Benoit Mandelbrot intrigued by geometry of nature. Interested all phenomena.

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1 Fractals Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. (Mandelbrot, 1983) Repeat tentative ideas from earlier - expand to better understand term fractal. Benoit Mandelbrot intrigued by geometry of nature. Interested all phenomena. Investigated cotton prices. Economists and random part. cotton prices contained orderly(regular) part Implied in long term, prices driven by real forces and in short term, prices are random. But the data did not agree! Too many large jumps, i.e., ratio of small price changes to large prices changes was much smaller than expected. In addition, was large tail (at high end) in distribution of price values. If prices changed at random, then would be distributed in normal distribution - the Bell curve. If true then probability of price change differing from mean change by 3 standard deviations extremely small - cotton data did not fit Bell curve! 1

2 Economics rules of road, articles of faith and prejudices imply small, transient changes have nothing in common with large, long term changes - that fast fluctuations are random - small scale, short term changes are just noise and thus they are unpredictable. Long term changes, however, are determined by deep macroeconomic forces. Dichotomy false. Mandelbrot showed all changes bound together - one pattern across all scales. Showed cotton prices, which did not obey normal distribution, produced symmetry from point of view of scaling - a kind of scale independence. Each particular price change was random and unpredictable, but sequence of changes was independent of scale. Mandelbrot also studied transmission of data - in particular errors in transmission. Mandelbrot found, contrary to intuition, can never find time during which errors were scattered continuously - within any burst of errors, no matter how short, there would always be periods that were completely error free. Mandelbrot found geometric relations between bursts of errors and error-free transmission periods. Relationship was same no matter time scale, namely, Cantor set(study later) - strange scattering of points arranged in clusters - infinitely many but yet infinitely sparse. Showed that transmission errors are Cantor set arranged in time. Investigated height of Nile river. Classified variations into 1. Noah effect discontinuous 2. Joseph effect persistence 2

3 Other researchers neglected changes of Noah type. Data showed persistence applied over centuries as well as over decades - another example of scale independence. Research showed that trends in nature are real, but can vanish as quickly as they arise. Discontinuities, bursts of noise, Cantor sets - where does all of this fit into classical geometry of lines, planes and circles. Lines, planes and circles are powerful abstraction of reality - this kind of abstraction eliminates complexity and prevents any understanding from being developed. As quote at beginning of chapter states: clouds spheres mountains cones lightning does not travel in straight lines Mandelbrot proposed - there is geometry associated with complexity and implies strange shapes that have meaning in real world. Euclidean objects are idealizations and cannot represent essence of geometry of an object in real world. Mandelbrot asked - how long is coastline of Norway? He said, in a sense, that coastline is infinitely long or that answer depends on size of your ruler. Common sense says that smaller ruler sees finer detail and measures longer length - but must be some limiting value! If coastline was Euclidean abstraction, limiting value would be equal to length. 3

4 Thinking about these ideas leads us to think about dimension. Ask - what is dimension of ball of twine? 1. from great distance - a point dimension is 0 2. from closer, fills spherical space dimension is 3 3. closer still and twine now seen and dimension now 1 (dimension is tangled up making use of 3D space) Says, effective dimension not always equal to 3. What about in between - say non-integer dimension? See later, non-integer dimensions associated with coastlines. Coastline = new kind of curve called fractal and length is infinite! Will learn how to determine dimension of these fractals - fractal dimension will be way of measuring quantities such as roughness or irregularity in object - twisting coastline, despite not being able to measure length, has certain degree of roughness. Will see fractal dimension remains constant over different scales. Simple Euclidean 1D line between 2 points fills no space at all. Will study fractal line between 2 points called Koch curve that has infinite length, but fits into finite area, completely filling it. Will be seen to be more than line, but less than plane and cannot be represented by Euclidean geometry. Its dimension > 1 but < 2 (actually ). Smooth curves are 1D objects whose length can be precisely defined between two points. A fractal curve has infinite variety of detail at each point along curve implying more and more detail as zoom in, which implies that length will be infinite. 4

5 Suddenly fractals everywhere. Found hidden in all kinds of data. Property of infinite detail at each point and detail looks like original called self-similarity. Self-similarity is symmetry across scale or recursion of patterns within patterns. At first sight, idea of consistency on new scales seems to provide less information because goes against tradition of reductionism(building blocks). Power of self-similarity begins at much greater levels of complexity. Will be matter of looking at whole, rather than the parts. Now for a detailed study of fractals. At that point will have developed necessary tools to deal with sciences of chaos/complexity. The Nature and Properties of Fractals Fractals are mathematical sets of point with fractional dimension. Fractals are not simple curves in Euclidean space. A Euclidean object has integral dimension equal to dimension of space the object is being drawn in. In addition, if Euclidean line connects 2 points in 3-dimensions and also stays within finite volume, then length of the line is finite. 5

