In the Classroom
Disorder and Chaos: Developing and Teaching an Interdisciplinary Course on Chemical Dynamics Steven G. Desjardins Department of Chemistry, Washington and Lee University, Lexington, VA 24450;
[email protected] In this article we describe an interdisciplinary course on nonlinear dynamics developed for nonscience majors. We have taught this course, Disorder and Chaos, every year since 1989 and added a laboratory component in 2002. With typical enrollments of 50–70, this course is one of the more popular selections for a general education laboratory science at Washington and Lee University. As with any course with a long history, the content has evolved considerably, resulting in the current, more finely tuned version. Nonetheless, some basic ideas have been present since the beginning, even if they only became obvious to us after many iterations. Although taught by chemists, this course is interdisciplinary by nature, using examples from physics (pendulums), biology (population dynamics), and chemistry (reaction kinetics). While this course may not be appropriate for every curriculum, an abbreviated version emphasizing the chemical kinetics portion could be taught as a freestanding module in a more general nonmajors course. Basic Principles of the Course The general themes and methodologies that have emerged during the development of this course include the four briefly described below. Universality of Dynamical Concepts and Terminology Since the 1970s, researchers from many diverse fields of science and mathematics began to create the generalized subject of dynamics (1–5). Although this field has appeared under the headings of nonlinear dynamics, chaos theory, and fractal geometry, a common language has emerged to describe how any physical (or social) system evolves with the passage of time. This universality, usually framed with differential or difference equations, is an important component of the physical and biological sciences. Theoreticians studying the behavior of these equations in the absence of the associated physical models have developed a general terminology, with terms such as fixed point and strange attractor that allow scientists from different fields to exchange model-based methodologies and software with significant independence from subject-based jargon. The Nature of Mathematical Science The use of mathematics in science courses can create pedagogical difficulties for chemistry and biology majors, and this is exacerbated in nonmajor courses. The difficulty is that many fundamental ideas of modern science are expressed primarily in mathematical terms, and certainly much experimental verification is based on a comparison between the predicted and measured values. The use of ordinary language to describe phenomena not directly experienced—for example, atoms— can be elusive even with a heavy dose of jargon. An example from our course is the concept of entropy. Even though there
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are numerous attempts to explain entropy in words, the ultimate rigorous basis for this concept rests in a set of thermodynamic equations that involve heat and temperature or microstates and probability distributions. It is these equations that produce the predictions that allow for definitive comparison with experimental results. Although a conceptual understanding of entropy as a measure of disorder or degraded energy is important to the thinking of scientists, we rely on the rigorous underpinnings of the equations. Further, the ordinary language versions of scientific concepts are more likely to experience some drift in their precise meanings during written or verbal discussions. This can be especially tricky for students majoring outside of the sciences. Using Mathematical Software Because many of the modern dynamical results are based on numerical simulation, it is fitting to try to overcome the aforementioned pedagogical difficulties with computer–based solutions. Powerful software is currently available for symbolic or numerical mathematical modeling. In the earliest versions of our course, we used computer programs written in Quick Basic to evaluate systems of differential equations numerically and display graphical versions of the solutions. More recently, however, we have switched to less proprietary packages, such as Excel and Maple. In the earlier Basic programs the mathematics was hidden so that students only saw plots; with Excel, students must explicitly implement a numerical method for a given set of equations. Although this is initially intimidating, the resulting hands-on experience with the equations gives a much richer exposure to the mathematics. Translating an Idea into an Equation Many students blame their difficulty in the mathematical sciences on their mathematical ability, for example, solving and manipulating equations. It is often the case, however, that the real problem students have is translating the scientific concepts into mathematical equations. It is therefore increasingly important to present problems in which the transition from concept to equation is presented in a clear and systematic fashion. In particular, we emphasize the construction of a mathematical model of a natural phenomenon as the essential step in translating a scientific hypothesis into mathematical form. Chemical kinetics provide an excellent example of this, as the idea of “the more we have the faster it goes” translates relatively easily into an equation. As expected, some students are better than others at picking up this transition. In a nonmajors course, however, it is sufficient to make students comfortable with the notion of expressing a concept as an equation. Many courses on chemical dynamics have been developed over the years. A classic example is the course for first-year students developed by George Hammond and Harry Gray over 40
Journal of Chemical Education • Vol. 85 No. 8 August 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom
years ago (6). Most of these courses have been targeted at science majors and emphasize the use of dynamical methods to describe the detailed features of chemical processes. Our course is very different in that it also uses many examples from outside of chemistry and instead emphasizes how chemical systems display dynamical properties similar to that of pendulums and insect populations. In addition, we present this material using the modern dynamical language developed by the mathematicians and physicists to describe chaotic systems. Rather than giving a future scientist a set of mathematical and experimental tools, we try to show a group of nonscientists that chemical systems can show sufficient dynamical complexity to describe the changing world they see in everyday life.
