Friday, November 8, 2019

A Town Hall About The Health Risks of 5G Cell Phone Technology

5G Town Hall
Rochester, Michigan
November 7, 2019.
Yesterday I participated in a town hall meeting in Rochester, Michigan to discuss the new 5G cell phone technology. I was invited to attend in part because of my contributions to the book Intermediate Physics for Medicine and Biology, which discusses the health risks of electromagnetic fields.

When preparing for the event, I created a list of frequently asked questions (well, these were the questions I thought people would ask). Not wanting to waste this effort, I reproduce my FAQ below.

The event was....interesting. I was impressed by the passion of these concerned citizens, who packed a large room on a cold Thursday evening and for over two hours asked questions and voiced their opinions (mostly voiced their opinions). I’ve taught plenty of apathetic 20-something-year-olds who don’t engage with the lecture or challenge what I say, so I found this feisty crowd refreshing. Unfortunately, I was not convinced by their claims of dire health effects from 5G technology, and they were not convinced by me. The most disturbing moment was when I said something along the lines of “if you want to know more about the risks of cancer, consult the National Cancer Institute” and the response was a chorus of “No, No, No!” Goodness, if we can’t trust the National Cancer Institute to understand cancer, who can we trust? But no one threw a tomato at me, so I’ll call the evening a success. The FAQ below summarizes my view on this matter.

FAQ: 5G Cell Phone Health Effects

What is 5G?

5G is the fifth generation of technology for cell phones. It uses higher frequencies of electromagnetic radiation than 4G technology (up to 300 GHz, with a wave length of 1 mm).

What does IPMB say about the health risks from cell phones?

Section 9.10.5 of Intermediate Physics for Medicine and Biology addresses possible health risks from microwaves, mobile phones, and wi-fi. Russ Hobbie and I cite a 2005 review by John Moulder, Ken Foster, and their colleagues, which concludes that “Overall, a weight-of-evidence evaluation shows that the current evidence for a causal association between cancer and exposure to RF [radiofrequency] energy is weak and unconvincing.” We also cite “an exhaustive (390 page) report…by the International Committee on Non-Ionizing Radiation Protection (Vecchia et al. 2009)” that reaches a similar conclusion. Next we write that “Sheppard et al. (2008) evaluated all the proposed mechanisms for radio-frequency interactions with biological molecules and processes…[and conclude that] the principal mechanism for biological effects, and the only well-established mechanism, is the heating of tissues.” Finally, “Foster and Moulder (2013) reviewed the current state of research [on the health effects of wi-fi, and conclude that the evidence provides] ‘no basis to anticipate any biological effects.’”

The 2013 review by Foster and Moulder summarizes my opinion on this topic.
“Impossibility” arguments are difficult to sustain in biology; but the lack of a generally-accepted mechanism by which low-level (below ICNIRP and IEEE limits) RF fields in the GHz frequency range could produce biological effects, after many years of sustained efforts to uncover such mechanisms, makes it increasingly unlikely that any mechanism will be found.

What have Foster and Moulder said lately about health risks from wi-fi?

Ken Foster and John Moulder are experts on the interaction of electromagnetic radiation with the body. They are skeptical about many claims of health risks caused by power-line, cell-phone, and wi-fi radiation. Both are now emeritus professors; Foster at the University of Pennsylvania and Moulder at the University of Wisconsin. Intermediate Physics for Medicine and Biology cites Foster and Moulder several times, but the 5th edition of IPMB was published in 2015. What have they said lately?

Foster published a post two months ago in a Scientific American blog looking specifically at 5G technology and health risks. His conclusion: “So far, at least, there’s little evidence of danger.”

The most interesting development is a critique of Foster and Moulder’s 2013 review by Martin Pall, an emeritus professor at Washington State University. He concludes that “there are seven repeatedly found Wi-Fi effects which have also been shown to be caused by other similar EMF [electromagnetic field] exposures. Each of the seven should be considered, therefore, as established effects of Wi-Fi.” Pall’s central hypothesis is that cell phone radiation affects calcium ion channels, which if true could trigger a cascade of biological effects. In a rebuttal, Foster and Moulder (2018) write
Pall (2018) criticizes our 5-year-old review of studies related to Wi-Fi and health (Foster and Moulder 2013). We respond to his critique, and also note weaknesses in his selection and interpretation of studies on biological and health effects of Wi-Fi type signals...

Having examined the additional papers that Pall cites, we reaffirm our earlier conclusion: a number of studies have reported bioeffects of Wi-Fi exposures, but technical limitations make many of them difficult to interpret and artifacts cannot be excluded. We are not aware of any health-agency warnings about health risks of Wi-Fi technology. Despite some level of public controversy and an ongoing stream of reports of highly variable quality of biological effects of RF energy (e.g. articles in a recent special issue of the Journal of Chemical Neuroanatomy, Volume 75, 2016) health agencies consistently conclude that there are no proven hazards from exposure to RF fields within current exposure limits (even as they consistently call for more research).
My advice is to read the review, the critique, and the rebuttal, and then draw your own conclusion. You may find many of the technical details difficult to understand (I do), but you will better appreciate the complexity if these issues, and the difficulty in drawing definite conclusions from imperfect data. I don’t agree with Pall’s claims.

What does Bob Park have to say about the health risks from cell phones?