6 Fractal is line that stays within finite volume, but length = infinity. Implies complex structure at all levels of magnification. In comparison, Euclidean line eventually looks like straight line at some magnification level. To describe this property of fractal, generalize concept of dimension. Dimensionality D of space defined as number of coordinates needed to determine unique point in space. Defined this way, only allowed values for D are non-negative integers 0, 1, 2, 3,... Several ways concept of dimension redefined - still takes on non-negative integer values for systems described above, but also takes on non-negative real number values. Adopt simplified version of Hausdorff dimension called box counting or capacity dimension. In box-counting scheme, dimension of object determined by asking how many boxes are needed to cover object. Here appropriate boxes for coverage are lines, squares, cubes, etc. Size of boxes is repeatedly decreased and dimension of object is determined by how number of covering boxes scales with length of side of box. 6

7 fractal: Irregular shape with self-similarity - has infinite detail - has no derivative. Wherever chaos, turbulence, and disorder are found, fractal geometry is at play. fractal dimension: Measure of geometric object - can take on fractional values. Usually meant Hausdorff dimension. Fractal dimension now more general term for measure of how fast length, area, or volume increases with decrease in scale. The Concept of Dimension Have used concept of dimension in two ways: 1. 3 dimensions of Euclidean space (D = 1, 2, 3) 2. Number of variables in a dynamic system Fractals(irregular geometric objects) require third meaning. 1 dimension, consider line of length. Need ` 1 box(a line) of length ` 2 boxes of length `/ m boxes of length `/2 m... If we define length of the box, then m = m th m = ` 2 m 7

8 Thus, number of boxes N( m ) scales (grows) as N( m )= ` m In two dimensions, consider square of side `. Need 1 box(square now) of area 4 boxes of area (`/2) m boxes of area (`/2 m ) 2... Thus, number of boxes `2 N( m ) scales (grows) as N( m )= Generalizing to D integer dimensions, we have ` m 2 N( m )= ` m D Now after some algebra (details of algebra not important - final result isimportant). ` log (N( m )) = log m D = D log = D(log (`) log m ) ` m! D = log (N( m)) log (`) log ( m ) 8

9 Define dimension D by D = lim m!1 log (N( m )) log (`) + log 1 m As m!1, log (`) becomes negligible compared to other terms and we have D = lim m!1 log (N( m )) 1 log m =lim!0 log (N( )) log (1/ ) where N( ) = number p dimensional cubes of side needed to completely cover set. Hausdorff or fractal dimension D of set of points in p dimensional space. Wow! Very mathematical/tedious derivation. Make clear meaning of result by examples. Examples: (1) Single point: only one cube is required. This means that as expected. N( ) =1! log N( ) =0! D =0 9

10 (2) A line of length ` : Number of cubes of side required = number of line segments of length. Therefore and D =lim as expected. log `!0 log 1 N( ) =` log 1 + log (`) =lim!0 log 1 =lim!0 1+ (3) A square of area `2 : Number of cubes of side required = number of squares of side. Therefore D =lim log!0 log `2 2 1 =lim!0 2+2 N( ) =`2 2 2 log 1 + log (`2) =lim!0 log 1! log (`) =2 log 1 log (`) log 1! =1 as expected. Definition of dimension works for Euclidean objects and clearly makes sense. 10

11 Alternative way to think about it. Always good to look at a new concept in several ways! The Hausdorff Dimension If take object residing in Euclidean dimension D and reduce linear size by 1/r in each spatial direction, its measure (length, area, or volume) would increase to N = r D times original. Pictured in figure below. Consider get N = r D, take log of both sides and log (N) =D log (r) Solve for D to get D = log (N) log (r) 11

12 Point of exercise: Examined this way, D need not be an integer, as in Euclidean geometry. Could be a fraction, as in fractal geometry. Generalized treatment of dimension named after German mathematician, Felix Hausdorff. Proved useful for describing natural objects and for evaluating trajectories of chaotic dynamic systems. The length of a coastline Mandelbrot began treatise on fractal geometry by considering question: How long is coast of Britain? Coastline is irregular, so measure with straight ruler, (figure below), provides estimate. Estimated length L, equals length of ruler, s, multiplied by N, number of such rulers needed to cover measured object. In figure below measure part of coastline twice, ruler(unit of measurement) on right is half that used on left. But estimate on right is longer. If scale on left is one, have six units, but halving unit gives us 15 rulers (L = 7.5), not 12 (L = 6). If halved scale again, get similar result, a longer estimate of L. In general, as ruler gets diminishingly small, length gets infinitely large. Concept of length, begins to make little sense. 12

