The Logistic Map

In Section 10.8 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the logistic map, a difference equation that can describe phenomena such as population dynamics. We are by no means the first to use the logistic map to illustrate deterministic chaos. Indeed, it has become the canonical example of chaos since Robert May published “Simple Mathematical Models With Very Complicated Dynamics” in 1976 (Nature, Volume 261, Pages 459–467). This paper has been cited nearly 2500 times, implying that it has had a major impact.

Russ and I write the logistic equation as (Eq. 10.36 in our book)

x_j+1 = a x_j (1 – x_j)

where x_j is the population in the j^th generation. Our first task is to determine the equilibrium value for x_j.

The equilibrium value x* can be obtained by solving Eq. 10.36 with x_j+1 = x_j = x*:

x* = a x* (1 – x*) = 1 – 1/a.

Point x* can be interpreted graphically as the intersection of Eq. 10.36 with the equation x_j+1 = x_j as shown in Fig. 10.22. You can see from either the graph or from Eq. 10.37 that there is no solution for positive x if a is less than 1. For a = 1 the solution occurs at x* = 0. For a = 3 the equilibrium solution is x* = 2/3. Figure 10.23 shows how, for a = 2.9 and an initial value x₀ = 0.2, the values of x_j approach the equilibrium value x* = 0.655. This equilibrium point is called an attractor.

Figure 10.23 also shows the remarkable behavior that results when a is increased to 3.1. The values of x_j do not come to equilibrium. Rather, they oscillate about the former equilibrium value, taking on first a larger value then a smaller value. This is called a period-2 cycle. The behavior of the map has undergone period doubling. What is different about this value of a? Nothing looks strange about Fig. 10.22. But it turns out that if we consider the slope of the graph of x_j+1 vs x_j at x*, we find that for a greater than 3 the slope of the curve at the intersection has a magnitude greater than 1.

Usually, when Russ and I say something like “it turns out”, we include a homework problem to verify the result. Homework 34 in Chapter 10 does just this; the reader must prove that the magnitude of the slope is greater than 1 for a greater than 3.

One theme of Intermediate Physics for Medicine and Biology is the use of simple, elementary examples to illustrate fundamental ideas. I like to search for such examples to use in homework problems. One example that has great biological and medical relevance is discussed in Problems 37 and 38 (a model for cardiac electrical dynamics based on the idea of action potential restitution). But when reading May’s review in Nature, I found another example that—while it doesn’t have much direct biological relevance—is as simple or even simpler than the logistic map. Below is a homework problem based on May’s example.

Section 10.8

Problem 33 ½ Consider the difference equation

     (a) Plot x_n+1 versus x_n for the case of a=3/2, producing a figure analogous to Fig. 10.22.

     (b) Find the range of values of a for which the solution for large n does not diverge to infinity or decay to 0. You can do this using either arguments based on plots like in part (a), or using numerical examples.

     (c) Find the equilibrium value x* as a function of a, using a method similar to that in Eq. 10.37.

     (d) Determine if this equilibrium value is stable or unstable, based on the magnitude of the slope of the x_n+1 versus x_n curve.

     (e) For a = 3/2, calculate the first 20 values of x_n using 0.250 and 0.251 as initial conditions. Be sure to carry your calculations out to at least five significant figures. Do the results appear to be chaotic? Are the results sensitive to the initial conditions?

     (f) For one of the data sets generated in part (e), plot x_n+1 versus x_n for 25 values of n, and create a plot analogous to Fig. 10.27. Explain how you could use this plot to distinguish chaotic data from a random list of numbers between zero and one.

Ernest Rutherford

Who is the greatest physicist never mentioned by name in the 4th edition of Intermediate Physics for Medicine and Biology? Russ Hobbie and I allude to Newton, Maxwell, Faraday, Bohr, Einstein, and many others. But a search for the name “Rutherford” comes up empty. In my opinion, Ernest Rutherford is the greatest physicist absent from our book. Ironically, he is also one of my favorite physicists; a colorful character who rivals Faraday as the greatest experimental scientist of all time.

Asimov's Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

Rutherford (1871–1937) was born in New Zeeland, and attended Cambridge University in England on a scholarship. His early work was on radioactivity, a subject discussed in Chapter 17 of our textbook. Asimov’s Biographical Encyclopedia of Science and Technology states

[Rutherford] was one of those who, along with the Curies, had decided that the rays given off by radioactive substances were of several different kinds. He named the positively charged ones alpha rays and the negatively charged ones beta rays… Between 1906 and 1909 Rutherford, together is his assistant, Geiger, studied alpha particles intensively and proved quite conclusively that the individual particle was a helium atom with its electrons removed.

