distribution F2 of T X ) s given by

To cany out the second iteration, we repeat the process. The energy T~T (T(x)) will have the same distribution as (Yl +Y2)/2, where l and Y2 are indepenand each is distributed as T(X). In other words, if we let XI , X2, X 3 , and X4independent and distributed as XI , then l is distributed as (XI X^)/2, and Ydistributed as (X3 +X^)/2. The energy is distributed as T*(x) = (Xi X2 X3 X4

After n iterations the energy per particle will have the same distribution as T ((Xi - -+Xy)/2 , where the Xi's are independent and distributed as X. This expresfor Tn(X) s exactly the expression that appears in the strong law of large numbers page 71 . Therefore the strong law tells us that the limiting energy of each particlas n is

X1(w) +Xp(w)lim T n X (w)) = lim 2 = E(X l) 1, almost surely,11 n wwhere E(Xl) is the expected value of the initial distribution Thus, after n iteration

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this random process, the energies of almost all particles converge to unity. In terms ofdistribution functions, we say that in the space of all "potential" actualizations of thisiterative random process, almost surely, or with probability 1, the limiting distributionof energy will be a step function with a jump of size 1 at t = 1 (Fig. 5 .

Notice that for this simplest redistribution law (1) the redistribution operator T isa simple linear operator and (2) even so, the strong law of large numbers is needed todetermine the limiting behavior.More Complicated Redistribution Laws Stan proposed more interesting laws ofredistribution. The redistribution operator T for each of these laws is nonlinear, anddifferent techniques are needed to analyze the system. For example, after pairing theparticles, choose a number a between 0 and 1 at random. Then instead of givingeach particle one-half the total energy of the pair, let us give one particle a times thetotal energy of the pair and give the other particle (1 a imes the total energy. Theenergy T(X) will then have the same distribution as U (XI +X2), where U is uniformlydistributed on [O, 11 (that is, all values between 0 and 1 are equally probable) and U ,X I ,and X2 are independent. What happens to this system under iteration is a much more

R NDOM L W FOR ENERGYREDISTRIBUTION

LIMITING ENERGY DISTRIBUTION

complicated matter. For one thing, unlike the redistribution operator in the simplestcase, the operator T is now highly nonlinear and the law of large numbers is notavailable as a tool. A new approach is required. To get an idea of what to expect,Stan first used the computer as an experimental tool. From these studies he correctlyguessed the limiting behavior (Ulam 1980): no matter what the initial distribution ofenergy is, we have convergence to the exponential distribution (Fig. 6 .

Let me indicate how Blackwell and I proved this conjecture. We used a classical Imethod of moments together with an analysis of a quadratic recursion. For now let us tassume that a stable limiting distribution exists and let X have this distribution. ThenT(X) = U(X1 X2) has the same distribution. So, calculating m n he nth moment ofX (that is, the expected value of Xn) , we have

mn E ( x n ) E (T(x)") = E ((u(x, +x2))")= E ( u n ( x ,+x2)I7).By independence and the binomial theorem

Fig. 6. Consider a system identical to theone described in Fig. 5 except that the totalenergy of an interacting pair is redistributedrandomly between the members of the pair.In particular assume that one particle re-ceives a randomly chosen fraction of thetotal energy and the other particle receives1= E(U")E ((x, +x ~) ~ ) -E E ( x ~ - P ) he remainder. The system still reaches an 1 limiting energy distribution one equal to 0for t < 0 and equal to 1 t or t > 0

Since X1 and X2 are independent, the expected value of each product is equal to theproduct of the expected values, E (X < X;p ) E (X )E (Xl -p). Substituting this intothe equation above and using the definition of moments, we have

Using the fact that mo = 1 we solve for mn.

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This is a quadratic recursion formula. Substituting the initial condition ml = 1, we fthat m2 = 2 and m = 6. An induction argument shows that mn n for all nn is the nth moment of the exponential distribution Of course, our assumption is ta stable distribution and all its moments exist. It takes some work to prove that assumption is indeed true and that no matter what initial distribution one starts wthe distribution of the iterates converges to the exponential.

It should not be too surprising that our result agrees in its general form wthe Boltzmann distribution of statistical mechanics. After all, both are derived frsimilar assumptions. The Boltzmann distribution is derived from the assumptions tI ) energy and the number of particles are conserved, (2) all energy states are equ

probable, and 3) the distribution of energy is the most probable distribution. In problem we also assumed conservation of energy and number of particles. Moreovtaking in our redistribution law to be the uniform distribution makes all energy staequally probable. The difference is that the iteration process selects the most probadistribution with no a priori assumption that the most probable distribution willreached.