13 The Cantor Set Consider figure below. In diagram, points in interval (0,1) tracked for 4 iterations. Therefore, D = lim K!1 log (2 K ) log Dimension of Cantor set is not an integer! st iteration, middle third of points leave interval (0, 1). Middle thirds of the two in segments leave interval and so on. Bottom row(after infinite iterations) is Cantor set. Cantor set has some stunning properties. For one thing, in infinite limit it has zero measure (it is the empty line). But set is far from empty, in fact, has an uncountable number of points. To cover the set at step K we need K 1 3 log (2) = log (3) =0.631 N(K) =2 K cubes of side = K = lim K!1 K log (2) K log (3) 13

14 The Koch Snowflake Koch curve is constructed by recursion as exhibited in figures below. At each step middle-third of each segment is replaced with V shaped bulge. Curve turns out to have infinite length, while enclosing (together with its natural base: original line segment) a finite area. Demo Koch Snowflake. Begin with straight line of length 1, called initiator. Then remove middle third of line, and replace it with two lines that each have same length (1/3) as remaining lines on each side. This new form is called the generator, because it specifies rule that is used to generate a new form. Rule says to take each line and replace it with four lines, each one-third the length of original. 14

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16 Dimension of Koch snowflake given by D = lim n!1 log (4 n ) log 1 3 n = lim n!1 n log (4) n log (3) = log (4) log (3) = Get dimension between 1 and 2. Clearly, final length of the fractal line lim n!1 4 3 n!1 is infinite even though it stays within a finite area of the 2-dimensional Euclidean plane. Sierpinski Triangle - Figures below illustrate construction of Sierpinski triangle by iteration: 16

17 Calculating the dimension. D = log (N) log (r) = log (3) log (2) = Again get number between 1 and 2. IFS Fractals One method of generating fractals is to choose dilation(rescaling) and translation transformations at random. Say, for example, t map given by: xn e f xn+1 y n+1 = a c b d y n + or x n+1 = ax n + by n + e y n+1 = cx n + dy n + f where the [a,b,c,d] dilation matrix and the [e,f] translation vector are chosen randomly. A fern can be generated using appropriate matrix choices: run fern.m. 17

18 Demos: Fern I and Fern II Sierpinski.m Barnsley.m Sierpinski Arrowhead Beech Branch Bush Elm Branch Plant Tree 1 Tree 2 Tree 3 Koch Snowflake. 18

19 Fractal Dimension Program Illustrate program that carries out box dimension calculations on complex fractal structures. Idea of covering fractal with boxes shown below. Program simply chooses a box size, covers the fractal and counts the number of boxes. It then reduces the box size and repeats until the box is small. At that point it can recognize the limit of plot log N(r) versus log r and determine the slope which corresponds to the dimension. If apply the program to Koch snowflake which looks like 19

20 we get result The log(n)/log(r) graph looks like: Even more dramatic, apply program to fractal tree(created on computer) as shown below 20

21 and get result The log(n)/log(r) graph looks like: What about a picture of a real tree? I found a picture of a real tree and put it into program. 21

22 and result is The log(n)/log(r) graph looks like: 22

23 Complex Maps To see self-similar nature of fractals even more dramatically study 2-dimensional complex maps. In particular, look at Mandelbrot fractal boundary and cube roots of 1 via Newtons rule(called Newton attractor). Mandelbrot Sets Key mapping is quadratic map given by: z n+1 = z 2 n + c where z n are complex numbers and c = complex parameter. Turns out that if z n > 2 further iteration of this quadratic map is unbounded, i.e., for different values of c, trajectories either stay near origin or escape to infinity. When calculations are done we stop iteration when this condition is satisfied. When investigating these maps, one iterates equation as follows: (a) start with complex value for c inside rectangular boundary (b) iterate equation until either (1) z n > 2 or (2) preset number of iterations exceeded In case (1) color the starting point (c) in red. In case (2) leave the point (c) black. Boundary between two regions is Mandelbrot fractal as shown below. 23

24 Alternately, in case (1) color starting point (c) by number of iterations it took to escape. In case (2) leave point (c) a single color (say dark blue). Boundary between two regions is still Mandelbrot fractal as shown below left, but wide variety of spectacular images can be generated by clever choice of colormap. or choosing different colormap (no change in information) get 24