Rutherford’s interest in alpha particles led to something greater still. In 1906, while still at McGill in Montreal, he began to study how alpha particles are scattered by thin sheets of metal… From this experiment Rutherford evolved the theory of the nuclear atoms, a theory he first announced in 1911…

For working out the theory of radioactive disintegration of elements, for determining the nature of alpha particles, [and] for devising the nuclear atom, Rutherford was awarded the 1908 Nobel Prize in chemistry, a classification he rather resented, for he was a physicist and tended to look down his nose at chemists…

Rutherford was … the first man ever to change one element into another as a result of the manipulations of his own hands. He had achieved the dream of the alchemists. He had also demonstrated the first man-made “nuclear reaction”…

He was buried in Westminster Abbey near Newton and Kelvin.

Rutherford also measured the size of the nucleus. To explain his alpha particle scattering experiments, he derived his famous scattering formula (see Chapter 4 of Eisberg and Resnick for details). He found that his formula worked well except when very high energy alpha particles are fired at low atomic-number metal sheets. For instance, results began to deviate from his formula when 3 MeV alpha particles are fired at aluminum. The homework problem below explains how to estimate the size of the nucleus from this observation. This problem is based on data shown in Fig. 4-7 of Eisberg and Resnick’s textbook.

Section 17.1

Problem ½  An alpha particle is fired directly at a stationary aluminum nucleus. Assume the only interaction is the electrostatic repulsion between the alpha particle and the nucleus, and the aluminum nucleus is so heavy that it is stationary. Calculate the distance of their closest approach as a function of the initial kinetic energy of the alpha particle. This calculation is consistent with Ernest Rutherford’s alpha particle scattering experiments for energies lower than 3 MeV, but deviates from his experimental results for energies higher than 3 MeV. If the alpha particle enters the nucleus, the nuclear force dominates and the formula you calculated no longer applies. Estimate the radius of the aluminum nucleus.

Rutherford, Simple Genius,
by David Wilson.

To learn more about Ernest Rutherford and his groundbreaking experiments, I recommend the book Rutherford: Simple Genius by David Wilson.

In addition to his fundamental contributions to physics, I have a personal reason for liking Rutherford. Academically speaking, he is my great-great-great-great-grandfather. My PhD advisor was John Wikswo, who got his PhD working under William Fairbank at Stanford. Fairbank got his PhD under Cecil Lane, who studied under Etienne Bieler, who worked for James Chadwick (discoverer of the neutron), who was a student of Rutherford’s.

Ernest Rutherford died (needlessly) on October 19, 1937; 75 years ago today.

The Gaussian integral

One of my favorite “mathematical tricks” is given in Appendix K of the 4th edition of Intermediate Physics for Medicine and Biology. The goal is to calculate the integral of the Gaussian function, e^-x2, or bell shaped curve. (This is often called the Gaussian Integral). The indefinite integral cannot be expressed in terms of elementary functions (in fact, “error functions” are defined as the integral of the Gaussian), but the definite integral integrated over the entire x axis (from –∞ to ∞) is amazingly simple: the square root of π. Here is how Russ Hobbie and I describe how to derive this result:

Integrals involving e^-ax2 appear in the Gaussian distribution. The integral

can also be written with y as the dummy variable:

There can be multiplied together to get

A point in the xy plane can also be specified by the polar coordinates r and θ (Fig. K.1). The element of area dxdy is replaced by the element rdrdθ:

To continue, make the substitution u = ar², so that du = 2ardr. Then

The desired integral is, therefore,

Of course, if you let a =1, you get the simple result I mentioned earlier. Isn’t this a cool calculation?

To learn more, click here. For those of you who prefer video, click here.

Evaluation of the Gaussian Integral.

https://www.youtube.com/watch?v=nqNzKeVCYBU

Asimov's Biographical Encyclopedia
of Science and Technology,
by Isaac Asimov.

The integral and function are named after the German mathematician Johann Karl Friedrich Gauss (1777–1855). Asimov’s Biographical Encyclopedia of Science and Technology (2nd Revised Edition) says

Gauss, the son of a gardener and a servant girl, had no relative of more than normal intelligence apparently, but he was an infant prodigy in mathematics who remained a prodigy all his life. He was capable of great feats of memory and of mental calculation. There are those with this ability who are only average or below-average mentality, but Gauss was clearly a genius. At the age of three, he was already correcting his father’s sums, and all his life he kept all sorts of numerical records, even useless ones such as the length of lives of famous men, in days. He was virtually mad over numbers.

