From Order to Chaos

The following sections are included:

• Acknowledgements

• CONTENTS

• Publications of Leo Kadanoff

This book is a selection of my research and popular essays, with particular emphasis on works which review or discuss in a general way some scientific or technical question. The papers are all about the world of science, or rather about the different worlds in which a scientist works. In my own work I can see at least four different kinds of things which might be meant when one talks about the worlds of a scientist…

This section contains a collection of essays about apparently disconnected subjects in field theory, condensed matter physics, and hydrodynamics. There is, however, a thread of connection among all these different essays. In each case, we ask some kind of question about the relation between different levels of description of the physical world. This is a very natural question for a person trained as I am, in the area of condensed matter physics. My thesis advisers, Paul Martin and Roy Glauber, continually directed my attention to the relation between a microscopic description of reality and a macroscopic description. Thus a gas is composed of molecules, but it also obeys the laws of fluid mechanics. A microwave cavity contains not only photons but also an electric field. Or again, a fluid near its critical point is a bunch of molecules, but they can also be described by a scale-invariant field theory…

Some of the most interesting situations in physics, and indeed in other sciences as well, concern the connections between two “levels of reality.” How does the presumed world of strings connect with the more observable world of quarks and gluons? How do quantum problems “go to” their classical limits? How does the irreversibility of the macroscopic world connect with the known time-reversibility of microscopic description? In each of these cases, there is a tension between two levels of description. For each situation, different laws, formulations, conceptualizations, theories and experiments apply at each of the two levels…

The response of a system to an external disturbance can always be expressed in terms of time dependent correlation functions of the undisturbed system. More particularly the linear response of a system disturbed slightly from equilibrium is characterized by the expectation value in the equilibrium ensemble, of a product of two space- and time-dependent operators. When a disturbance leads to a very slow variation in space and time of all physical quantities, the response may alternatively be described by the linearized hydrodynamic equations. The purpose of this paper is to exhibit the complicated structure the correlation functions must have in order that these descriptions coincide. From the hydrodynamic equations the slowly varying part of the expectation values of correlations of densities of conserved quantities is inferred. Two illustrative examples are considered: spin diffusion and transport in an ordinary one-component fluid.

Since the descriptions are equivalent, all transport processes which occur in the nonequilibrium system must be exhibited in the equilibrium correlation functions. Thus, when the hydrodynamic equations predict the existence of a diffusion process, the correlation functions will include a part which satisfies a diffusion equation. Similarly when sound waves occur in the nonequilibrium system, they will also be contained in the correlation functions.

The description in terms of correlation functions leads naturally to expressions for the transport coefficients like those discussed by Kubo. The analysis also leads to a number of sum rules relating the dissipative linear coefficients to thermodynamic derivatives. It elucidates the peculiarly singular limiting behavior these correlations must have.

The following sections are included:

• Correlation Function Approach

• Green's Function Approach

• Specialization to a Metal

• The Superconductor

• REFERENCES

The basis of our microscopic picture of superconductivity is the use of a single wave function 〈ψ(r)〉 to describe the quantum-mechanical state of a macroscopic number of Cooper pairs.¹ This wave function was originally introduced in the phenomenological theory of Landau and Ginzburg.² After the pairing idea was derived by Bardeen, Cooper, and Schrieffer (BCS),³ one could see that the phase of this wave function was related to the “superfluid” velocity and the vector potential by…

Recent work by K. G. Wilson, A. A. Migdal and others has led to a statistical mechanical treatment of systems of interaction quarks and strings. This work is summarized here. The major topics discussed include boson and fermion variables in statistical mechanics; descriptions of local and gauge symmetries; exact solutions of one-dimensional problems with nearest-neighbor interactions; exact solutions of two dimensional problems with plaquette interactions; Wilson's model of quarks and strings; asymptotic freedom and trapping for this model; the effect of a phase transition in this system; approximate recursion relations of the Migdal form. Finally, all this is put together to give a partial argument for the simultaneous existence of asymptotic freedom and trapping O₂ in the quark-string case. Arguments are developed which distinguish this case from the superficially analogous example of quantum electrodynamics.