Robert Park is an emeritus professor of physics at the University of Maryland, and was the director of public information at the Washington office of the American Physical Society. He is the author of Voodoo Science, and wrote a weekly column titled “What’s New” debunking pseudoscience. His health has not allowed him to contribute to the recent discussion about the risks of cell phones and 5G technology (and, oh, how we miss him). Here’s what he wrote in “What’s New” on Sunday, May 6, 2012.
1. ALBERT WHO? DEAD PHYSICIST DISPELS MOBILE-PHONE MYTH. According to news reports last week: "There is still no evidence of harm to health from mobile-phone technologies," or other wireless devices such as Wi-Fi. A study for the UK's Health Protection Agency (HPA) is said to be the most complete review yet and new evidence is still being examined, according to Professor Anthony Swerdlow of the Institute of Cancer Research, who chaired the study. I once had a rubber stamp made that said: “More research is needed,” since its found at the end of every science paper. The unanswered question is why anyone thought microwave radiation might be a cancer agent in the first place? Cancer is linked to the formation of mutant strands of DNA. More than 100 years ago in his 1905 paper on the photoelectric effect, Albert Einstein predicted an abrupt threshold for photoemission at about 5 eV, just above the lovely blue limit of the visible spectrum, demonstrating wave-particle duality. He was awarded the 1923 Physics Nobel Prize [actually, Einstein received the Nobel Prize in 1921, and Robert Millikan received it in 1923, in part for his experimental work on the photoelectric effect]. Its also the threshold for the emission of invisible ultraviolet radiation that causes hideous skin cancers. The cancer threshold, is therefore, 1 million times higher than the microwaves band. The same enormous mistake was made in the 1980s when epidemiologists falsely warned that exposure to power line emission can cause cancer. Power lines abruptly stopped causing cancer in 1997 after the U.S. National Cancer Institute conducted a better study. Its painful to witness this sad history being replayed with mobile-phone radiation. Aside: My apologies to regular readers who have heard this 20 times before, but it has not gotten through to everyone.
Park’s argument remains as true now as eight years ago, except that a 300 GHz photon has an energy of one-thousandth of an electron volt, which is “only” a few thousand times less than the threshold for photoemission or UV skin cancer (and is about 25 times smaller than thermal energy, 0.025 eV). The possibility of DNA damage—the underlying cause of cancer—remains extraordinarily remote, but not as ridiculously remote as it was for cell phone technology ten years ago.

Electromagnetic radiation is considered a possible carcinogen. What’s that mean?

Something that is “possibly carcinogenic to humans” doesn’t probably cause cancer. It probably doesn’t cause cancer, but we can’t say for certain. RF radiation was not placed into two more threatening categories: “probably carcinogenic to humans” and “carcinogenic to humans.”

The website Science-Based Medicine states that
Despite the negative evidence to date, in 2011, the International Agency for Research on Cancer classified EMF [low-frequency electromagnetic fields] as a “possible” carcinogen. They have a low threshold for this category, which is rather long. It requires limited evidence of carcinogenic potential in humans and inadequate evidence in animals. This is the, “Probably should do more research just to be sure, but basically don’t worry about it,” category.

Who was Eleanor Adair, and what did she think about microwaves?

I have written about Eleanor Adair before in this blog. She was a leading expert on the interaction of microwaves with biological tissue, and was skeptical of any health hazards claims. A New York Times interview included this exchange:
Q. If I were to say to people, “Hey there’s this really cool idea: Why heat your whole house when you could use microwaves to heat yourself?” they would say: “You’ve got to be kidding. Don’t you know that microwaves are dangerous? They can even cause cancer.” What do you say to people who respond like that?

 A. I try to educate them in exactly what these fields are. That they are part of the full electromagnetic spectrum that goes all the way from the radio frequency and microwave bands, through infrared, ultraviolet, the gamma rays and all that.

And the difference between the ionizing X-ray, gamma ray region and the microwave frequency is in the quantum energy. The lower you get in frequency the lower you get in quantum energy and the less it can do to the cells in your body.

If you have a really high quantum energy such as your X-rays and ionizing-radiation region of the spectrum, this energy is high enough that it can bump electrons out of the orbit in your cells and it can create serious changes in the cells of your body such that they can turn into cancers and various other things that are not good for you.

But down where we are working, in the microwave band, you are millions of times lower in frequency and there the quantum energy is so low that they can’t do any damage to the cells whatsoever. And most people don’t realize this.

Somehow, something is missing in their basic science education, which is something I keep trying to push. Learn the spectrum. Learn that you’re in far worse shape if you lie out on the beach in the middle of summer and you soak up that ultraviolet radiation than you are if you use your cell phone.

Any new data about health effects of electromagnetic fields in the last few years?

A recent article examining the “Occupational Exposure to High-Frequency Electromagnetic Fields and Brain Tumor Risk in the INTEROCC Study: An Individualized Assessment Approach,” (Vila et al., 2018) provides the following highlights
• Evidence on health effects of long-term occupational exposure to high-frequency EMF remains weak
• Individualized cumulative occupational RF [radiofrequency, 10 MHz–300 GHz] and IF [intermediate frequency, 3 kHz–10 MHz] exposure indices were assigned to study subjects
• No clear associations with RF or IF EMF and glioma or meningioma risk were observed
• The possible role of RF magnetic fields on brain tumor promotion/progression should be further investigated.
As Bob Park said, everyone supports doing additional research (as do I). But I don’t see a lot to be worried about here.