25 Run programs for colormaps and zooming. If zoom far enough see entire repeating itself as all fractals do since they are self-similar. So what is a fractal? It is an irregular geometric object with an infinite nesting of structures at all scales. Why do we care about fractals? natural objects are fractals chaotic trajectories (strange attractors) are fractals - pictures assessing the fractal properties of an observed time series is informative complex systems that self-organize will have self-similar or fractal behavior at all scales complexity: While, chaos is study of how simple systems can generate complicated behavior, complexity is study of how complicated systems can generate simple behavior. Example of complexity is synchronization of biological systems ranging from fireflies to neurons. complex system: Spatially and/or temporally extended nonlinear systems characterized by collective properties associated with system as a whole that are different from characteristic behaviors of constituent parts. 25

26 Final Example - Cube Roots of 1 Another interesting map is called the Newton attractor. Start with an equation like z 3 1=0 which is solved via classical Newton iteration method for finding zeroes: z 3 1=0=f(z) z n+1 = z n f(z n ) df (z n )/dz = z n z 3 n 1 3z 2 n = 2 3 z n + 1 z 2 n z 0 Starts with some initial complex point. Iterate until answer stops changing (converges to one of three complex roots of 1). One might assume that plane will divide up into three pie shaped wedges such that all starting points in a given wedge are closest to one of the three roots(in its basin) and eventually end up at that closest root (basin attractor). In fact, this does not happen. The boundaries of wedges are nibbled at by interlaced basins of attraction for all roots producing a complex fractal boundary as shown below(remember magnetic pendulum). 26

27 If zoom in on boundary see complex self-similar structure of fractal strange attractor making up boundaries between basins as shown below. 27

28 Further zooming in shows self-similar structure of fractal boundary. 28

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30 Let me start with some further miscellaneous thoughts. Consider the origin "myth": Emergence vs. Reductionism More spectacular and bizarre than anything dreamed up by primitive tribe. Origin "myth" contains (at least) two examples of so-called emergence and of broken symmetry: "electroweak" transition responsible for "inflation". Emergence of protons/neutrons from primeval plasma of quarks and gluons. Physicists - like to use reductionism : to take apart and analyze into simpler and simpler components Instead of reductionism, one sees (in the real world) complex consequences "emerging" from simple laws. One sees, to borrow term from 19th century biology, "emergence at every scale" By 1932, fundamental laws for most sciences were reasonably well known. If reductionism true, why are we still working? Example of superconductivity(disappearance of electrical resistance in some metals discovered in 1911) instructive. Studied (unsuccessfully) for 46 years thereafter. Basic underlying theory of electrons in metals discovered missing years ( ) seem to me to be crucial proof of failure of reductionism(similarly: 30 years of string theory). 30

31 If principles of reductionism true - know basic laws - compute consequences - should have understood. Only was solution found. Not lack of computational power. Had ability to follow motion of all particles, computer output superconductivity - BUT not why. In fact, question is meaningless! Solution involved broken symmetry : underlying laws of macro-phenomena have symmetry not manifest in consequences of laws. Atom of gold cannot be shiny and yellow and conduct electricity - properties with meaning only for macroscopic sample. Nor can single atom of lead be superconducting: something which can only happen to macroscopic sample. Large collection of objects can behave completely differently - have different symmetry from anything that separate objects can themselves exhibit. Are broken symmetry ideas general? Go up hierarchy of intellectual endeavors: at each stage would be new concepts, not computable - often not imaginable from simpler substrate. One cannot deduce from simple facts of chemistry that self-replicating molecules are possible. In biology, begin to have some record of stages of emergence: sex, symbiosis, morphology, nervous system and on to greatest puzzle of all emergence of consciousness. 31

32 Already at stage of morphology, and many subsequent stages, we are confronted with Steve Gould s question - Would it have looked same if ran the tape twice? Answer: obviously not, so in some sense nothing is predictable from first principles. When history enters, determinism must leave! Yet one still has motivation for, at each stage, an explanation of each concept in terms of something simpler. I do not argue against reductionist program itself but against the arguments associated with it, which gives Theory of Everything status of a God Equation from which all knowledge is supposed to follow. I see the structure of the world as a hierarchy with levels separated by stages of emergence. This means that each is intellectually independent from its substrate. Reduction has value in terms of unifying sciences intellectually and strengthening their underpinnings but not as program for comprehending complexity of the world completely. In 1820, great French scientist Pierre Laplace staked out ultimate claim of physicists for intellectual hegemony over all science - and possibly a lot else. Philosophical position named Laplacean Determinism. Hubris remains within minds of many physicists, and it is with this scenario that otherwise extremely hardheaded governments continue to shell out billions for scientific instruments for probing depths of atom and universe. 32

