## Friday, June 25, 2010

Russ Hobbie and I discuss Fick’s laws of diffusion in Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology. The German scientist Adolf Fick (1829–1901) was a classic example of a researcher who was comfortable in both physics and physiology. He enrolled at the University of Marburg with the goal of studying mathematics and physics, but eventually switched to medicine, and earned an MD in 1852. Of particular interest to me is that he wrote a classic textbook titled Medical Physics (1856), which was one of the first books on this topic. I have not read this book, which almost certainly is written in German (although I am half German through my father’s side, I cannot speak or read the language). Nevertheless, I wonder if Intermediate Physics for Medicine and Biology might be a descendant of this text.

Fick was only 26 when he proposed his two laws of diffusion. The first law (Eq. 4.18a in our book)—similar to Ohm’s law for electrical current or Fourier’s law for heat conduction—states that the diffusive flux is proportional to the concentration gradient. The constant of proportionality is the diffusion constant, which Fick first introduced. Fick’s second law (Eq. 4.24) arises by combining his first law with the equation of continuity (Eq. 4.2) and is what we generally refer to as the diffusion equation. He tested his two laws by measuring the diffusion of salt in water. He even noticed the strong temperature dependence of the diffusion constant.

Fick contributed to physiology and medicine in several ways. He made the first successful contact lens, and he developed a method to measure cardiac output based on oxygen consumption and blood oxygen concentration. You can find more information about his life at http://www.corrosion-doctors.org/Biographies/FickBio.htm.

## Friday, June 18, 2010

### Myopia

Section 14.12 in the 4th edition of Intermediate Physics for Medicine and Biology discusses the physics of the eye. One topic related to vision that I have always found fascinating is myopia.
In nearsightedness or myopia, parallel rays come to a focus in front of the retina. The eye is slightly too long for the shape of the cornea… The total converging power of the eye is too great, and the relaxed eye focuses at some closer distance, from which the rays are diverging. Accommodation can only increase the converging power of the eye, not decrease it, so the unassisted myopic eye cannot focus on distant objects. Myopia can be corrected by placing a diverging spectacle or contact lens in front of the eye, so that incoming parallel rays are diverging when the strike the cornea.
The interesting thing about myopia is that, in contrast to far-sightedness (hypermetropy), you cannot correct it by accommodation. Before the invention of eye glasses in the late Middle Ages, if you were born with myopia then distant objects would always be a blur.

 Mornings on Horseback, by David McCullough.
When teaching Biological Physics (PHY 325) at Oakland University, I often end my discussion of myopia with a quote from David McCullough’s wonderful biography of Theodore Roosevelt, Mornings on Horseback. Roosevelt suffered from myopia and didn’t get his first glasses until he was a teenager. McCullough tells the story:
Then, at a stroke, the summer of 1872, he was given a gun and a large pair of spectacles and nothing had prepared him for the shock, for the infinite difference in his entire perception of the world about him or his place in it.

The gun was a gift from Papa—a 12-gauge, double-barreled French-made (Lefaucheux) shotgun with a lot of kick and of such simple, rugged design that it could be hammered open with a brick if need be, the ideal gun for an awkward, frequently absent-minded thirteen-year-old. But blasting away with it in the woods near Dobbs Ferry he found he had trouble hitting anything. More puzzling, his friends were constantly shooting at things he never even saw. This and the fact that they could also read words on billboards that he could barely see, he reported to his father, and it was thus, at summer’s end, that the spectacles were obtained.

They transformed everything. They “literally opened an entirely new world to me,” he wrote years afterward, remembering the moment. His range of vision until then had been about thirty feet, perhaps less. Everything beyond was a blur. Yet neither he nor the family had sensed how handicapped he was. “I had no idea how beautiful the world was… I could not see, and yet was wholly ignorant that I was not seeing.”