Dynamics, Equations of Motion, and Equations of State The most obvious part of any model is the equations, and it is important to carefully describe the relationship between the mathematics and the science. In this course we consider the time evolution or dynamics of a system as expressed by the equations of motion, which contain the state variables and time.
Week 1 M
Introduction: guidelines and overview
T
Basic definition and terminology for dynamical systems
W
Visualization of solutions with phase space plots
Course Outline
Th
Examples of dynamical systems
Although this course has undergone many revisions over the years, a synopsis of a typical syllabus is shown in Textbox 1. It is important to note that this course is taught during Washington and Lee’s six-week spring term during which students take only one or two courses. Consequently, this course is taught with twice the weekly contact hours of a longer term. For example, a three-credit course meets for six hours per week. In its current form, the course is organized around the concepts of dynamics with examples provided for each new idea. During the beginning of the course we present basic concepts and terminology of dynamics. Although this may seem obvious, it sets some critical groundwork about the nature of a mathematical science, namely, how and why we can model nature with mathematics. In particular, we emphasize the four points below.
F
First problem set
Measurements and Predictions Because the basis of any science is empirical verification of a hypothesis, it is appropriate to consider how mathematics can shape this process. Experimental results that can be expressed as numbers are highly prized in science because they can be well characterized and have considerable persuasive power. The theoretical counterpart of this process must also be able to cast its predictions as numbers. This leads to scientific models of nature whose ultimate forms are equations or algorithms. Although this may seem obvious to a practicing scientist, the rationale behind mathematical science needs to be discussed with students, both majors and nonmajors. State and State Variables The first step in the construction of a mathematical model is specifying the description or state of the system. In classical systems, the state is specified by a set of state variables representing physical properties of the system that can be measured, and thus, represented as numbers. Simple examples include describing the state of a pendulum with an angle and an angular velocity, or a chemical reaction with the concentrations of all chemical species. An important consideration is how comprehensive the state description needs to be to describe the phenomenon in question. For a typical kinetics problem, the concentrations and rates suffice. However, if we consider a biochemical or industrial process, other factors—such as temperature or mixing efficiency—may become important.
Week 2 M
Driven and dissipative systems
T
Difference equations: standard map, logistic map
W
Difference equations: stability criteria
Th
Nature of randomness
F
Second problem set
Week 3 M
Laboratory connection (BZ reaction)
T
Flows and differential equations: growth and decay
W
Flows and differential equations: oscillations
Th
Third problem set
F
Review
Week 4 M
Midterm
T
Higher dimensional flows
W
Higher dimensional flows: analysis
Th
Higher dimensional flows: examples
F
Scaling laws
Week 5 M
Attractors: Poincare sections and Lyapunov exponents
T
Fractal geometry
W
Fractal geometry
Th
Basins of attraction
F
The Mandelbrot and Julia sets
Week 6 M
Fourth problem set
T
Complexity, entropy, and the arrow of time
W
Randomness
Th
Probability and determinism
F
Review
Textbox 1. Course outline by week indicating when topics are introduced and when problem sets and exams are given.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 8 August 2008 • Journal of Chemical Education
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These equations predict future states of the system based on the current state. More generally, we refer to the combination of the set of state variables and the equations of motion as a dynamical system. At this point students need to understand the difference between the equations and the solutions to these equations. Although the equations of motion contain the dynamics, it is the solution to these equations that explicitly display the predictions of the model. One possibility is to solve the equations exactly to produce a “closed-form” solution. This kind of solution can express the dynamics in toto as another set of equations. If these equations cannot be solved exactly, we can generate a numerical solution using an algorithm. In this case, we generate future states sequentially and never possess the entire dynamics. As an example, we compare the exact solutions for the pendulum in terms of