Bill Curry concluded that 5G technology “is likely to be a serious health hazard.” Well?

I’ve written about physicist Bill Curry and his claims previously in this blog. That post begins
A recent article by William Broad in the New York Times—titled “The 5G Health Hazard That Isn’t”—tells the sad story of how unfounded fears of radiofrequency radiation were stoked by one mistaken scientist.

What is electromagnetic hypersensitivity?

Some people claim they’re extremely sensitive to weak electromagnetic fields. The SkepDoc Harriet Hall wrote a blog post titled “Myths About Electromagnetic Hypersensitivity and Multiple Chemical Sensitivity.” She begins
As if we didn’t have enough things to worry about already, now we are being told to fear our toasters. A typical headline trumpets “The Effects of Invisible Waves.” We are increasingly exposed to electromagnetic radiation from cell phones, cell phone towers, wireless Internet routers, cordless phones, and power lines. Other sources ... are our household appliances: TVs, hairdryers, light bulbs, and yes, your trusty toaster. These invisible villains are said to lead to a variety of symptoms, including poor sleep, fatigue, heart palpitations, headache, nausea, dizziness, memory impairment, prickling and burning sensations, along with skin rashes. They’ve even been blamed for depression, anxiety, colds, digestive disorders, and chronic pain. It’s called electromagnetic hypersensitivity or EHS.
Hall concludes
The symptoms described by “electromagnetic hypersensitivity” sufferers can be severe and are sometimes disabling. However, it has proved difficult to show under blind conditions that exposure to EMF can trigger these symptoms. This suggests that “electromagnetic hypersensitivity” is unrelated to the presence of EMF.
I recommend you read the entire article.

IPMB cited a point-counterpoint article that suggests cell phones are dangerous. True?

Point-counterpoint articles appear every month in the journal Medical Physics. They are a wonderful teaching tool, allowing students to consider and discuss questions at the cutting edge of medical physics. The one cited in Chapter 9 of IPMB is by Khurana, Moulder, and Orton (2008), titled “There is Currently Enough Evidence and Technology Available to Warrant Taking Immediate Steps to Reduce Exposure of Consumers to Cell-Phone-Related Electromagnetic Radiation.” Every point-counterpoint article pits one researcher against another, arguing opposing sides of the claim made in the article title. In this case, Vini Khurana supports the proposition, and John Moulder opposes it; Colin Orton is the moderator. I encourage you to read the article for yourself. I agree with Moulder’s conclusion that
weak epidemiological evidence of an association of mobile phone use with brain cancer incidence, when combined with the biophysical implausibility of a causal link and the strongly unsupportive animal studies, does not support the case that regulation of mobile phone use is urgently needed.

Last year I saw an article that says 5G cell phone radiation is unsafe! What’s up?

The Nation published an article titled “How Big Wireless Made Us Think That Cell Phones Are Safe: A Special Investigation. The Disinformation Campaign—and Massive Radiation Increase—Behind the 5G Rollout,” by Mark Hertsgaard and Mark Dowie. David Gorski published a critique of this article for the website Science-Based Medicine. He writes
The Nation indulges in fear mongering about cell phones and cancer An article published last week in The Nation likens wireless telephone companies to tobacco and fossil fuel episodes in their tactics of spreading fear, misinformation, and doubt regarding the science of cell phone radiation and health. To produce this narrative, the investigation’s authors rely on unreliable sources and cherry pick scientific studies, ignoring the scientific consensus that cell phone radiation almost certainly doesn’t cause cancer, all the while disingenuously claiming that they aren’t taking a position on the health effects of radio waves.
Read The Nation article and the critique and decide for yourself. At the least, you’ll learn how physics can be applied to medicine and biology.

What’s the bottom line regarding the risk of cancer from 5G cell phones?

Electromagnetic radiation consists of packets of energy called photons. The energy of a photon increases with the frequency of the radiation. Cancer is caused when DNA is damaged by very-high-frequency photons, such as x-rays (ionizing radiation). If the frequency is in the range of 300 GHz, the energy of a photon is far too small to disrupt bonds in DNA. It is, in fact, smaller than the energies associated with thermal motion of molecules. So, photons associated with 300 GHz radiation cannot cause cancer by damaging DNA. Of course, you could have a whole lot of 300 GHz photons, and they might pool their effort and together have enough energy to break bonds. We have a word for that: heat. 300 GHz radiation can heat tissue, but such heating is well understood and easily measured; Cell phone radiation is too weak to cause a significant temperature increase. So, we are left with no plausible mechanism for health risks from cell phone radiation. Perhaps some secondary effect could make your body less able to fight off cancer once it is induced by other mechanisms, but no one really knows how that might occur. Moreover, the epidemiological evidence suggests there is little risk. Cell phone use has increased dramatically since the turn of the century, but the incidence of brain cancer hasn’t increased. The National Cancer Institute says “The only consistently recognized biological effect of radiofrequency radiation in humans is heating... There are no other clearly established effects on the human body from radiofrequency radiation.” I wouldn’t say it’s impossible that cell phones put you at risk for cancer, but it’s unlikely. In my opinion, it’s exceedingly unlikely. We have many other things to worry about instead.


Foster KR, Moulder JE (2013) “Wi-Fi and Health: Review of Current Status of Research,” Health Physics, Volume 105, Pages 561-575.

Foster KR, Moulder JE (2018) “Response to Pall, ‘Wi-Fi is an Important Threat to Human Health’,Environmental Research, Volume 445-447, Pages 445-447.