How such a condition could possibly have gone undetected for so long is something of a mystery, but once discovered it did much to explain his awkwardness and the characteristic detached look he had, those large blue eyes “not looking at anything present.”
I am a lover of history, and a big fan of David McCullough. A couple of his books with a scientific or engineering bent are Path Between the Seas: The Creations of the Panama Canal and The Great Bridge: The Epic Story of the Building of the Brooklyn Bridge. His purely historical books, such as 1776 and John Adams, are also excellent.

To learn more, see the information about myopia on the website for the American Optometric Association. An modern option for correcting myopia that was not available in Roosevelt’s time is laser surgery to reshape the cornea.

## Friday, June 11, 2010

Last week in this blog, I wrote that the “Gibbs Paradox” deserved an entire entry of its own. Well, here it is. Russ Hobbie and I mention the Gibbs Paradox in a footnote in Section 3.18 (The Chemical Potential of a Solution) in the 4th edition of Intermediate Physics for Medicine and Biology. When calculating the entropy of mixing (where a solute and solvent are intermixed), we derived an expression for the number of ways N particles can be distributed among N sites. If we assume the solute particles are indistinguishable, there is only one way. The footnote then reads
The fact that there is only one mircostate because of the indistinguishability of the particles is called the Gibbs paradox. For an illuminating discussion of the Gibbs paradox, see Casper and Freier (1973).
 Fundamentals of Statistical and Thermal Physics, by Frederick Reif.
The Gibbs Paradox is examined in more detail by Frederick Reif in his landmark textbook Fundamentals of Statistical and Thermal Physics. (Indeed, Chapter 3 of Intermediate Physics for Medicine and Biology follows a statistical approach similar to Reif’s analysis, and even more similar to the discussion in his introductory textbook—a personal favorite of mine—Statistical Physics, Berkeley Physics Course Volume 5). Reif considers “a gas consisting of N identical monatomic molecules of mass m enclosed in a container of volume V.” When he calculates the entropy, S, of the gas, he obtains
S = N k [ln V + 3/2 ln T + σ]     (7.2.16)
where k is Boltzmann’s constant, T is the absolute temperature, and σ is a constant independent of N, T, and V. He then ends the section with the provocative statement “This expression for the entropy is, however, not correct,” which leads to his discussion (Sec. 7.3) of the Gibbs paradox. Reif continues
The challenging statement at the end of the last section suggests that the expression (7.2.16) for the entropy merits some discussion… [The expression] for S is clearly wrong since it implies that the entropy does not behave properly as an extensive quantity. Quite generally, one must require that all thermodynamic relations remain valid if the size of the whole system under consideration is simply increased by a scale factor α, i.e., if all its extensive parameters are multiplied by the same factor α. In our case, if the independent extensive parameters V and N are multiplied by α, the mean energy… is indeed properly increased by this same factor, but the entropy S in (7.2.16) is not increased by α because of the term N ln V.

Indeed, (7.2.16) asserts that the entropy S of a fixed volume V of gas is simply proportional to the number N of molecules. But this dependence on N is not correct, as can readily be seen in the following way. Imagine that a partition is introduced which divides the container into two parts. This is a reversible process which does not affect the distribution of systems over accessible states. Thus, the total entropy ought to be the same with, or without, the partition in place; i.e.

S = S' + S"     (7.3.1)

where S' and S" are the entropies of the two parts. But the expression (7.2.16) does not yield the simple additivity required by (7.3.1). This is easily verified. Suppose, for example, that the partition divides the gas into two equal parts, each containing N' molecules of gas in a volume V'. Then the entropy of each part is given by (7.2.16) as

S' = S" = N' k [ln V' + 3/2 ln T + σ]

while the entropy of the whole gas without partition is by (7.2.16)

S = 2 N' k [ ln (2 V') + 3/2 ln T + σ] .

Hence

S – 2 S' = 2 N' k ln(2 V') – 2 N' k ln V' = 2 N' k ln2

and is not equal to zero as required by (7.3.1).