trigonometric functions, a spreadsheet with numerically generated values of position and velocity, and a graph of both solutions. This approach also makes the practical point that graphical representations of the mathematics are often the most useful way to interpret the dynamics regardless of the method of solution. Phase Space The usefulness of graphical methods is a theme running throughout this course. If we simply plot the value of a state variable versus time we generate a time series, whereas if we plot the state variables against each other we generate a phase portrait or phase space plot. The time series is useful for direct prediction of the value of physical property at some point in time. The phase space plot, however, is useful as a conceptual tool in the analysis of the overall dynamics. The coordinates of each point on the phase space plot completely specify the state of the system at an instant in time. Plotting sequential points in phase space leads to graphical representations of entire dynamics known as trajectories (continuous time) or orbits (discrete time). For example, if a trajectory goes to a single point, we have reached a fixed point. If the system is a chemical reaction, this fixed point represents chemical equilibrium. If the trajectories form closed figures, we have periodic dynamics or limit cycles. This is a less familiar type of chemical behavior although many obvious examples exist from biochemistry, such as the citric acid cycle. Expectations for Mathematical Knowledge It is obviously important to carefully choose the level of mathematics at which the course proceeds. For example, we begin with one-variable systems using difference equations such as, (1) xn 1 R x n 1 xn
where x is the dynamical or state variable, R is a parameter, and subscript n is a time index (2–5). A commonly used application of this equation is from population biology, where x is an animal population and R is a growth parameter. Depending on the value of R, the rabbit population can reach equilibrium, oscillate, or undergo deterministic chaos. These equations can easily be solved by iteration using Excel and form the basis of many homework problems. It is important that students are able to actually solve these equations on their own, as this convinces
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them that they have some ability to actually do science without the teacher giving them the answer. Alternatively, we present physical systems that can be described using differential equations, namely equations of the form: dx f x
dt
(2) These are easily generalized to systems of equations, such as pendulums from physics and chemical kinetics. For example, we begin with a discussion of simple exponential growth or decay, described by the equation: dx qkx dt
(3) where x is the amount of a substance, k is a growth–decay parameter, and t is time. Although analytical solutions to these equations are well known, we ask students to solve them on their own using numerical methods. In particular, we use the method of Euler, which simply replaces the derivative with a ratio of differences:
%x qkx %t
By letting Δx = xn+1 − xn , we can rearrange this to give an iterative solution:
xn 1 xn q k xn %t
(4)
This method transforms the differential equation into a difference equation that may be solved by the already familiar method of iteration. In addition, we use this discussion to introduce and elaborate on the derivative as the mathematical expression of a rate of change or velocity. As students work through each stage of this process on their own they are able to both learn the concepts and demonstrate their ability to do scientific problems. This course also covers the basic concepts of fractal geometry and noninteger dimensionality (7). These ideas have proven to be a popular topic mainly because of the visually compelling nature of fractals. Fractals naturally arise in the context of dynamical systems, as many processes will either generate objects with a fractal character (such as a tree) or have fractal attractors in phase space plots. There are numerous good derivations of the mathematical form of the fractal dimension. In the course our emphasis is on the concept of scaling laws, which relate how an object’s properties change as it changes size. For example, the mass of a three-dimensional object will change as the cube of one of its three linear dimensions, whereas the mass of a twodimensional object, such as a rug, will increase as the square of one of its two linear dimensions. For a fractal object, the power relating the measurement of a bulk property to a linear measure of its size will be a noninteger. This noninteger exponent is thought of as the fractal dimension. Aside from examples from social sciences like economics, fractals also provide the basis for some interesting laboratory projects.