Khurana VG, Moulder JE, Orton CG (2008) “There is Currently Enough Evidence and Technology Available to Warrant Taking Immediate Steps to Reduce Exposure of Consumers to Cell-Phone-Related Electromagnetic Radiation,” Medical Physics, Volume 35, Pages 5203-5206.

Moulder JE, Foster KR, Erdreich LS, McNamee JP (2005) “Mobile Phones, Mobile Phone Base Stations and Cancer: A Review,” International Journal of Radiation Biology, Volume 81, Pages 189-203.

Pall ML (2018) “Wi-fi is an Important Threat to Human Health,” Environmental Research, Volume 164, Pages 405-416.

Sheppard AR, Swicord ML, Balzano Q (2008) “Quantitative evaluation of mechanisms of radiofrequency interactions with biological molecules and processes,” Health Physics, Volume 95, Pages 365-396.

Vecchia P, Matthes R, Ziegelberger G, Lin J, Saunders R, Swerdlow A (2009) “Exposure to High Frequency Electromagnetic Fields, Biological Effects and Health Consequences (100 kHz – 300 GHz),” Munich: International Commission on Non-ionizing Radiation Protection.

Vila, J, Turner MC, Gracia-Lavedan E, Figuerola J, Bowman JD, Kincl L, Richardson L, Benke G, Hours M, Krewski D, McLean D, Parent M-E, Sadetzki S, Schlaefer K, Schlehofer B, Schuz J, Siemiatycki J, Tongeren M, Cardis, E (2018) Occupational exposure to high-frequency electromagnetic fields and brain tumor risk in the INTEROCC study: An individualized assessment approach. Environment International, Volume 119, Pages 353-365.

Friday, November 1, 2019

Perrin, Einstein, and Avogadro's Number

Brownian Movement and Molecular Reality,  by Jean Perrin (1910),  translated by Frederick Soddy, superimposed on Intermediate Physics for Medicine and Biology.
Brownian Movement and Molecular Reality,
by Jean Perrin (1910),
translated by Frederick Soddy.
Chapter 4 of Intermediate Physics for Medicine and Biology includes a homework exercise (Problem 12) about Jean Perrin’s experiment to determine Avogadro’s number. Perrin measured the equilibrium distribution of small particles suspended in water as a function of height, fit his data to a Boltzmann factor to determine the Boltzmann constant, and then calculated Avogadro’s number via the gas constant. I like that homework problem because it combines a mini history lesson with a physics exercise, and the numbers aren’t made up; they came from Perrin’s book Brownian Movement and Molecular Reality.

Perrin didn’t use just one method to determine Avogadro’s number; he used several. Below I present a new homework problem describing another technique of Perrin’s. Again I draw data from his book.
Section 4.6

Problem 12 ½. Jean Perrin used a relationship between diffusion and viscosity to determine Avogadro’s number. He recorded the variance of the displacement, σ2, as a function of time, t, for small particles suspended in water. The particles had a radius, a, of 0.212 μm, and the viscosity of water, η, was 0.0012 N s/m2 at a temperature, T, of 17 °C.

(a) Use the data below and Eq. 4.77 to estimate the diffusion constant, D, of the particles.
         t  (s)    σ2  (μm2)
          30       45
          60       86.5
          90     140
        120     195
Either use the least squares method of Sec. 11.1 to fit the data, or estimate an average value of D by trial and error.
(b) Use the Einstein relationship, Eq. 4.23, to determine Boltzmann’s constant, kB, from the diffusion constant found in part (a).

(c) Use your result from part (b), along with the gas constant, R, and Eq. 3.31, to calculate Avogadro’s number, NA. Your result may not be the same as the currently accepted value of NA, but it should be close.
For those of you who don’t have your copy of IPMB at your side, Eq. 4.77 is
Eq. 4.23 is
and Eq. 3.31 is

‘Subtle is the Lord...’:  The Science and Life  of Albert Einstein,  by Abraham Pais, superimposed on Intermediate Physics for Medicine and Biology.
‘Subtle is the Lord...’:
The Science and Life
of Albert Einstein,
by Abraham Pais.
Although Perrin was the first to perform this experiment, Albert Einstein initially proposed the idea during his miraculous year, 1905. The story behind this method can be found in Abraham Pais’s magnificent biography ‘Subtle is the Lord…’. Pais writes
One never ceases to experience surprise at this result, which seems, as it were, to come out of nowhere: prepare a set of small spheres which are nevertheless huge compared with simple molecules, use a stopwatch and a microscope, and find Avogadro’s number.
During the first decade of the twentieth century, the research by Perrin and Einstein confirmed the existence of atoms.

I’ll give Perrin the last word by quoting from the conclusion of Brownian Movement and Molecular Reality.
I think it is impossible that a mind, free from all preconception, can reflect upon the extreme diversity of phenomena which thus converge to the same result, without experiencing a very strong impression, and I think that it will henceforth be difficult to defend by rational arguments a hostile attitude to molecular hypotheses.

Friday, October 25, 2019

One Hundred Books About Physics for Medicine and Biology

When I was in high school, I became intrigued by St. John’s College and their Great Books program. I had their brochure, which included a list of the books to read each year; the most famous works of western civilization.