33 Remarkable achievements of these branches of physics have made scarcely any difference at all to any practical achievement of Laplace s dream. In fact, have only made it less achievable than it seemed in Laplace s day. As we learn more about how world works, the more we find it depends on set of principles which separate operation of complex macroscopic world from underlying hardware of fundamental physics. Simple laws, rules, mechanisms can, when applied to very large assemblages, lead to qualitatively new consequences. Thus, the remarkable simplicity and universality which my colleagues find in the underlying quantum physics of the universe - and the even greater simplicity for which they are striving is increasingly irrelevant to genuine complexity of world in which we live. Something else more important is going on when we change from the atomic to the macroscopic scale. Seems mysterious that simplest properties of everyday materials have little direct connection to properties of atoms of which they are made, or to very general laws which rule atoms. When one puts a large number of atoms of copper together they become metal: something conceptually new, a property that was not evident in separate atoms themselves. What happens is called broken symmetry: idea is that the state of a collection of particles can choose not to have same symmetry as underlying laws which govern those particles. 33

34 Important lessons are two: 1) totally new physics can emerge when systems get large enough to allow them to break the symmetries of underlying laws 2) those emergent properties can be completely unexpected and intellectually independent of underlying laws Term emergence arose in problem of how life evolved from non-life. No agreed answer - in fact, very few things one can say. See in every living object, no matter how primitive, necessity for permanent, stable structure. Permanent structures play two roles: most important, but less obvious, is energetic: life - including replication involves use of stored energy and doing of work. This can be done by some biological structure - it does require structure! Second is stable storage of genetic information. Given these the exponential growth mechanism of Darwinian evolution can do almost everything from then on. But how it all actually started to happen is beyond knowing now. Really hard questions come at next stage: Life has learned - in order - three things: first, to act on own behalf; second, and almost implicit in that action, to be free to do so, i.e., to have free will; and third, to be conscious, whatever that is. I say almost implicit - after all, plant acts on own behalf by sending out branches and roots and leaving useless ones to die, and no one "accuses" plant of having free will. 34

35 That is a big hint: somehow whole process of searching for light and nutrients gradually became virtual rather than real, and result was evolution of brain function. However, in the end one has conscious objects, relating and competing, and the rules of the game become only marginally influenced by underlying chemistry and genetics. From these emerge ever more complex further worlds: language, religion, money, art, politics,... Efforts of scientists to understand emergence of free will/consciousness so far totally in vain. Predictability is certainly an illusion at even the Darwinian level of evolution and from there on as we will see. The Reverend Thomas Bayes and Needles in Haystacks In 1759 Reverend Thomas Bayes first wrote down the chain rule for probability theory P(A B)P(B)=P(B A)P(A) Bayes had no idea that this simple formula might have far-reaching consequences, but due to efforts of Harold Jeffreys, Bayesian statistics is now taught to statistics students. Unfortunately not taught to nutritionists or experimental physicists. These methods are the correct way to do inductive reasoning from necessarily imperfect experimental data. P(hypothesis data) = P(data hypothesis)p(hypothesis) P(data) What Bayesianism does is focus attention on question one wants to ask of data. Says, in effect, how do these data affect my previous knowledge of situation? 35

36 Essence is to clearly identify possible answers, assign reasonable a priori probabilities to them and then ask which answers made more likely by data. It s particularly useful in testing simple null answers. Consider, question of looking for needle in haystack. Set it up: two sources of hay one with no needles at all; one with up to 9 needles per stack. Assign probability1/2 to case where buying from needle-free source. (Represents null hypothesis in this example.) If dealing with potentially needly hay, assume that p = 1/2 x 1/10 for 0, 1,..., 9 needles in any stack (p=1/20 for each case - 10 cases : p=1/2)) Search for needles in one stack, and find none. What do I now know? Know that this outcome had p = 1/2 for needle-free hay, p = 1/20 for needly hay; Hence probability of this outcome is 10 times as great if hay is needle free. New a posteriori probability of null hypothesis is therefore(by Bayes) 10/11=(1/2)((1/2+1/20) rather than 1/2. Clearly should buy this hay. Now suppose you were an ordinary statistician: you would simply say (after the measurement) that the expected number of needles per stack is 0±2.25, and to get to 90% certainty one must search at least ten more haystacks! 36