We can go further and replace by any random variable with a symmetric disbution on [0 1]. The symmetric condition insures that the particles are indistinguishaWe call the distribution of the redistribution law. Again, one obtains a quadrrecursion formula. Blackwell and I analyzed this formula and showed that for evsuch the system tends toward a stable limiting distribution. In other words, theran attractive fixed point in the space of all distributions. Moreover, there is a one-to-correspondence between the stable limiting distribution and the redistribution law yields it.Momentum Redistribution. There is a corresponding momentum problem. Assuwe have a vast number of indistinguishable particles (all of unit mass) with some inidistribution of momentum. Let us assume that the particles interact in pairs as folloAt each step in the evolution of the system, pair all the particles at random and let total momentum of each pair be redistributed between the members of the pair accordto some law of redistribution that is independent of the pairs. Of course, we wishconserve energy and momentum. These conservation laws place severe constraintsthe possibilities. If v l and v2 are the initial velocity vectors of two particles in a pand v\ and v i are the velocity vectors after collision, then by momentum conservat

and by energy conservationI l 11v2112= llv/ 2 llv2112.

Consider this process in the center-of-mass frame of reference. Let A; be the fractof the total kinetic energy that particle i has after collision and let u be the unit vecin the direction of the velocity of particle i. Then

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From these equations it follows that i = A = 2 and v2 = vi. What this meansis that all we can do is choose in the center-of-mass frame a new direction vectorfor one of the two colliding particles. Everything else is then determined . The otherparticle goes in the opposite direction, and the total kinetic energy in the center-of-mas s frame is divided evenly between the two particles. Thu s, the only element ofrandomn ess is in how the new direction vector is chosen. If all directions are assumedto be equiprobable, then it can be shown that no matter what the initial distribution ofvelocity is, the system tends under iteration to a limiting distribution that is the standardnormal distribution in three-dimensional Euclidean space SR3 We have thus rederivedthe Max well-Boltzmann distribution of velocities. Here again we can go further andconsider m ore comp licated redistribution laws.

Suppose one allows ternary collisions instead of binary collisions. Then there aremore degrees of freedom, and the problem again becomes interesting mathematically.The results of our analysis show that the situation is much like the redistribution ofenergy in that the limiting distribution of velocity depends on the law of redistributionof velocity.

Problem 2 Geom etry Invariant Measures and Dynamical SystemsThe intimate relationship among geometry, measures, and dynamical systems that

was elucidated in the last century continues to deepen and hold our attention today.Poincare made several monumental contributions to this development in his treatiseLes Mithodes Nouvelles de la Micanique Celeste. One major issue he consideredconcerned the stability of motion in a gravitational field such as that of our solar system.Would small perturbations from any given set of initial orbits lead to a collision of theplanets? tremendous amount of work had been done on this dynamica l system,but the governing system of differential equations remained unsolved. Faced with thissituation, Poincare made a wonderful flanking maneuver by introducing qualitativemethods that involved measures.

For the setting consider the motion of N bodies and the corresponding phase spac eS , whose 6N coordinates code the position and momentum of each of the N bodies.The phase space is a subset of Euclidean 6N-space and each point of S corresponds toa state of the system. Consider T, the time-one map of S That is, if s is the initialstate of the system, then T(s ) is the state of the system one time unit later. Now,various notions of stability can be given in terms of the properties of T . One of theseis recurrence, or, as Poincar6 said, stabilit6 la Poisson. A state s is said to berecurrent provided that if the system is ever in s , then it will return arbitrarily close tos infinitely often. Formally, s is recurrent provided that f or every open region U abouts there are infinitely many positive integers n such that Tn (s ) is in U Poisson hadearlier attempted to show this kind of stability for the restricted three-body problem.Poincar6 used the fundamental tenet of measure theory, countable additivity, to provethat the set of all points in the phase space for which recurrence does not occur is ofmeasure zero.Recurrence Theorem: Let B = (s S 1s is not recurrent). Then B has measure zero.Poincare's proof of this theorem (see The Essenc e of Poincare's P roof of the Re-Los lamos Science Special ssue 987