It is shown that “clock” type models in two-dimensional statistical mechanics possess order and disorder variables ϕ_n and χ_m with n and m falling in the range 1,2,…,p. These variables respectively describe abelian analogs to charged fields and the fields of 't Hooft monopoles with charges q = n/p and topological quantum number m. They are related to one another by a dual symmetry. Products of these operators generate, via a short-distance expansion, para-fermion operators in which rotational symmetry and the internal symmetry group are tied together. The clock models in two dimensions are shown to be an ideal laboratory where these ideas have a very simple realization.

In the last 50 years, a new approach to physics has arisen almost equal in importance to the two “old” branches of theory and experiment. This new type of effort is computational physics: It involves the use of computers large and small to simulate the behavior of physical systems and to work out the consequences of physical ideas, as expressed in mathematical form…

One argument often used to justify society's support of pure science is that contemporary science is producing great and enduring structures that will be passed on to future generations as a major portion of the legacy of our age. The analogy to medieval cathedral building is frequently pressed. In this column, I point to a few of the results of physics that are likely to endure and remain important, not to just a few specialists in physical law, but to people in general…

This column is about a particularly elegant idea for an experiment in neutrino physics. Of course any observation of neutrinos is a wonderful accomplishment. But the proposed experiment to be discussed here has the further charm that it makes essential use of the quantum mechanical properties of low-temperature matter. The matter, in this case, is liquid helium, and the experiment will rely on low-lying quantum excitations (called quasiparticles) as messengers to carry the energy transferred from the neutrinos to the helium…

This section describes the development of ideas of scaling and universality as they relate to phase transitions and critical phenomena. In the late 1960s and early 1970s, a group of physicists and chemists changed the way scientists look at problems in statistical mechanics and related fields. Looking back at this work almost thirty years later, I still feel proud, pleased, and somewhat surprised that I could play a role in such an achievement…

A model for describing the behavior of Ising models very near T_c is introduced. The description is based upon dividing the Ising model into cells which are microscopically large but much smaller than the coherence length and then using the total magnetization within each cell as a collective variable. The resulting calculation serves as a partial justification for Widom's conjecture about the homogeneity of the free energy and at the same time gives his result sν′ = γ′ + 2β.

This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the “critical indices” which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the “scaling law” concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.

In recent years, there has been considerable progress in understanding the characteristic divergences in thermodynamic derivatives which appear as one approaches the critical point for a “second order” phase transition. (Reviews of recent developments in this area can be found in Refs. 1–5.) However, non-equilibrium properties such as transport coefficients remain much less well understood. A flurry of recent experimental and theoretical papers, taken together, lead me to believe that considerable progress in this area should be expected within the next few years. In this note, I would like to present my view of the current situation and the directions in which I expect to see advances…

Consider a magnetic system near its critical point. The entire volume of the system will be filled by << droplets >> or micro-domains which represent local fluctuations in the order parameter.

The following sections are included:

• Introduction

• Scaling derived from universality

• Dimensional analysis

• Back to universality

• Operator algebras

The following sections are included:

• Introductory Note

• Critical Points and Thermodynamic Singularities
  • The Ising model
  • Other problems
  • Critical singularities
  • The importance of correlation functions
  • Droplet picture of correlation behaviour

• Mean Field Theory
  • Results
  • Generalizations
  • Failure of mean field theory

• The Roots of The Theory: Universality
  • Fields and operators
  • Statement of universality hypothesis
  • A simple example
  • Additional universality statements

• The Roots of The Theory: Scaling
  • The irrelevance of the length scale
  • A formal description of the length transformation
  • Fixed points
  • Deviations from the fixed point
  • Scaling results for the free energy
  • A simple example of the theory

• Scale Transformations for Correlation Functions
  • General discussion
  • Formal theory
  • Connections with universality
  • Consequences

• References

The renormalization group theory of second-order phase transitions is described in a form suitable for presentation as part of an undergraduate statistical physics course.