This paradox was first discussed by Gibbs and is commonly referred to as the “Gibbs paradox.” Something is obviously wrong in our discussion; the question is what.
Reif then analyzes in more detail the implications of removing the partition between the two sides of the box. He finds that
The act of removing the partition has thus very definite physical consequences. Whereas before removal of the partition a molecule of each subsystem could only be found within a volume V', after the partition is removed it can be located anywhere within the volume V = 2 V'. If the two subsystems consisted of different gasses, the act of removing the partition would lead to diffusion of the molecules throughout the whole volume 2V' and consequent random mixing of the different molecules. This is clearly an irreversible process; simply replacing the partition would not unmix the gases. In this case the increase in entropy in (7.3.2) would make sense as being simply a measure of the irreversible increase of disorder resulting from the mixing of unlike gases [the entropy of mixing that Russ and I calculated].

But if the gases in the subsystems are identical, such an increase of entropy does not make physical sense. The root of the difficulty embodied in the Gibbs paradox is that we treated the gas molecules as individually distinguishable, as though interchanging the positions of two like molecules would lead to a physically distinct state of the gas. This is not so. Indeed, if we treated the gas by quantum mechanics (as we shall do in Chapter 9), the molecules would, as a matter of principle, have to be regarded as completely indistinguishable. A calculation of the partition function would then automatically yield the correct result, and the Gibbs paradox would never arise. Our mistake has been to take the classical point of view too seriously. Even though one may be in a temperature and density range where the motion of molecules can be treated to a very good approximation by classical mechanics, one cannot go so far as to disregard the essential indistinguishability of the molecules.
Just how different must molecules be before they should be considered distinguishable?… In a classical view of nature two molecules could, or course, differ by infinitesimal amounts… In a quantum description this troublesome question does not arise because of the quantized discreteness of nature… Hence the distinction between identical and nonidentical molecules is completely unambiguous in a quantum-mechanical description. The Gibbs paradox thus foreshadowed already in the last [19th] century conceptual difficulties that were resolved satisfactorily only by the advent of quantum mechanics.
Several good American Journal of Physics articles discuss the Gibbs phenomenon. Pesic examines Gibb’s own writings to trace his thoughts on the issue (“The Principle of Identicality and the Foundations of Quantum Theory: I. The Gibbs Paradox,” American Journal of Physics, Volume 59, Pages 971–974, 1991), and Landsberg and Tranah study in more detail in role of the Gibbs paradox for quantum mechanics (“The Gibbs Paradox and Quantum Gases,” American Journal of Physics, Volume 46, Pages 228–230, 1978). Finally, Casper and Freier (the authors of the paper cited in our footnote) analyze the Gibbs paradox by comparing macroscopic and microscopic points of view (“‘Gibbs Paradox’ Paradox,” American Journal of Physics, Volume 41, Pages 509–511, 1973).

You know, there is a lot of physics in that little footnote on page 68 of Intermediate Physics for Medicine and Biology.

## Friday, June 4, 2010

### The Gibbs Phenomenon

In chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Fourier analysis, a fascinating but very mathematical subject. One of the most surprising results of Fourier analysis is the Gibbs phenomenon, which we describe at the end of Sec. 11.5 (Fourier Series for a Periodic Function).
Table 11.4 shows the first few coefficients for the Fourier series representing the square wave, obtained from Eq. 11.34… Figure 11.16 shows the fits for n = 3 and n = 39. As the number of terms in the fit is increased, the value of Q [measuring the least squares fit between the function at its Fourier series] decreases. However, spikes of constant height (about 18% of the amplitude of the square wave or 9% of the discontinuity in y) remain. These are seen in Fig. 11.16. These spikes appear whenever there is a discontinuity in y and are called the Gibbs phenomenon.
You have to be amazed by the Gibbs phenomenon. Think about it: as you add terms in the sum, the fit between the function and its Fourier series gets better and better, but the overshoot in amplitude does not get any smaller. Instead, the region containing ringing near the discontinuity gets narrower and narrower. If you want to see a figure like our Fig. 11.16 presented as a neat animation, take a look at http://www.sosmath.com/fourier/fourier3/gibbs.html. Also, check out http://ocw.mit.edu/ans7870/18/18.06/javademo/Gibbs/ for an interactive demo that will let you include up to 200 terms in the Fourier series.