Journal of Chemical Education • Vol. 85 No. 8 August 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom
Learning Outcomes The three sections below outline the expectations for student learning regarding fundamental components of the course: concepts and terminology, problem solving, and laboratory work. Concepts and Terminology As with any technical discipline, students must learn the specialized terminology and basic concepts on which physical models are based. In our experience with this course, this is best done with short essays. Because many students struggle at first to rigorously express technical ideas in writing, both homework problems and laboratory assignments coupled with timely feedback are important to developing these skills before confronting essays on exams. See List 1 for the learning outcomes we have identified for this aspect of the course. Problem Solving With basic concepts in hand, students must see how these ideas are used to explain specific phenomena. In science major courses problem solving is often emphasized at the expense of concepts; the reverse is often true in nonmajor courses. The approach in this course is based on a weekly take-home problem set in which students are allowed one class period to work in groups with the instructors available for help. Students are encouraged to discuss the approach to solving each problem during the class. After this point students must write their own answers although the instructors are still available for help. See List 2 for the problem-solving learning outcomes. Laboratory Work The laboratory work is this course is not designed to teach techniques. Rather, the assignments are designed so students see how these theoretical concepts appear in real life and investigate some of the difficulties in collecting data for comparison with predictions. See List 3 for the lab work learning outcomes. The Laboratory Experience We added a laboratory component to the course in 2001 with the support of a grant from the Keck Foundation. In the earliest versions students attended a weekly computer session where they performed assignments based on computer simulations. In the new format students performed laboratory-based projects in addition to significantly expanded computer exercises. Note that we use the term “project” rather than “experiment”. This is because students were not following a standard scientific method model of observation, hypothesis, and so forth. Instead, they are being presented with a specific dynamical system, such as an electronically tracked pendulum or an oscillating chemical reaction in a spectrophotometer, and being asked to collect data and analyze the dynamics. Thus, the “physical” laboratory was designed to illustrate the reality of the phenomena being studied and to consider the difficulties of actual data collection and display. The computer simulation projects, on the other hand, allow more opportunity for “what if ?” experimentation. This combination provides a balanced exposure to the empirical method given the constraints of time, skill, and equipment.
The computer projects fall into two types. The first type involves using software that simulates a particular system, such as the Lorenz attractor (1, 3–5). Students must decide on a systematic way to explore the dynamics by adjusting the parameters of the model and creating the appropriate plots. The second type involves being presented with a mathematical description of a system and evaluating the dynamics using a spreadsheet program. This second type of project involves translating the model to a form that can be solved by iteration and making more extensive decisions about which plots to generate to capture the system’s behavior. Aside from the well-known effectiveness of these “student active” approaches, we have noted that students are very responsive to learning the details of using a spreadsheet program and see this as a practical skill. List 1. Concepts and Terminology Learning Outcomes for Students 1.
Give concise and accurate definitions for technical terminology.
2.
Describe basic principles from dynamics in prose.
List 2. Problem-Solving Learning Outcomes for Students 1.
Write qualitative descriptions of how basic concepts apply to a specific example. For example, a student should be able to describe how a population at equilibrium is different from a population in a cycle.
2.
Evaluate the behavior of mathematical models expressed as difference equations using spreadsheet programs.
3.
Translate basic differential equations into difference equations using Euler’s method.
4.
Display the results of numerical solutions graphically and interpret these graphs.
5.
Find fixed points of difference and differential equations using analytical or graphical methods.
6.
Determine the fractal dimension of a simple fractal analytically or graphically, depending on the data.
List 3. Laboratory Work Learning Outcomes for Students 1.
Describe the qualitative dynamics of the physical system in question, with an emphasis on describing which variables are being measured and which are being controlled.
2.
Collect data from a given experiment and discuss any difficulties or uncertainties with this process in qualitative terms.
3.
Create the appropriate graphical representations of the data and interpret these graphs.
4.
Draw reasonable conclusions about the results.
5.
Organize the efforts in the previous four outcomes into a coherent scientific report.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 8 August 2008 • Journal of Chemical Education
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The laboratory projects involve either observation of a dynamical system or analysis of a phenomenon with a fractal character. Examples of the first type are:
• A damped and driven pendulum whose motion is directly measured electronically and displayed on a computer as either a phase space plot, or a time series. These devices are available commercially (8).
• A dripping faucet, which was one of the earliest systems studied in the history of nonlinear dynamics (9). We have used a commercially available drop counter (intended for automated titrations) to measure the time between drops.