37 Thus, it is very important to focus on what question one wants to ask - namely, whether you have any reason to believe that there is any effect at all. In physical experiments, one is often measuring something where we know there is finite answer, but we do not know how big. In this case Bayesian approach is same as conventional rules, since we have no viable null hypothesis. But exist many very interesting measurements where one does not know whether effect one is testing exists, and where the real question is whether or not the null hypothesis - read simplest theory - is right. Then Bayes can make very large difference. Emerging Physics We will see in class the triumph of emergence over reductionism: that large objects such as ourselves are, in myriad ways, product of principles of organization and of collective behavior which in no meaningful sense can be reduced to the behavior of our elementary constituents. Physics: The Opening to Complexity In minds of the public, or even of scientists from unrelated fields, physics is mainly associated with extremes: big bangs and big bucks; cosmic/subnucleonic scales; matter in most rarified form - single trapped atoms; or measurements of great precision to detect phenomena - dark matter, proton decay, neutrino masses - which may well not be there at all. The intellectual basis for this kind of science has been expounded by many. 37

38 Buzzword is reductionism" - idea that goal of physics is solely to discover fundamental laws which all phenomena must obey. Many physicists now working at another frontier between the mysterious and the understood: the frontier of complexity. At this frontier, watchword is not reductionism but emergence. Emergent complex phenomena are not in violation of microscopic laws, but do not appear as logical consequences of these laws. That this is case will be illustrated by many examples which show that a complex phenomenon can follow laws independently of the detailed substrate in which it is expressed. Principle of emergence is as pervasive a philosophical foundation of modern science as is reductionism. It underlies, for example, all of biology and much of geology. It represents an open frontier for the physicist. It is this frontier that this class will discuss. Is Complexity Physics? Is It Science? What is It? Questions like this are being asked nowadays by faculty members in university departments of physics - and many other disciplines as well. They find themselves having to assess graduate work or applicants for jobs or tenure in strange fields like neural networks, self-organized criticality applied to earthquakes or landforms, learning systems, self-organization and even such older fields as nonlinear dynamics. 38

39 Easy way out in these decisions is to give nod to more orthodox and traditional work. So it is a matter of some urgency to realize that complexity is a part of physics - or even vice versa. For most of its history; and for most of its practitioners, physics, as we have said, is the ultimate reductionist subject. Physicists reduce matter to molecules, to atoms, to nuclei and electrons, to nucleons and so on - always attempting to reduce complexity to simplicity. Found, in all of physics, only four forces, reduced to three; now string theorists tell us all three boil down to single quantum gravitational, supersymmetric, utterly featureless universal interaction. With maturation of physics, new and different set of paradigms began to develop that pointed another way -- toward developing complexity out of simplicity. Emphasized in concept of broken symmetry" the ability of large collections of simple objects to abandon their own symmetry as well as the symmetries of forces governing them and to exhibit an emergent property of new symmetry. Emergent is philosophical term from 19th-century debates about evolution, implying properties that do not preexist in system or substrate. Life and consciousness, are emergent properties. Now must include dynamical instabilities, deterministic chaos, spontaneous pattern formation, fractals and the beginning of self-organization. Everything jumbled together by remarkable idea of Bak, Wiesenfeld and Tang (will study) that many of phenomena of nature exhibit scaling laws determined by self-organized behavior like that at critical points of standard phase transitions(will talk about). 39

40 As one probes deeper into origin of universe or interior of quark, it has never been questioned that one is doing physics. By contrast, traditional, reductionist physicists and, for sure, funding agencies can be left vaguely disturbed or hostile as new fields lead us up hierarchy of complexity toward sciences such as geology, developmental biology, computer science, artificial intelligence or even economics. There can be a somewhat surprising lack of understanding of what those working in these new fields are doing. If broken symmetry, localization, fractals and strange attractors are not fundamental what are they? Joining together into general subject all various ideas about ways new properties emerge science of complexity. Whole Truths False in Part If I had to give my own definition of complexity science, I would say it is the search for general concepts, principles and methods for dealing with systems which are so large and intricate that they show autonomous behavior which is not reducible to properties of the laws and concepts, as opposed to reductionist assumption for study of detail. Complexity is, at this time, an enormous, rapidly growing and diversifying field. Complexity is a subject which has an annoying propensity to define itself, almost in spite of efforts of many to avoid the word as a vague, undefinable generalization, welding together a wealth of different ideas. 40