The twin concepts of Scaling and Universality have played an important role in the description of statistical systems. Hydrodynamics contains many applications of scaling including descriptions of the behavior of boundary layers (Prandtl, Blasius) and of the fluctuating velocity in turbulent flow (Kolmogorov, Heisenberg, Onsager).

Phenomenological theories of behavior near critical points of phase transitions made extensive use of both scaling, to define the size of various fluctuations, and universality to say that changes in the model would not change the answers. These two ideas were combined via the statement that elimination of degrees of freedom and a concomitant scale transformation left the answers quite unchanged. In Wilson's hands, this mode of thinking led to the renormalization group approach to critical phenomena.

Subsequently, Feigenbaum showed how scaling, universality, and renormalization group ideas could be applied to dynamical systems. Specifically, this approach enabled us to see how chaos first arises in those systems in which but a few degrees of freedom are excited. In parallel Libchaber developed experiments aimed at understanding the onset of chaos, the results of which were subsequently used to show that Feigenbaum's universal behavior was in fact realized in honest-to-goodness hydrodynamical systems. More recently, Gemunu Gunaratne, Mogens Jensen, and Itamar Procaccia have indicated that they believe that a different (and weaker) universality might hold for the fully chaotic behavior of low dimensional dynamical systems.

Dynamically generated situations often seem to show kinds of scaling and universality quite different from that seen in critical phenomena. A technical difference which seems to arise in these intrinsically dynamical processes is that instead of having a denumerable list of different critical quantities, each with their critical index, instead there is continuum of critical indices. This so-called multifractal behavior may nonetheless show some kinds of universality. And indeed this might be the kind of scaling and universality shown by those hydrodynamical systems in which many degrees of freedom are excited.

Science includes the critical application of ideas to real world situations. In the 1950s, 60s and 70s, I found many applied problems very interesting. While I was still in college, I worked for the Guided Missile Division of Republic Aviation Corporation, helping with the proposal stage of the design of robot aircraft. In graduate school and thereafter, I worked for the AVCO corporation doing heat transfer calculations related to the design of guided missiles and space probes. I found the latter job extremely instructive. It enabled me to interact with some scientists whose paths I would not normally have crossed, including Hans Bethe, Jim Keck and Arthur Kantrowitz. I was very impressed by the intellectual vitality of the work at AVCO, particularly the branch at Everett, Massachusetts…

The following sections are included:

• Introduction

• A First Cut—The Centralia Study

• Data Gathering for Kankakee

• Presentation of Kankakee Data

• A Hypothetical Example: The Location of a Regional Expressway

• A Technical Note on the Display System

Forrester's model is described and critically analyzed with a view to understanding the relationship between his conclusions and his normative scheme. He claims that his model produces “counterintuitive” results; it is argued here that the main results follow directly from his goals. An alternative method of evaluating the results of the model is proposed, based upon the model's calculation of the city's attractiveness for various groups. This alternative normative scheme leads to quite different conclusions from those reached by Forrester.

This paper is intended to be a case study in the use of simulation models to test public policy alternatives. Suggested programs for dealing with urban poverty are tested with the aid of different models which describe the linked growth of housing, population, and industry in an urbanized area. The tested programs are: 1) training, to provide the unskilled with job skills; 2) job provision, to make extra jobs for skilled workers; 3) clearance, to eliminate “excess” housing and thereby free land which may be used by industry. The models used are: the original Forrester [1] model, which treats a single city as a unit in an unchanging national environment; an extension of this model to include all the central cities of the nation and describe the migration between these areas; and finally a complete revision of the Forrester model to obtain a simulation of the national economy including both cities and suburbs.

The three different models give very different results. The original model, which focuses upon applying programs to a single city, strongly indicates that clearance is the only one of these programs which is effective in eliminating urban poverty. The second model indicates that when applied throughout the nation, both the clearance and training programs are effective, but job provision is ineffective. However, the third model uses its more complete picture of the national economy to conclude that job provision can indeed be effective in reducing poverty. In fact, the third model suggests that a combination of training, clearance, and the provision of more jobs will reduce poverty with a minimum of undesirable side effects. This analysis is intended to show that the choice of focus for the model will strongly affect the policy conclusions reached.