In the spirit of St. John’s, below I list one hundred Great Books about physics applied to medicine and biology. Read all these and you will have obtained a liberal education in biological and medical physics. One book you won’t find on this list is Intermediate Physics for Medicine and Biology. I’m going to assume you’ve already read IPMB and my goal is to suggest books to supplement it.

Where to begin? I’ll assume you have taken a year of physics and a year of calculus. Once you have these prerequisites, start reading.
  1. Powers of Ten. First an overview that’s easy and fun. It provides an intuitive feel for the relative sizes of things. 
  2. The Machinery of Life. Although I’m assuming you’ve studied some physics and math, I’m not assuming you have much background in biology. This book provides a gentle introduction to biochemistry. Plus, it has those wonderful drawings by David Goodsell
  3. The Art of Insight in Science and Engineering. Remember: We seek insight, not just facts.
  4. Physical Models of Living Systems. IPMB is about modeling in medicine and biology. Philip Nelson’s little book gets us started building models. 
  5. The Feynman Lectures on Physics. I know, I’ve already studied introductory physics, but The Feynman Lectures are special. You don’t want to miss them, and they contain some biology too.
  6. Air and Water. Now we get to our main topic: physics applied to biology. Mark Denny’s book covers many topics found in the first half of IPMB.
  7. Physics with Illustrative Examples from Medicine and Biology. This classic three-volume set covers much of the same ground as IPMB.
  8. The Double Helix. To further strengthen your background in biology, read James Watson’s first-person account of how he and Francis Crick discovered the structure of DNA. It’s a required text for any student of science, and is an easy read.
  9. The Eighth Day of Creation: The Makers of the Revolution in Biology. After warming up with The Double Helix, it’s time to dig deeper into the history and ideas of modern biology. Physicists play a large role in this book, and it’s wonderfully written.
  10. Biomechanics of Human Motion. Chapter 1 in IPMB covers statics applied to the bones and muscles of the body. It’s our first book that focuses in detail on a specific topic.
  11. Structures, or Why Things Don’t Fall Down. A delightful book about mechanics, including some biomechanical examples. It’s one of the most enjoyable books on this list. Don’t miss the sequel, The New Science of Strong Materials.
  12. Biomechanics: Mechanical Properties of Living Tissue. We need a book about biomechanics that treats tissue as a continuous medium. YC Fung’s textbook fills that niche.
  13. A Treatise on the Mathematical Theory of Elasticity. This book is long and technical, and may contain more material than you really need to know. Nevertheless, it’s a great place to learn elasticity. I’m sure there are more modern books that you may prefer. Skip if you’re in a hurry.
  14. The Physics of Scuba Diving. An easy read about how hydrostatics impacts divers.
  15. Life in Moving Fluids. Fluid dynamics is one of those topics that’s critical to life, but is often skipped in introductory physics classes. This book by Steven Vogel provides an excellent introduction to the field of biological fluid dynamics.
  16. Vital Circuits. Another book by Vogel, which focuses on the fluid dynamics of the circulatory system. 
  17. Boundary Layer Theory. This large tome may be too advanced for the list, but I learned a lot from it. Skip if you need to move along quickly.
  18. Textbook of Medical Physiology. We need to get serious about learning physiology. This classic text is by Arthur Guyton, but any good physiology textbook will do. Not much physics here. The book contains more biology than we need, but physiology is too important to skip.
  19. e, The Story of a Number. A gentle introduction into calculus and differential equations, and a great history of the exponential function, the topic of IPMB’s second chapter.
  20. Quick Calculus. Yes, you already studied calculus. But we are about to get more mathematical, and this book will help you brush up on math you may have forgotten. If you don’t need it, move on. 
  21. Used Math. Finish your math review with this outline of mathematics essential for college physics.
  22. The Essential Exponential. It’s time to focus specifically on the exponential function and its properties, so important in biology and medicine.
  23. A Change of Heart. Chapter 2 of IPMB mentions the Framingham heart study. Read the story behind the project.
  24. On Being the Right Size. This is really an essay, but indulge me while I include it here among the books. J. B. S. Haldane is too fascinating of a writer to miss.
  25. Scaling: Why is Animal Size so Important? Knut Schmidt-Nielsen’s study of scaling, a key topic in Chapter 2 of IPMB.
  26. Lady Luck. Chapter 3 of IPMB requires us to know some probability, and Warren Weaver’s book is an engaging introduction.
  27. Statistical Physics. The first few sections of Chapter 3 in IPMB develop the ideas of statistical physics in a way reminiscent of Frederick Reif’s volume in the Berkeley Physics Course.
  28. An Introduction to Thermal Physics. For those who want a more traditional approach to thermodynamics, I recommend Daniel Schroeder’s textbook.
  29. Lehninger Principles of Biochemistry. Biological thermodynamics overlaps with biochemistry. Any good biochemistry book will do. They all contain more detail than you need, but a biological physicist must know some biochemistry.
  30. The Second Law. This delightful book by Peter Atkins will fill a hole in IPMB: a penetrating discussion about the second law of thermodynamics.
  31. Div Grad Curl and All That. Chapter 4 of IPMB uses vector calculus, and there is no better introduction to the topic.
  32. Random Walks in Biology. Howard Berg’s wonderful little book about diffusion.
  33. The Mathematics of Diffusion. John Crank’s intimidating giant tome about diffusion. Mathephobes shouldn’t bother with it; Mathephiles shouldn’t miss it.
  34. Conduction of Heat in Solids. Like the book by Crank, this ponderous textbook by Horatio Carslaw and John Jaeger presents all you ever want to know about solving the heat equation (also known as the diffusion equation).
  35. How Animals Work. Another delightful book by Schmidt-Nielsen that considers comparative physiology, and topics in Chapter 5 of IPMB such as countercurrent heat exchange.
  36. The Nuts and Bolts of Life. A colorful book about the first dialysis machine.
  37. The Biomedical Engineering Handbook. Don’t read this encyclopedia-like multi-volume handbook in one sitting. Yet it provides dozens of examples of how physics is applied to medicine. Ask your library to buy this set and the next one.
  38. Encyclopedia of Medical Devices and Instrumentation. The title should be Case Studies: How Physics is Applied to Medicine.
  39. Plant Physics. IPMB doesn’t say much about plants, but physics impacts botany as well as zoology.
  40. Nerve, Muscle, and Synapse. Bernard Katz’s excellent, if somewhat dated, introduction to all the electrophysiology you need for Chapter 6 of IPMB.
  41. The Conduction of the Nervous Impulse. Read about the Hodgkin-Huxley model from the pen of Alan Hodgkin himself.
  42. From Neuron to Brain. A modern introduction to neuroscience.