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To summarize, the N -body problem is a classical dynamical system in which thetime-one map T is a continuous one-to-one map of the phase space X onto itself.The inverse map, T 1 , s also continuous. Thus, T is a homeomorphism There is anatural measure on the phase space X that is invariant under T. From one point ofview, this measure is the volume element corresponding to the dimension of the phasespace. From another viewpoint the natural invariant measure expresses the fact that thesystem is a Hamiltonian system. In the phase space X a surface S of constant energyforms an invariant set, and again there is an invariant measure on S corresponding toour ordinary notion of volume. The set B of all points that are not recurrent is alsoan invariant set with respect to T. However, it is not at all clear that we can definesome natural invariant measure on B that is both nonzero and invariant under T. Manydynamical systems being studied today live on invariant sets that, like B, are notmanifolds. Instead they are pathological sets, sets that at one time were thought tobe the private domain of the purest and most abstract mathematicians. The examplesrange from Cantor sets to nowhere-differentiable curves to indecomposable continua.Many of these pathological invariant sets are strange attractors of dynamical systems;the system is attracted in the sense that it will eventually end up on the set from anystarting point. (The discovery of one of the first strange attractors is described in thesection Cubic Maps and Chaos of the article Iteration of Maps, Strange Attractors,and Number Theory-An Ularnian Potpourri. )

Properties of Invariant Sets Let us now indicate some of the problems andtechniques used in studying such sets in the context of dynamical systems We willconsider discrete dynamical systems, that is, systems in which the time evolution isdescribed by discrete steps. We consider a function T that maps a space X into itselfand the iterates of T , that is, T ,T , r 3 , where Tn+' x) = T T (x)) We areinterested in an invariant set-a subset M of X such that T(M) M. The simplestinvariant set consists of a fixed point x such that T(x) = x; a more complicated invariantset is a periodic orbit, a set consisting of the points x, T(x), . . T -~(x),and Tn(x)= x.Invariant sets are further classified according to how points near the invariant set behaveunder T. An invariant set M is called an attractor if there is a region U surroundingM such that if x U, then Tn(x) gets closer and closer to M as n increases. On theother hand, M is called a repeller if there is a region surrounding M such that ifx e (U M , then Tn(x) is not in M for n sufficiently large. For example, if X is thereal number line, then 0 is an attracting fixed point for T(x) =x/2 and a repelling fixedpoint for T^(x) 2 x The intrinsic properties of an invariant set are also of interest.For example, one might want to know whether there is a point x of M such that theorbit of x , that is, x, T(x), T ~(x ) , , is dense in M . If T is an irrational rotation ofthe plane, then the unit circle is invariant and the orbit of every point on the circle isdense in the circle. Another possibility is that T is topologically mixing on M; that is,for every region U of M there is some n such that M c Tn(U).

One central problem we will look at in some depth is the construction of naturalor useful invariant measures for the sets M . In particular we want a measure p suchthat p(X M ) = 0 and p (T- '(B)) = d B ) for each measurable subset B of M. That is,the measure is zero for points outside the invariant set M and is invariant with respectto the inverse of T.Los Alamos Science Special ssue 1987

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n invariant set for the transfermatlon defined in Fig. 7 eonistat ll theclosed subintervals of [0 1] that re not

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We conclude that the Hausdorff dimension of is log 2 og 3 Of course, this is onlya heuristic argument (because we cannot cancel Hf f( C) unless H a C ) is positive andfinite), but it can be justified.

Returning to our example T(x) = (3/2)(1 \2x [) , we have shown that theinvariant set M is Cantor's middle-third set and that the Hausdorff dimension of M is

= log 21 log 3 In fact p = H , Hausdorff's volume element in dimension a is aninvariant measure on M .

Our analysis of this example is typical of the analyses of many discrete dynarn-ical systems. We found an invariant set M that is constructed by an algorithm thatanalyzes the behavior of points near M. The first application of the algorithm yieldsnonoverlapping closed regions J l . n he second yields nonoverlapping subregionsJi . . Jin n each Ji , and so forth. Finally, the invariant set M is realized as

In this example the construction is self-similiar; that is, there are scaling ratiostl , . , n such that a region at iteration k, Jil...k and a subregion at level k 1, Ji ... k+,are geometrically similar with ratio ti,, (In our example ti = ti = 113.) When suchsimilarity ratios exist, one can use a fundamental formula due to P. A. P. Moran forcalculating the Hausdorff dimension of the invariant set.Theorem: If M = flzl uiSn il. k) , then dim(M) = a where a is the solution oft ? + . . - + t E = 1. Moreover, 0 < Hff(M)< oo.That is, is the Hausdorff dimension of M , and Ha is a well-defined finite measureon M.andom Cantor Sets One of my current interests centers on analyzing the invariant

sets obtained when the dynarnical system experiences some sort of random perturba-tion. The perturbation introduces a perturbation in the algorithm used to construct theinvariant set. Thus we randomize the algorithm, and the scaling ratios ti , t2, , ,,,instead of having fixed or deterministic values as before, are now random variables thathave a certain probability distribution. One theorem of Williams and mine (Mauldinand Williams 1986) is that the Hausdorff dimension of the final perturbed set is,with probability 1, the solution of

where E (t? t? . - s the expected value of the sum of the ath powers of the scalingratios. Note that this formula reduces to Moran's formula in the deterministic case.