The following sections are included:

• Introduction

• A model of labour market segmentation

• Probabilistic mobility models

• Specification of the labour mobility model

• Initialisation and calibration

• Model output

• Summary and future directions

This should be a good time for physics. Our President, several governors and many industrial leaders have eulogized science as significant both for its own sake and also in helping our nation maintain its long-term competitive position. But despite all this public goodwill, US physics continues to decay…

In recent years, physics has seen much of its support base disappear. In Eastern Europe the decline of governments and the changes in the economy have moved interest away from all activities with long-range payoffs. For different reasons, a short-term focus has begun to dominate American life also. Here, all the props for science have begun to weaken. Government has become unpopular. The military has started to shrink. Corporations are concerned with tomorrow's stock value and the next quarterly income statement, and have lost interest in promoting applied research. Antiscientific threads have become evident in many parts of popular thinking. Congress enjoys exposing the misbehavior of some of the leading figures in the biological sciences. Both the animal rights movement and the environmental movement have considerable antiscientific components. Science is in low regard…

In the last dozen or so years, my interests have turned to yet another field of science: the description of complexity. My own view of this field starts with a major intellectual problem: We know that the laws of physics are rather simple in structure. Newton's laws or the Schrödinger equation or even string theory are described by a rather simple system of equations. One's expectation might be that such simplicity in formulation should lead to simplicity in outcome. However, all our experience in life contradicts any expectation of simple outcomes. The world is wondrously complicated and bewilderingly diverse. How can it be that from simple beginnings one gets complex endings…

Hydrodynamic systems often show an extremely complicated and apparently erratic flow pattern of the sort shown in figure 1. These turbulent flows are so highly time-dependent that local measurements of any quantity that describes the flow—one component of the velocity, say—would show a very chaotic behavior. However, there is also an underlying regularity in which the motion can be analyzed (see figure 1 again) as a series of large swirls containing smaller swirls, and so forth. One approach to understanding this turbulence is to ask how it arises. If one puts a body in a stream of a fluid—for example, a piece of a bridge sitting in the stream of a river—then for very low speeds (figure 2a) the fluid flows in a regular and time-independent fashion, what is called laminar flow.¹ As the speed is increased (figure 2b), the motion gains swirls but remains time-independent. Then, as the velocity increases still further, the swirls may break away and start moving downstream. This induces a time-dependent flow pattern—as viewed from the bridge. The velocity measured at a point downstream from the bridge gains a periodic time-dependence like that shown in figure 2c. The parameter that characterizes these changes in the flow pattern is the dimensionless Reynolds number ℜ, which is the product of the velocity and density times a characteristic length (the size of the bridge pier, for example) divided by the viscosity. As ℜ is increased still further, the swirls begin to induce irregular internal swirls as in the flow pattern of figure 2d. In this case, there is a partially periodic and partially irregular velocity history (see the second column of figure 2d). Raise ℜ still further and a very complex velocity field is induced, and the v(t) looks completely chaotic as in figure 2e. The flow shown in figure 1 has this character.

The following sections are included:

• Introduction
  • Definition of chaos: complexity and order

• Chapter I: Simple laws, complex outcomes
  • Practical predictability

• Chapter II: Routes to chaos
  • A model of chaos
  • The model in mathematical form
  • Order and chaos
  • Period doubling and the onset of chaos
  • Universality and contact with experiment

• Chapter III: The geometry of chaos
  • Well-developed turbulence
  • Attractors, strange and otherwise
  • Chaos on strange attractors
  • Tracing chaos through time

The simplest example of the onset of chaos in a Hamiltonian system is provided by the “standard” or Chirikov–Taylor model. As a nonlinearity parameter, k, is increased the long term behavior of the momentum, p, is examined. At k = 0, p is conserved. For k < k_c, for all starting points, p is of bounded variation. For some starting points its behavior is periodic, for others quasi-periodic, for others chaotic. At some critical value of k, unbounded chaotic variation first appears. A scaling analysis to describe this onset is described.