  43. Electricity and Magnetism. This book by Ed Purcell is part of the Berkeley Physics Course. The first of three physics books about electricity and magnetism.
  44. Introduction to Electrodyamics. David Griffiths’s text competes with Purcell’s for my favorite electricity and magnetism book.
  45. Classical Electrodynamics. John David Jackson’s famous graduate-level physics text may be more electricity and magnetism than you want, but how could I leave it off the list?
  46. Galvani’s Spark. A history of neurophysiology.
  47. Shattered Nerves. A fascinating look at using electrical stimulation to compensate for neural injury. A history of neural prostheses.
  48. Bioelectricity: A Quantitative Approach. The first, and probably easiest, of three bioelectricity textbooks.
  49. Bioelectromagnetism. Jaakko Malmivuo and Robert Plonsey’s big book about bioelectricity.
  50. Bioelectricity and Biomagnetism. Another big tome. Ramesh Gulrajani’s alternative to Malmivuo and Plonsey.
  51. The Art of Electronics. In order to understand voltage clamping and other electrophysiological methods, you need to know some electronics. This book is my favorite introduction to the topic. 
  52. Mathematical Handbook of Formulas and Tables. Chapter 6 contains lots of mathematics, and the next three books are references you may want. This Schaum’s Outline contains most of the math you’ll ever need. It’s cheap, light, and easy to use. Keep it handy.
  53. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. No one would sit down and read this handbook straight through, but “Abramowitz and Stegun” is invaluable as a reference.
  54. Table of Integrals, Series, and Products. “Gradshteyn and Ryzhik” is the best integral table ever. Let the library buy it, but have them to keep it in the reference section so you can find it quickly. 
  55. Numerical Recipes. If you want to solve the equations of the Hodgkin-Huxley model, you need to program a computer. This book is great for finding the needed numerical methods.
  56. Numerical Methods that Work. Forman Acton’s book is more chatty than Numerical Recipes, but full of insight.
  57. Machines in our Hearts. Chapter 7 of IPMB examines the heart. Read this history of pacemakers and defibrillators to put it all in perspective.
  58. Cardiac Electrophysiology: From Cell to Bedside. This multi-author, multi-edition work contains everything you always wanted to know about the electrical properties of the heart, but were afraid to ask.
  59. Cardiac Bioelectric Therapy. Another multi-author collection, with several excellent chapters about the bidomain model.
  60. When Time Breaks Down. Art Winfree’s unique analysis of the electrical properties of the heart.
  61. Electric Fields of the Brain. Paul Nunez’s book about the electroencephalogram from the perspective of a physicist.
  62. Iron, Nature’s Universal Element. Why people need iron and animals make magnets.
  63. The Spark of Life. An accessible introduction to electrophysiology and ion channel diseases.
  64. Ion Channels of Excitable Membranes. The definitive textbook about ion channels, by Bertil Hille.
  65. Voodoo Science. Some of the topics in Section 9.10 of IPMB about possible effects of weak electric and magnetic fields make me yearn for this hard-hitting book by Bob Park.
  66. Dynamics: The Geometry of Behavior. Chapter 10 of IPMB covers nonlinear dynamics. This beautiful book introduces dynamics using pictures.
  67. From Clocks to Chaos. Leon Glass and Michael Mackey introduce the idea of a dynamical disease.
  68. Nonlinear Dynamics and Chaos. Steven Strogatz’s classic; my favorite book about nonlinear dynamics.
  69. Mathematical Physiology. An award-winning textbook about applying math to biology.
  70. Mathematical Biology. Another big fine textbook for the mathematically inclined.
  71. The Geometry of Biological Time. A quirky book by Art Winfree, more wide-ranging than When Time Breaks Down.
  72. Data Reduction and Error Analysis for the Physical Sciences. Many of the ideas about least squares fitting discussed in Chapter 11 of IPMB are related to analyzing noisy data.
  73. The Fourier Transform and its Applications. The Fourier transform is the most important concept in Chapter 11. Ronald Bracewell’s book is a great place to learn about it.
  74. Introduction to Membrane Noise. Louis DeFelice’s book explains how to deal with noise.
  75. Naked to the Bone. A historical survey of medical imaging.
  76. Medical Imaging Physics. A book by William Hendee and E Russell Ritenour, at a level similar to IPMB but dedicated entirely to imaging. Also see its partner, Hendee's Radiation Therapy Physics.
  77. Foundations of Medical Imaging. A big, technical book about imaging.
  78. Theoretical Acoustics. Not much biology here, but a definitive survey of acoustics to back up Chapter 13 of IPMB.
  79. Physics of the Body. This book discusses many topics, including hearing.
  80. Musicophilia. An extraordinary book by Oliver Sacks about the neuroscience of hearing.
  81. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. My choice for a modern physics textbook, with much information about the interaction of light with matter.
  82. The First Steps in Seeing. Robert Rodieck’s incredible book about the physics of vision.
  83. The Optics of Life. This masterpiece by Sonke Johnsen walks you through optics, examining all the biological applications. A great supplement to Chapter 14 of IPMB.
  84. From Photon to Neuron. A study of light, imaging, and vision.
  85. Introduction to Physics in Modern Medicine. Suzanne Amador Kane’s nice introduction to physics applied to medicine, covering many topics in the last half of IPMB.
  86. Introduction to Radiological Physics and Radiation Dosimetry. Frank Herbert Attix wrote the definitive textbook about how x-rays interact with tissue, a topic covered in Chapter 15 of IPMB.
  87. Radiobiology for the Radiologist. The go-to reference for how cells and tissues respond to radiation.
  88. Molecular Biology of the Cell. The classic textbook of cell biology.
  89. Radiation Oncology: A Physicists Eye View. Explains how to treat cancer using radiation.
  90. The Physics of Radiation Therapy. Faiz Khan’s in-depth study of radiation therapy.
  91. The Atomic Nucleus. An classic about nuclear physics, providing background for Chapter 17 of IPMB. You could replace it with one of many modern nuclear physics textbooks.
  92. The Immortal Life of Henrietta Lacks. A fascinating study of how a women treated for cancer using radioactivity ended up providing science with an immortal cell line.
  93. Strange Glow. How radiation impacts society.
  94. The Radium Girls. This book is about women poisoned by radium-containing paint (lip, dip, paint). It reminds us why we study medical physics.
  95. Magnetic Resonance Imaging: Physical Properties and Sequence Design. All you need to know about MRI.
  96. Principles of Nuclear Magnetic Resonance Microscopy. Paul Callaghan’s view of magnetic resonance imaging.
  97. Echo Planar Imaging. Advanced MRI techniques.
  98. Biological Physics. IPMB is not strong in covering physics applied to cellular and molecular biology. Here are three great books to fill that gap.
  99. Cell Biology by the Numbers. I love the quantitative approach to biology.
  100. Physical Biology of the Cell. How physicists view biology. 
Don’t see your favorite listed? Here’s my call to action: Add your recommendations to the comments below.