Some people consider him to have been one of the three great mathematicians of all time, the others being Archimedes and Newton.

The Truth About Terahertz

In Chapter 14 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Terahertz Radiation.

For many years, there were no good sources or sensitive detectors for radiation between microwaves and the near infrared (0.1-100 THz; 1 THz = 10¹² Hz). Developments in optoelectronics have solved both problems, and many investigators are exploring possible medical uses of THz radiation (“T rays”). Classical electromagnetic wave theory is needed to describe the interactions, and polarization (the orientation of the E vector of the propagating wave) is often important. The high attenuation of water to this frequency range means that studies are restricted to the skin or surface of organs such as the esophagus that can be examined endoscopically. Reviews are provided by Smye et al. (2001), Fitzgerald et al. (2002), and Zhang (2002).

(By the way, apologies to Dr. N. N. Zinovev, a coauthor on the Fitzgerald et al. paper, whose last name is spelled incorrectly in our book.) 

In the September 2012 issue of the magazine IEEE Spectrum, Carter Armstrong (a vice president of engineering at L-3 Communications, in San Francisco) reviews some of the challenges facing the development of Terahertz radiation. His article “The Truth About Terahertz” begins

Wirelessly transfer huge files in the blink of an eye! Detect bombs, poison gas clouds, and concealed weapons from afar! Peer through walls with T-ray vision! You can do it all with terahertz technology—or so you might believe after perusing popular accounts of the subject.

The truth is a bit more nuanced. The terahertz regime is that promising yet vexing slice of the electromagnetic spectrum that lies between the microwave and the optical, corresponding to frequencies of about 300 billion hertz to 10 trillion hertz (or if you prefer, wavelengths of 1 millimeter down to 30 micrometers). This radiation does have some uniquely attractive qualities: For example, it can yield extremely high-resolution images and move vast amounts of data quickly. And yet it is nonionizing, meaning its photons are not energetic enough to knock electrons off atoms and molecules in human tissue, which could trigger harmful chemical reactions. The waves also stimulate molecular and electronic motions in many materials—reflecting off some, propagating through others, and being absorbed by the rest. These features have been exploited in laboratory demonstrations to identify explosives, reveal hidden weapons, check for defects in tiles on the space shuttle, and screen for skin cancer and tooth decay.

But the goal of turning such laboratory phenomena into real-world applications has proved elusive. Legions of researchers have struggled with that challenge for decades.

Armstrong then explores the reasons for these struggles. The large attenuation coefficient of T-rays places severe limitations on imaging applications. He then turns specifically to medical imaging.

Before leaving the subject of imaging, let me add one last thought on terahertz for medical imaging. Some of the more creative potential uses I’ve heard include brain imaging, tumor detection, and full-body scanning that would yield much more detailed pictures than any existing technology and yet be completely safe. But the reality once again falls short of the dream. Frank De Lucia, a physicist at Ohio State University, in Columbus, has pointed out that a terahertz signal will decrease in power to 0.0000002 percent of its original strength after traveling just 1 mm in saline solution, which is a good approximation for body tissue. (Interestingly, the dielectric properties of water, not its conductive ones, are what causes water to absorb terahertz frequencies; in fact, you exploit dielectric heating, albeit at lower frequencies, whenever you zap food in your microwave oven.) For now at least, terahertz medical devices will be useful only for surface imaging of things like skin cancer and tooth decay and laboratory tests on thin tissue samples.

Following a detailed review of terahertz sources, Armstrong finishes on a slightly more optimistic note.

There is still a great deal that we don’t know about working at terahertz frequencies. I do think we should keep vigorously pursuing the basic science and technology. For starters, we need to develop accurate and robust computational models for analyzing device design and operation at terahertz frequencies. Such models will be key to future advances in the field. We also need a better understanding of material properties at terahertz frequencies, as well as general terahertz phenomenology.

Ultimately, we may need to apply out-of-the-box thinking to create designs and approaches that marry new device physics with unconventional techniques. In other areas of electronics, we’ve overcome enormous challenges and beat improbable odds, and countless past predictions have been subsequently shattered by continued technological evolution. Of course, as with any emerging pursuit, Darwinian selection will have its say on the ultimate survivors.

Terahertz radiation is such a big field that one year ago the IEEE introduced a new journal: IEEE Transactions on Terahertz Science and Technology. In the inaugural issue, Taylor et al. examine “THz Medical Imaging.”