41 For my money complexity is a state of mind, embracing any study of a realistic system which negates the strong reductionist (once called constructionist) point of view which assumes everything follows from fundamental laws, and emphasizes appearance of emergent phenomena of all kinds, at least intellectually independent of the microscopic substrate in which they appear. This attitude is often mistaken for "holism" or for a rejection of scientific reductionism but it is neither. Many of its ideas appeared either as parts of computer science or as results of computer investigations, but many did not, and in my opinion the greatest problem the field faces is maintaining its sometimes tenuous connection with the non-virtual world, especially considering the seductive nature of computer work and the fascination of the public with the images the computer produces. Complexity science mainly results from the creative tension between two intellectual traditions: creative side of computer science, and natural science of complex systems. I believe firmly that complexity, however defined, is the scientific frontier. 41

42 More Thoughts Before Proceeding... Consider a collection of electrons, or a pile of sand grains, a bucket of fluid, an elastic network of springs, an ecosystem, or the community of stock-market dealers. Each of these systems consists of many components that interact through some kind of exchange of forces or information. In addition to these internal interactions, the system may be driven by some external force: an electric or a magnetic field, gravitation (in case of sand grains), environmental changes, and so forth. The system will now evolve in time under the influence of the external driving forces and the internal interaction forces. What happens? Is there some simplifying mechanism that produces a typical behavior shared by large classes of systems, or will the behavior always depend crucially on the details of each system? Paper by Bak, Tang, and Wiesenfeld (1987) contained the hypothesis that, indeed, systems consisting of many interacting constituents may exhibit some general characteristic behavior. The seductive claim was that, under very general conditions, dynamical systems organize themselves into a state with a complex but rather general structure. The systems are complex in the sense that no single characteristic event size exists: there is not just one time and one length scale that controls the temporal evolution of these systems. 42

43 Although the dynamical response of the systems is complex, the simplifying aspect is that the statistical properties are described by simple power laws. Moreover, some of the exponents may be identical for systems that appear to be different from a microscopic perspective. The claim by Bak, Tang, and Wiesenfeld (BTW) was that this typical behavior develops without any significant tuning of the system from the outside. Further, the states into which systems organize themselves have the same kind of properties exhibited by equilibrium systems at the critical point. Therefore, BTW described the behavior of these systems as self-organized criticality (SOC). The hope was that here was a dynamical explanation of why so many systems in nature exhibit complex spatial and temporal structures. Self-organized criticality became a candidate for the sought-after theory of complexity. One reason for the intense interest SOC has received is that it combines two fascinating concepts - self-organization and critical behavior - to explain a third, no less fascinating and fashionable, notion: complexity. Phenomena in very diverse fields of science have been claimed to exhibit SOC behavior. It started out with sandpiles, earthquakes, and forest fires. Next came electric breakdown, motion of magnetic flux lines in superconductors, water droplets on surfaces, dynamics of magnetic domains, and growing interfaces. The idea was soon suggested to apply to economics, and SOC models have more recently been proposed as ways of understanding biological evolution. It is useful to understand what the name self-organized criticality is meant to imply. The term consists of two parts. Self-organization has for many years been used to describe the ability by certain nonequilibrium systems to develop structures and patterns in the absence of control or manipulation by an external agent. 43

44 Examples include the growth of patterns in chemical reactions and, to make an ambitious leap, the development of structure in biological systems. The word criticality has a very precise meaning in equilibrium thermodynamics. It is used in connection with phase transitions (strictly speaking, continuous transitions). When the temperature of the system is precisely equal to the transition temperature, something extraordinary happens. For all other temperatures, one can disturb the system locally and the effect of the perturbation will influence only the local neighborhood. However, at the transition temperature, the local distortion will propagate throughout the entire system. The effect decays only algebraically rather than exponentially. Although only nearest neighbor members of the system interact directly, the interaction effectively reaches across the entire system. The system becomes critical in the sense that all members of the system influence each other. The critical behavior of thermodynamic systems is well understood. BTW suggested that a large group of systems behaves very much like thermodynamical systems right at the phase transition temperature. Moreover, dynamical systems will drive themselves into states characterized by algebraic correlations - unlike systems in thermodynamical equilibrium, for which tuning is essential. What kind of systems will evolve into a SOC dynamical state? A separation of time scales is required. The process connected with the external driving of the system needs to be much slower than the internal relaxation processes. 44