• The Belousov–Zhabotinsky (BZ) reaction. This is a wellstudied example of an oscillating chemical reaction. We use one of the basic “recipes” for this reaction from the laboratory books of Shakhashiri (10). In lab, we measure the “position” of the reaction by using a peristaltic pump to move a continuous sample of the reaction mixture into a flow-through cell in a UV–visible spectrophotometer.
The latter set of projects look at physical fractals generated by several different processes. These fractals are presented as “footprints” of the dynamical processes involved. Examples include:
• Electrochemical deposition of copper (11, 12). This rather elegant experiment creates a delicate fractal as copper metal is deposited on a piece of plexiglass by the reduction of ions in solution.
• Viscous fingers (13). These patterns are generated by letting one fluid diffuse into another.
• A ball of crumpled paper (14). This is a simple yet fun experiment determining the fractal dimension of crumpled balls of various kinds of paper.
• Natural images. In this project, students use digital cameras to find “natural fractals”. These images are converted to high contrast grayscale using image processing software and the fractal dimension is determined using commercially available software (15).
Conclusions Over the years this course has captured the imaginations of many students with its unusual combination of abstract mathematical concepts and mundane physical examples. Aside from the learning outcomes listed, we believe we have succeeded in showing students the power of mathematics in science and demonstrating their own ability to solve problems using these methods. As useful as they may find the knowledge gained, we sincerely hope that the most persistent result of their experience is a sense of wonder at the ability of people to understand the world they inhabit.
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Acknowledgments I wish to thank my colleague Michael Pleva for his many years of service with me in teaching this course, and Lisa Alty and Frank Settle for proofreading this document. I would also like to thank Philip Trimmer, chemistry technician, and Chad Ellis, a former student now at Carnegie-Mellon University, for their help in developing laboratory experiments. This work was supported by grants from the W. M. Keck Foundation, the Dr. Scholl’s Foundation, and Washington and Lee University. Literature Cited 1. Gleick, J. Chaos: Making a New Science; Penguin Books: New York, NY, 1987. 2. Devaney, R. L. An Introduction to Chaotic Dynamical Systems, 2nd ed.; Persus Books: Reading, MA, 1989. 3. Kaplan, D.; Glass, L. Understanding Nonlinear Dynamics; Springer-Verlag: New York, 1995. 4. Baker, G. L.; Gollub, J. P. Chaotic Dynamics: An Introduction, 2nd ed.; Cambridge University Press: New York, 1996. 5. Williams, G. W. Chaos Theory Tamed; Joseph Henry Press: Washington, DC, 1997. 6. Hammond, G.; Gray, H. J. Chem. Educ. 1968, 45, 354–356. 7. Barnsley, M. F. Fractals Everywhere, 2nd ed.; Academic Press: San Diego, CA, 1993. 8. PASCO Web site for Chaos/Driven Harmonic Accessory. http:// store.pasco.com/pascostore/showdetl.cfm?&DID=9&Product_ ID=1698&Detail=1 (accessed Apr 2008). 9. Crutchfield, J. P.; Farmer, J. D.; Packard; N. H.; Shaw, R. S. Sci. Am. 1986, 255 (6), 46–57. 10. Shakhashiri, B. Z. Chemical Demonstrations: A Handbook for Teachers of Chemistry, Vol. 2; The University of Wisconsin Press: Madison, WI, 1985. 11. 4.1–Growing Rough Patterns: Electrodeposition. http://argento. bu.edu/ogaf/html/chp41.htm (accessed Apr 2008). 12. Williams, H. T.; Goodwin, L.; Desjardins, S. G.; Billings, F. T. Phys. Lett. A 1998, 250, 105–110. 13. 4.4–Viscous Fingering. http://polymer.bu.edu/ogaf/html/chp44. htm (accessed Apr 2008). 14. Ko, R. H.; Bean, C. P. The Physics Teacher 1991, Feb, 78–79. 15. Benoit Fractal Analysis System Home Page. http://www.trusoft. netmegs.com/ (accessed Apr 2008).
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Aug/abs1078.html Abstract and keywords Full text (PDF) Links to cited URLs and JCE articles Supplement Excel spreadsheet that illustrates the use of the Euler method to calculate the trajectories for the Lorenz equations
Journal of Chemical Education • Vol. 85 No. 8 August 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