In a dynamical system described by a map, it may be that a “strange” sets of points is left invariant under the mapping. The set is a repeller if points placed in its neighborhood move away. An escape rate is defined to describe this motion. An alternative method of evaluating the escape rate, based on the consideration of repulsive cycles, is proposed. In the several cases examined numerically and analytically, the escape rate is shown to agree with the proposed formula.

We study an experimental orbit on a two-torus with a golden-mean winding number obtained from a forced Rayleigh-Bénard system at the onset of chaos. This experimental orbit is compared with the orbit generated by a simple theoretical model, the circle map, at its golden-mean winding number at the onset of chaos. The “spectrum of singularities” of the two orbits are compared. Within error, these are identical. Since the spectrum characterizes the metric properties of the entire orbit, this result confirms theoretical speculations that these orbits, taken as a whole, enjoy a kind of universality.

We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f(α). We apply this formalism to the 2^∞ cycle of period doubling, to the devil's staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.

Why all the fuss about fractals? Physical Review Letters complains that every third submission seems to concern fractals in some way or another. Corporate research labs such as Exxon's and IBM's expend perceptible fractions of their entire basic-research budgets on the study of fractal systems. Perhaps a half-dozen conferences during the past year were devoted to the subject. Why?…

Many different physical situations can be described by multifractal distributions. A general framework is presented. Several specific examples are discussed.

This paper is an expository treatment of recent work on using complex analytical methods for understanding the stability of hydrodynamic flow patterns in two-dimensional or almost two-dimensional geometries. We want to know the instabilities which might arise when a more viscous fluid is displaced by a less viscous one and also how surface-tension effects can restore the stability of non-trivial flow patterns.

In the first section, we describe the physical situation, restate the description in terms of partial differential equations, and summarize our state of knowledge about the solutions to the equations and the physical phenomena that arise. In two-dimensional problems, one can often make considerable progress by using calculational methods based upon analytic functions of complex variables. Section 2 describes how these methods can be used to obtain exact solutions for zero surface tension, while Sec. 3 sets up the interface equations for nonzero surface tension. Finally, the fourth section uses complex-variable methods to describe the stabilization of finger-like flow patterns.

The physicist is largely concerned with abstracting simple things from a complex world. Newton found simple laws that could predict planetary motion. Results from atomic physics have demonstrated that the Schrödinger equation is a correct description of almost all observed atomic behavior. In fact, the hydrogen atom is used as a metaphor to describe the search for simple and illuminating examples in all areas of the physical sciences…

The practice of physics, and indeed of all the sciences, has changed greatly because of the existence and ready availability of computers. Naturally, practice has changed more rapidly than education. Nonetheless, in the last few years the computer revolution has resulted in some revision in the style and content of undergraduate physics instruction…

Experimental and theoretical studies of Rayleigh-Bénard convection at high Rayleigh numbers (10⁸ < Ra < 10¹³) were performed by Bernard Castaing, Gemunu Gunaratne, François Heslot, Leo Kadanoff, Albert Libchaber, Stefan Thomae, Xiao-Zhong Wu, Stéphane Zaleski, and Gianluigi Zanetti (J. Fluid Mech. (1989)). The results of these studies are further examined in the light of visualization in Rayleigh-Bénard flow in water (Steve Gross, Giovanni Zocchi, and Albert Libchaber, C.R. Acad Sci. Paris t. 307, Série 2, 447 (1988)). The previously developed theory is shown to be incomplete in leaving out many of the structures in the flow. We take special note of the coherent flow throughout the entire water tank. Despite the many omissions of geometrical structures from the scaling analysis, most of the order of magnitude estimates seem right.

Please refer to full text.

A simple physical system might be defined as one that obeys simple laws. But simplicity of the rules of the game does not necessarily imply triviality of outcome. On the contrary, the action of elementary laws on many particles over long periods of time will often give rise to interesting structures and events. (We ourselves are probably results of this principle.) Thus, all the richness of structure observed in the natural world is not a consequence of the complexity of physical law, but instead arises from the many-times repeated application of quite simple laws…

The following sections are included:

• SUMMARY

• REFERENCES