45 The prototypical example is an earthquake. The stress in the earth s crust is built up on the scale of years owing to the motion of the tectonic plates. The stress is subsequently released in a few seconds or minutes during an earthquake. The separation of time scales is intimately connected with the existence of thresholds and metastability. It is the existence of a threshold that ensures the separation of time scales. Think again about solid friction - or earthquakes. Say you want to push your piano across the floor. You slowly increase the force you apply to the piano. At first, nothing happens; the piano is stuck. The stress between the floor and the bottom of the piano builds up as the applied force increases. At a certain point, the friction forces between the floor and the piano are not able to sustain the applied force any longer. The piano does a rapid jump ahead, and the stress in the piano-floor interface is released. The applied force drops, the piano is stuck again, and the cycle starts over. The applied force has to build up in order to overcome a certain threshold. This occurs over a time scale much longer than the short time interval it takes the piano to jump ahead. During the build-up phase, energy is gradually stored. 45

46 This energy is then released, or dissipated, nearly instantaneously in the moment the piano moves forward. If no threshold for motion existed - that is, if the piano were, say, standing on ice - then the piano would move ahead continuously and the energy would be dissipated at the same rate as it is pumped into the system. The actual friction force that the piano must overcome at a given moment will depend on the microscopic details of how the rough piano bottom interlocks with the rough surface of the floor. This means that there are many different states in which the piano will remain stuck even in the presence of an applied force. All these states are metastable. The friction forces induce strain in the floor as well as in the piano, and this strain corresponds to a certain amount of stored elastic energy. Thus, despite the piano-floor system being in a stable (i.e., time-independent) state, the system is not in the lowest energy state. It is in one of many metastable states. Among all the metastable states, some are of particular importance: the set of configurations visited by the piano as it performs the jerky motion. These states are marginally stable. A slight increase in the applied force can lead to almost any response. Sometimes the increase in the force will be able to bring the piano forward by a small jump. At other times the increase in applied force might not even be able to make the piano move, and at still other times the same amount of increase of the driving force might induce a large jump forward. 46

47 Bak, Tang, and Wiesenfeld originally envisaged the marginally stable states as characterized by the lack of any typical time or length scale. This is precisely the case for the configurations of a thermodynamic system at the critical temperature. The lack of a typical scale leads to algebraic correlation functions. One finds, as anticipated, that the distribution functions describing the frequency with which various events occur in the SOC state exhibit power laws. The Gutenberg-Richter law for the distribution of energy releases in earthquakes is a power law. If E is the energy released during an earthquake, then the probability for an earthquake of that size is given by P(E ) ~ E -B. This kind of distribution is seen again and again in SOC model systems. For example, in toy models of sandpiles one finds that the distribution of lifetimes of the avalanches as well as the distribution of avalanche sizes follow power law behavior. The original ambition of the BTW paper was to explain why spatial fractals and fractal time series, known as 1/f fluctuations, are so ubiquitous in nature. The properties of fractals have been studied intensively over the last one or two decades. Despite these investigations, very little is known about why fractals are formed. What aspects of the evolution or dynamics of macroscopic systems are responsible for the formation of fractals? Many materials form crystalline structures, for example, metals and common kitchen salt. 47

48 We know why this is the case: the principle of lowest energy selects the ordered crystalline phase. Fractals are certainly not the lowest energy configuration that can be selected in thermodynamic equilibrium, hence some kind of dynamical selection must occur. How can SOC possibly be an explanation of 1/f noise and of fractals? The speculation by BTW was as follows. A signal will be able to evolve through the system as long as it is able to find a connected path of above-threshold regions. When the system is either driven at random or started out from a random initial state, regions that are able to transmit a signal will form some sort of random network. This network will be modified, or correlated, by the action of the internal dynamics induced by the external drive. The dynamics stop every time the internal dynamics have relaxed the system, so that all local regions are below threshold. The slow external drive will eventually bring some region above threshold once again, and the internal relaxation will restart. The result is a complicated, delicately interwoven web of regions that are coupled dynamically. When we continue to drive the system after this marginally stable SOC state has been reached, we will see flashes of action as the external perturbation manages to spark off activity through different routes of the system. The intricate nature of the combined operation of the external drive and the internal relaxation of the threshold dynamics makes it natural to imagine that the network of connected dynamical paths has some sparse percolation like geometry. 48

49 It could well be that the structure of this dynamical network has a fractal geometry; at least this was the suggestion of BTW. If the activated regions consist of fractals of various sizes, then the duration of the induced relaxation processes traveling through these fractals can also be expected to vary greatly. It is well known that many different-acting time scales can, under certain circumstances, lead to 1/f noise. Bak, Tang, and Wiesenfeld imagined that this is precisely what happens in SOC systems. 49