I didn’t end up going to St. John’s College and studying the Great Books. Instead, I attended a more traditional school, the University of Kansas. I loved KU, and I have no regrets. But sometimes I wonder...

Friday, October 18, 2019

Entry Region and Deviations from Poiseuille Flow

Chapter 1 of Intermediate Physics for Medicine and Biology analyzes viscous flow in a tube, also known as Poiseuille flow. Russ Hobbie and I derive the well-known parabolic distribution of speed: motionless at the wall—the no-slip boundary condition—and maximum at the center. In Section 1.20 we consider departures from the parabolic flow profile.
The entry region causes deviations from Poiseuille flow in larger vessels. Suppose that blood flowing with a nearly flat velocity profile enters a vessel, as might happen when blood flowing in a large vessel enters the vessel of interest, which has a smaller radius. At the wall of the smaller vessel, the flow is zero. Since the blood is incompressible, the average velocity is the same at all values of x, the distance along the vessel. (We assume the vessel has a constant cross-sectional area.) However, the velocity profile v(r) changes with distance x along the vessel. At the entrance to the vessel (x = 0), there is a very abrupt velocity change near the walls. As x increases, a parabolic velocity profile is attained. The transition or entry region is shown in Fig. 1.35. In the entry region, the pressure gradient is different from the value for Poiseuille flow. The velocity profile cannot be calculated analytically in the entry region. Various numerical calculations have been made, and the results can be expressed in terms of scaled variables (see, for example, Cebeci and Bradshaw 1977). The Reynolds number used in these calculations was based on the diameter of the pipe, D = 2Rp, and the average velocity. The length of the entry region is

L = 0.05DNR,D = 0.1RpNR,D = 0.2RpNR,Rp.       (1.63)
IPMB’s Figure 1.35 is shown below.

Figure 1.35 from Intermediate Physics for Medicine and Biology.

This figure first appeared in the 3rd edition of IPMB, for which Russ was sole author. I got to wondering how he created it.

