Nonlinear Physics for Beginners

The following sections are included:

• Contents

Two things stand out in the physics department of San Jose State University. First, our teaching load is 12 units plus five office hours per week. Second, undergraduate research and publications with students as coauthors are very much encouraged. To get some fun out of these challenging demands and to maintain my own vitality as a research physicist, in the Fall of 1988 onwards, I created two new courses in nonlinear physics to teach. At that time, there were no suitable textbooks; reviews and research papers were used as teaching materials or recommended reading. There was much excitement in the classroom, for both the students and the instructor, mostly due to the freshness and novelty of the material we learned together…

The following sections are included:

• A Quiet Revolution

• Nonlinearity

Many spatial structures in nature result from the self-assembly of a large number of identical components. To be efficient, the self-assembly process takes advantage of and occurs via some simple prescriptions, which we call the principles of organization. The two simplest principles are the principle of regularity and the principle of randomness. With the former, the components arrange themselves in a periodic or quasiperiodic regular fashion, resulting in crystals, alloys, a formation of soldiers in a parade, etc. Examples of structures (or nonstructures) resulting from the latter are those in gases and the distribution of animal hairs…

In the realm of science, chaos is a technical word representing the phenomenon that the behavior of some nonlinear systems depend sensitively on the initial conditions (Section 10.1). This usage of the word obviously differs from that adopted in our daily lives, in which chaos is synonymous to “a state of utter confusion” (Fig. 3.1). (The word work is another example of this kind of free borrowing by the scientists. When you breathe heavily after carrying a heavy object up ten stories in the school building and back to the same spot, and your teacher says that you have done zero work, then you know you and your teacher are not speaking the same language.) Chaos as envisioned by the artists is sampled in Fig. 3.2…

Solitons are spatially localized waves traveling with constant speeds and shapes. They are special solutions of some partial differential equations (Section 11.1).

In some nonlinear media, such as a layer of shallow water or an optical fiber, under suitable conditions, the widening of a wave packet due to dispersion could be balanced exactly by the narrowing effects due to the nonlinearity of the medium. In these cases, it is possible to have solitons. For example, the equation describing wave propagation in shallow water is given by the Korteweg-deVries equation…

One can hardly fail to notice the striking similarity between the ramified patterns formed by rivers, trees, leaf veins and lightning. These branching patterns are different from compact patterns observed in snowflakes, clouds and algae colonies. How does nature generate these patterns? Is there a simple principle or universal mechanism behind these pattern-forming phenomena? These are the profound questions that interest lay people and experts alike. Athough final answers to these questions are still lacking, tremendous progress has been made in the last 15 years…

Cellular automata are discrete dynamical systems whose evolution is dictated by local rules. In practice, they are usually realized on a lattice of cells, with a finite number of discrete states associated with each cell, and with local rules specifying how the state of each cell should be updated in discrete time steps (Section 14.4.1). Because of the discreteness of all the quantities involved, cellular automata calculations obtained from computers are exact. Note that a cellular automaton is just a computer algorithm and is not a real machine…

The fact that there is one and only one doctoral degree, the Doctor of Philosophy (Ph.D.), but not the Doctor of Physics or Doctor of Economics, attests to the fact that not too long ago, science was considered and studied as a whole. There was no division of social and natural sciences, not to mention no fragmentation of the natural science into physics, chemistry, biology, etc. As suspected by some, this compartmentalization of science is due more to administrative convenience than to the nature of science itself…

Nonlinear science involves the interplay of order and disorder, as well as the simple and the complex. Technically, what makes the fascinating outcomes possible is nonlinearity. It is the nonlinearity that makes the behavior of the systems nontrivial and interesting, and the world as complex as it is…

Most of the papers reprinted here are reviews or essays written by pioneers. They are selected for their pedagogical values and their accessibility to undergraduates. The others are research papers divided between experimental (Sections 9.2, 10.2, 11.2, 12.3 and 12.4) and theoretical/computational (Sections 9.3, 9.6, 10.4, 11.3, 12.2 and 13.5). With perhaps the exception of one or two in the latter category, all these papers are reasonably easy to read.

It is our experience and conviction that for a good education in any subject, nonlinear physics in particular, there is no substitute for reading articles by the original contributors. However, like looking at an abstract painting, it is not always necessary to understand the content in order to be inspired by it.

The following sections are included:

• Fractal Growth Processes
  • Fractals and scale invariance
  • Growth models
  • Attempts at theory
  • Fractals and snowflakes

• Fractal geometry in crumpled paper balls
  • ACKNOWLEDGMENTS

• FRACTAL OF LARGE SCALE STRUCTURE IN THE UNIVERSE
  • Acknowledgments
  • References

• The Devil’s staircase
  • Origins of staircases
  • Experiments with dynamical systems
  • Long-range periodic structures
  • References

• Multifractal phenomena in physics and chemistry
  • Complex surfaces
  • Fluid flow in complex media
  • Generality

• Simple Multifractals with Sierpinski Gasket Supports
  • NTRODUCTION
  • A SIMPLE MULTIFRACTAL
  • GENERALIZATIONS
  • COMPUTER GENERATION OF MULTIFRACTALS
  • ACKNOWLEDGMENTS
  • APPENDIX: DERIVATION OF EQUATION (9)
  • REFERENCES

The following sections are included:

• Chaos

• Chaos in a dripping faucet
  • Introduction
  • Basic concepts and the period-doubling route to chaos
  • The dripping faucet experiment
  • Conclusions
  • Acknowledgments
  • References

• Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics
  • Chaotic Attractors
  • The Evolution of Chaotic Attractors
  • Universality
  • Fractal Basin Boundaries
  • Conclusion
  • REFERENCES AND NOTES

• Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series
  • Forecasting for a chaotic time series
  • Forecasting with uncorrelated noise
  • The embedding dimension
  • Problems and other approaches
  • Time series from the natural world
  • Conclusion

• CONTROLLING CHAOS
  • Dynamical systems
  • Sensitivity and orbit complexity
  • Control based on unstable periodic orbits
  • Methods for controlling chaos
  • Controlling chaos experimentally
  • New directions
  • References

• Quantum Chaos

• Now random is a coin toss?
  • Contemporary results
  • Chaotic orbits
  • A simple model
  • Algorithmic complexity theory
  • How random is a coin toss?
  • Consequences
  • References

The following sections are included:

• SOLIIONS
  • Nonlinearity and solitons
  • Nonlinear Schradinger equation
  • Sine-Gordon equation
  • Notional spins
  • Particle physics
  • Further reading

• Soliton Propagation in Liquid Crystals

• Possible relevance of soliton solutions to superconductivity

The following sections are included:

• Dendrites, Viscous Fingers, and the Theory of Pattern Formation
  • Dendritic Solidification of a Pure Substance
  • Viscous Fingering
  • Pattern Selection
  • Special Features of the Solvability Theory
  • Dendritic Growth Rates
  • Couder’s Bubbles
  • Stability and Sidebranching
  • Snowflakes
  • REFERENCES

• Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy
  • Relation between noise and tip splitting
  • Local anisotropy and dendritic growth
  • Discussion

• Oblique Roll Instability in an Electroconvective Anisotropic Fluid

• Critical Behavior in the Transitions to Convective Flows in Neraatic Liquid Crystals
  • REFERENCES

• Chemical Waves
  • Geometric Forms of Waves
  • Properties of Chemical Waves
  • Conclusions
  • REFERENCES AND NOTES

The following sections are included:

• More Is Different
  • References

• Cellular automata as models of complexity
  • Cellular automata
  • Applications
  • Mathematical approaches
  • Entropies and dimensions
  • Information propagation
  • Thermodynamics
  • Formal language theory
  • Computation theory

• Catastrophes and Self-Organized Criticality
  • Suggestions for further study
  • Acknowledgments
  • References

• ACTIVE-WALKER MODELS: GROWTH AND FORM IN NONEQUILIBRIUM SYSTEMS
  • Ants with a scent
  • Multiwalkers
  • A moving worm
  • Abnormal growth
  • Rough surfaces
  • Suggestions for further study
  • Acknowledgments
  • References

• Active Walks and Path Dependent Phenomena in Social Systems
  • Introduction
  • Active Walks
  • increasing Returns in Economics
  • Discussions
  • References

The projects presented in Part III are the works of students from San Jose State University. Here are the stories behind their creations.

In my teaching of the freshman physics course on mechanics, after finishing the chapter on oscillations I would usually show the class the one-hour videotape of the Nova program on Chaos, which was followed by one lecture using the logistic map as an illustrative example. The idea was to emphasize to the students that not every oscillation in the world was small and simple harmonic and, of course, to introduce them to something new and exciting that was not yet in their textbooks. (That video program was splendidly made, in color, and guaranteed to excite anybody from all walks of life. The program included many demonstrations from the experts in chaos. For example, near the beginning, the chaos game was demonstrated by Michael Barnsley.) On one of these occasions, the day after I showed the video, a student in my freshman class, Prasanna Pendse, walked into my office, inserted his floppy disk into the obsolete Apple He computer belonging to my officemate, and–voila!–the chaos game just jumped out of the screen. Prasanna's effort, as inspired by the Nova progam, was recorded in Section 14.1.1…

The following sections are included:

• Fractals
  • The Chaos Game and Sierpinski Gasket
  • Iteration Maps and the Sierpinski Fractals
  • Calculating the Box Dimension
  • Diffusion-Limited Aggregates in Radial Geometry
  • The Dielectric Breakdown Model with Noise Reduction

• Chaos
  • The Tent Map
  • The Waterwheel

• Pattern Formation
  • Biased Random Walks
  • Surface Tension and the Evolution of Deformed Water Drops
  • Ising-like Model of Ferrofluid Patterns

• Cellular Automata
  • One-Dimensional Totalistic Cellular Automata
  • Two-Dimensional Cellular Automata: Formation of Clusters

The following sections are included:

• Curve Length and the Scaling Parameter
  • Introduction
  • The Simplest Case: A sinusoidal Curve
  • Conclusions

• Analysis of the Back-Propagating Neural Network for the XOR Problem
  • Introduction
  • Dynamics
  • Conclusions
  • Appendix A: Computer Program for Back-Propagation
  • Reference

The following sections are included:

• Instabilities of Finite Water Columns
  • Experimental Procedure
  • Results
  • Discussions
  • References

• Viscous Fingering in Optical Cement Displaced by Water
  • Introduction
  • Experiments
  • Results
  • Conclusions
  • References

• The Fractal Nature of Shock-Wave Induced Fractures
  • Introduction
  • Experimental Procedures and Results
  • Discussions
  • References

More often than not, people become physicists because they love physics. But that does not mean that physicists do not love or care about other things in the real world such as the environment, art and literature, and social justices. In fact, most of them do! For example, after the June 4, 1989 Tiananmen massacre, six of the 21 most wanted student leaders of the democracy movement were physicists, according to the October 1996 issue of APS News. [For a description of this movement, see, for example, Massacre in Beijing: China’s Struggle for Democracy, edited by Donald Morrison (Warner Books, 1989).]…

The following sections are included:

• Appendix Al Computer Program for Active Walk

• Appendix A2 Publications from Nonlinear Physics Group of SJSU

• Acknowledgments

• Index