The Gibbs phenomenon is important in medical imaging. The entry for the Gibbs phenomenon from the Encyclopedia of Medical Imaging is reproduced below.
Gibbs phenomenon, (J. Willard Gibbs, 1839-1903, American physicist), phenomenon occurring whenever a “curve” with sharp edges is subject to Fourier analysis. The Gibbs phenomenon is relevant in MR imaging, where it is responsible for so-called Gibbs artefacts. Consider a signal intensity profile across the skull, where at the edge of the brain the signal intensity changes from virtually zero to a finite value. In MR imaging the measurement process is a breakdown of such intensity profiles into their Fourier harmonics. i.e. sine and cosine functions. Representation of the profiles measured with a limited number of Fourier harmonics is imperfect, resulting in high frequency oscillations at the edges, and the image can therefore exhibit some noticeable signal intensity variations at intensity boundaries: the Gibbs phenomenon, overshoot artefacts, or “ringing.” The artefacts can be suppressed by filtering the images. However, filtering can in turn reduce spatial resolution.
Figures 12.24 and 12.25 of our book show a CT scan with ringing inside the skull and its removal by filtering, an example of the Gibbs phenomenon.

Josiah Willard Gibbs was a leading American physicist from the 19th century. He is particularly well known for his contributions to thermodynamics. Gibbs appears at several places in Intermediate Physics for Medicine and Biology. Section 3.17 discusses the Gibbs free energy, a quantity that provides a simple way to keep track of the changes in total entropy when a system is in contact with a reservoir at constant temperature and pressure. A footnote on page 68 addresses the Gibbs paradox (which deserves an entire blog entry of its own), and Problem 47 in Chapter 3 introduces the Gibbs factor (similar to the Boltzmann factor but including the chemical potential).

 Selected Papers of Great American Physicists.
The preface to Gibbs’ book on statistical mechanics is reproduced in Selected Papers of Great American Physicists: The Bicentennial Commemorative Volume of the American Physical Society 1976, edited by Spencer Weart. I recall being quite impressed by this book when in graduate school at Vanderbilt University. Below is a quote from Weart’s biographical notes about Gibbs.
Gibbs, son of a Yale professor of sacred literature, descended from a long line of New England college graduates. He studied at Yale, received his Ph.D. there in 1863—one of the first doctorates granted in the United States—tutored Latin and natural philosophy there, and then left for three decisive years in Europe. Up to that time, Gibbs had shown interest in both mathematics and engineering, which he combined in his dissertation “On the Form of the Teeth in Wheels in Spur Gearing.” The lectures he attended in Paris, Berlin and Heidelberg, given by some of the greatest men of the day, changed him once and for all. In 1871, two years after his return from Europe, he became Yale’s first Professor of Mathematical Physics. He had not yet published any papers on this subject. For nine years he held the position without pay, living on the comfortable inheritance his father had left; only when Johns Hopkins University offered Gibbs a post did Yale give him a small salary.

Gibbs never married. He lived out a calm and uneventful life in the house where he grew up, which he shared with his sisters. He was a gentle and considerate man, well-liked by those who knew him, but he tended to avoid society and was little known even in New Haven. Nor was he known to more than a few of the world’s scientists—partly because his writings were extremely compact, abstract and difficult. As one of Gibb’s European colleagues wrote, “Having once condensed a truth into a concise and very general formula, he would not think of churning out the endless succession of specific cases that were implied by the general proposition; his intelligence, like his character, was of a retiring disposition.” The Europeans paid for their failure to read Gibbs: A large part of the work they did in thermodynamics before the turn of the century could have been found already in his published work.