As an example suppose our randomly perturbed system produces Cantor subsetsof [0,1] as follows. First, choose x at random according to the uniform distribution on[0,1]. Then between x and 1 choose y at random according to the uniform distributionon [x , 11. We obtain two intervals Jl = [0,x ] and J2 = [y 11. Now in each of theseintervals repeat the same procedure (independently in each interval). We obtain twoOS Alamos Science pecial Issue 1987

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THE GOLDEN ME NFig. 9 (a) Consider a rectangle with sidesof length A and B, A < B. Let r denotethe ratio of A to B. Divide this rectangleinto a square of side A and a new rectan-gle. If the ratio of the lengths of the sides ofthe new rectangle, (B A)/A, also equalsr then both the original rectangle and thenew rectangle are golden rectangles and ris equal to the golden mean m. (The nu-merical value of m, ( 1 /2, is obtainedby solving the two simultaneous equationsr = A/ B and r = (B A)/A.) (b) The processof dividing a golden rectangle into a squareand a new golden rectangle can, of course,be continued indefinitely. It can be shownthat the logarithmic spiral given in polar co-ordinates by logp = m passes throughtwo opposite vertices of each successivelysmaller square. This fact may help explainwhy the Hausdorff dimension of the randomCantor sets described in the text is equal tothe golden mean.

subintervals of J i l l and Jiz, and two subintervals of J2, 21 and 722. Continue process. We will obtain a random Cantor set, and its Hausdorff dimension a is, probability 1, the solution of E t r t \ = 1, or

little calculus shows thatV^ 1a = - 2 , the golden mean

problem left for the reader: Why should the golden mean (Fig. 9 arise asdimension of these randomly constructed Cantor sets?Problem 3 omputer Experiments and Random Homeomorphisms

One topic Stan and I discussed several times was whether one could "randomdynamical systems in some way. Is it possible to define a probability measurea wide class of dynamical systems such that meaningful statements could be mfor instance, about the probability that a system would become turbulent or aboutexpected time to the "onset of chaos"? To get started on this very ambitious problwe discussed how we would go about generating homeomorphisms at random. simplicity, let us generate homeomorphisms of the unit interval [0,1] onto itself. Twe wish to build continuous, strictly increasing maps h with h(0) = 0 and h(1)One algorithm for doing this randomly follows.Set h(0) = 0 and h(1) = 1. Choose h(1/2) according to the uniform distribution[0,1]. Continue by choosing h(l/4) and h(3/4) according to the uniform distribuon [O, 1/21 and [1/2,1], respectively. In general, once the values of h(i/2") hbeen determined for i = 0, 1 , 2", choose h ((2i 1)/2'^+') according to the unifdistribution on [h(i/2"), h(i 1)/2"]. This simple algorithm is easily implementeda computer. (It needs no more than fifty lines of FORTRAN.) If the computer's randnumber generator is fairly good, general properties of these functions can be guesHowever, to show that this algorithm defines an associated probability measure P1, the set of all homeomorphisms of [0,1] onto [0,1], is no small task. First we n

to define a class of elementary events and the probabilities associated with them. elementary event in the sample space i comes naturally from the random algoritFor a positive integer n, consider the dyadic grid on [0,1] given by the points 12/2", (2" )/2". Over each grid point i 12" construct a "gate", an interval (aisuch that a, < bi < a;+l. An elementary event consists of all elements h of i that pthrough all the gates: ai < h(i/2") < bi9for i = 1, 2, 2" 1 (Fig. 10).

The probability assigned to an elementary event is defined by induction on n .example, if n = 1, an elementary event consists of all h that pass through a singate: a < h(1/2) < b. Since the random algorithm chooses h(1/2) uniformly, probability assigned to this event is the length of the interval, b . If n > 1,probability of an elementary event is determined from the conditional probabiligiven by the algorithm. For example, the distribution function of the random variah(3/4) is P (h(3/4)

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conditional probability that h (314) t , given that h (112) = s. It follows directly fromthe construction algorithm thatif 1 s t

P ( h ( 3 / 4 ) 5 t \ h ( l / 2 ) = s ) = - s ) / ( l - s ) i f s < t < 1if t < s .

= t + 1 ) ln(1 ).The distribution of h(3/4) is shown in Fig. 11.

The exact formulas for the probabilities assigned to various elementary events arequite complicated. What is required is to determine that probabilities of the form

CONSTRUCTION OFELEMENTARY EVENTSFig. 10 In the study of random homeomor-phisms described in the text, an elementaryevent is defined as the set of all homeomor-phisms h that pass through 2 1 gatesconsisting of open intervals (ai, b i ) over thegrid points i / 2 ( i = 1 , 2 , ..., 2 1). Theai9s and hi s are restricted by the conditionsa; < bi < a;+,. Shown here is one possibleset of gates for n = 2 and a member of thecorresponding elementary event.

satisfy Kolmogorov's consistency theorem. We have shown that these conditions are in-deed satisfied and therefore a probability measure P is defined on the homeomorphismsof [0,1]. To see what these homeomorphisms look like, we used the computer. Figure12 shows a few samples from our computer studies in which the values of h(i 12 ) arecomputed for n = 10.

S. Graf, S. C. Williams, and I studied this method in detail (Graf, Mauldin, and DISTRIBUTION FUNCTIONWilliams 1986). For example, we examined a large number of the computer studies FOR h ( 3 / 4 )and guessed that with probability the derivative of a random homeomorphism at theorigin is 0. This conjecture turned out to be correct. The argument is essentially the Fig. 11. As demonstrated in the text, F(t )~ h 3 / 4 ) < t) equals 0 for t < 0 andfollowing. First, since h is increasing and h(0) = 0, it is enough to show that equals t (1 ) ln(1 or t > 0. Shown

h(\/ 2 ) h(o)lim = lim 2 h(1/2 ) = 0.n 112 n + o oSecond, set

here is the graph of that distribution func-tion.

where n = 1,2 ,3 , . . It is intuitively clear and can be proved that , ,Qs,. .are independent random variables, all uniformly distributed on [O, 11. Set Xn = In Qn.The n 's are independent and identically distributed, and E (Xn = f ~ l ndt = .Therefore, by the strong law of large numbers,

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COMPUTER-GENERATEDRANDOM HOMEOMORPHISMSFig. 12. Each of the graphs here is a randomhomeomorphism passing through a set ofpoints / /21) h 2/2I0), /~ 102 3/2 ~~ ).The sets of points were generated by a com-puter according to the algorithm describedin the text. Such graphs provide experimen-tal data about the properties of the homeo-morphisms as a class.

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Multiplying both sides by n we have, with probability 1,oo= lim X lirn z l n < ' , lirn l n n f , .

n i 0 0 n m

Exponentiating we get0 lirn nX Pp lim 2"A(1/2'),

I I ~ n mp

which is what we wanted to show.We have also shown that, with probability 1, a random homeomorphism has a

derivative of 0 almost everywhere, that is, everywhere except for a subset of [0,1] withLebesgue measure 0. Consequently, with probability 1, a random homeomorphism isnot smooth. Therefore this approach will not yield answers to questions concerning thetransition from smooth to turbulent, or chaotic, behavior. As often happened with Stan'sproblems, the original question, which was motivated by physics, would eventuallybecome a purely mathematical problem.

By the way, our original studies on an Apple computer illustrate the pitfalls ofworking with numerical results. From looking at the graphs we guessed that the setof fixed points for these homeomorphisms is a Cantor set. When we were unable toprove this conjecture, Tony Wamock conducted more highly resolved computer studieson a Cray. The results suggested not that the fixed points are a Cantor set but ratherthat a high proportion of the random homeomorphisms have an odd number of fixedpoints (see the accompanying table). This time we guessed that, with probability 1, arandom homeomorphism has a finite odd number of fixed points. Indeed we were ableto prove this; however, the proof is too complicated to outline here.

A few closing comments on this problem. First, the procedure for generating arandom homeomorphism can also be viewed as a procedure for generating a distributionfunction at random. Thus, we have a probability measure on the space of probabilitymeasures This viewpoint was thought of and developed earlier by Dubins and Freed-man. Second, Stan and I did consider the generation of random homeomorphisms onother spaces. For example, the algorithm for generating homeomorphisms of the circlereads almost exactly like that for generating homeomorphisms of the interval. (How-ever, in that case we don't know whether there is a positive probability of generatinghomeomorphisms with no periodic points. This is an interesting possibility.) Third,it is possible to bootstrap oneself up from generating homeomorphisms of the intervalto generating homeomorphisms of the square, the cube, and so on. These possibilitiesare described in Graf, Williams, and Mauldin 1986. Finally Stan had some wild ideasabout "crossing" random homeomorphisms with something like Brownian motion toproduce flows at random.

That wildness was the joy of being with Stan Ulam. His boundless i

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