Momentum Transfer in Boundary Layers, by Tuncer Cebeci and Peter Bradshaw, superimposed on Intermediate Physics for Medicine and Biology.
Momentum Transfer in
Boundary Layers
by Cebeci and Bradshaw.
I checked out the book Momentum Transfer in Boundary Layers, by Cebeci and Bradshaw (1977), through interlibrary loan and found the part about entrance-region flow in their Section 5.7.1. They write
Figures 5.9 and 5.10 show the velocity profiles, u/u0, and the dimensionless centerline (maximum) velocity, uc/u0, as functions of 2x*/Rd in the entrance region of a pipe. Here x* = x/r0 and Rd = u0d/ν. Figure 5.10 also shows the measured values of the centerline velocity obtained by Pfenninger (1951). According to the results of Fig. 5.10, the centerline velocity has almost reached its asymptotic value of 2 at 2x*/Rd = 0.20. Thus the entrance length for a laminar flow in a circular pipe is

le/d = Rd/20       (5.7.12)
Their Fig. 5.9 is

A photograph of Figure 5.9 from Momentum Transfer in Boundary Layers, by Tuncer Cebeci and Peter Bradshaw (1977).
A photograph of Figure 5.9 from Momentum Transfer
in Boundary Layers
, by Cebeci and Bradshaw (1977).
I believe this is the figure Russ used to create his drawing. Clever guy that he is, he seems to have taken the traces, rotated them by 90°, and reflected them across the centerline so they extend from the upper to lower wall. The variable r in their Fig. 5.9 is the distance from the wall, and r0 is the radius of the vessel, so r/r0 = 1 at the center of the vessel and r/r0 = 0 at the wall (don’t ask me why they defined r in this odd way); x* is x/r0, or the distance along the length of the vessel in terms of the vessel radius; ν is the kinematic viscosity, equal to the coefficient of viscosity η divided by the density ρ; and finally Rd is the Reynolds number, defined using the vessel diameter, which Russ and I call NR,D.

At the vessel entrance (x* = 0), the flow is uniform across its cross section with speed u0. The curve corresponding to 2x*/Rd = 0.531 looks almost exactly like Poiseuille flow, with the shape of a parabola and a maximum speed equal to 2u0. For the curve 2x*/Rd = 0.101 the flow is close to, but not exactly, Poiseuille flow. Cebeci and Bradshaw somewhat arbitrarily define the entrance length as corresponding to 2x*/Rd = 0.2.

The example that Russ analyzed in the caption to Fig. 1.35 corresponds to a large vessel such as the brachial artery in your upper arm. A diameter of 4 mm and an entrance length of 240 mm implies a Reynolds number of Rd = 1200. In this case, the entrance length is much greater than the diameter, and is similar to the length of the vessel. If we consider a small vessel like a capillary, however, we get a different picture. A typical Reynolds number for a capillary would be du0ρ/η = (8 × 10-6 m)(0.001 m/s)(1000 kg/m3)/(3 × 10-3 kg/m/s) = 0.0027, which implies an entrance length of about one nanometer. In other words, the parabolic flow distribution is established almost immediately, over a distance much smaller than the vessel’s length and even its radius. The entrance length is negligible in small vessels like capillaries.

Tuncer Cebeci is a Turkish-American mechanical engineer who worked for the Douglas Aircraft Company and was chair of the Department of Aerospace Engineering at California State University Long Beach. He has authored many textbooks in aeronautics, and developed the Cebeci-Smith model used in computational fluid dynamics. Peter Bradshaw was a professor in the Department of Aeronautics at the Imperial College London and then at Stanford, and is a fellow of the Royal Society. He developed the “Bradshaw Blower,” a type of wind tunnel use to study turbulent boundary layers.

Cebeci and Bradshaw describe why they wrote Momentum Transfer in Boundary Layers in their preface.
This book is intended as an introduction to fluid flows controlled by viscous or turbulent stress gradients and to modern methods of calculating these gradients. It is nominally self-contained, but the explanations of basic concepts are intended as review for senior students who have already taken an introductory course in fluid dynamics, rather than for beginning students. Nearly all stress-dominated flows are shear layers, the most common being the boundary layer on a solid surface. Jets, wakes, and duct flows are also shear layers and are discussed in this volume. Nearly all modern methods of calculating shear layers require the use of digital computers. Computer-based methods, involving calculations beyond the reach of electomechanical desk calculators, began to appear around 10 years ago... With the exception of one or two specialist books... this revolution has not been noticed in academic textbooks, although the new methods are widely used by engineers.
This post illustrates how IPMB merely scratches the surface when explaining how physics impacts medicine and biology. Behind each sentence, each figure, and each reference is a story. I wish Russ and I could tell them all.

Friday, October 11, 2019

A Blog as Ancillary Material for a Physics Textbook

Today I’m attending the Fall 2019 Meeting of the Ohio-Region Section of the American Physical Society and the Michigan Section of the American Association of Physics Teachers, held at Kettering University in Flint, Michigan. Flint is just 45 minutes north of Oakland University, so this is a local meeting for me.

At the meeting I’ll present a poster titled “A Blog as Ancillary Material for a Physics Textbook.” As you can probably guess, the blog I’m referring to is the one you’re reading now. My poster is shown below.

My poster for the Fall 2019 Meeting of the Ohio-Region Section of the American Physical Society
and the Michigan Section of the American Association of Physics Teachers
The poster begins with my meeting abstract.
Nowadays, textbooks come with many ancillary materials: solution manuals, student guides, etc. A unique ancillary feature is a blog. A blog allows you to keep your book up-to-date, to expand on ideas covered only briefly in your book, to point to other useful learning materials such as websites, articles and other books, and to interact directly with students using your book.
Then I address the question “Why write a blog associated with a textbook?” My reasons are to
  • Keep your book up-to-date. 
  • Present background material. 
  • Offer links to related websites, videos, and other books. 
  • Try out new material for future editions. 
  • Provide a direct line of communication between you and your readers. 
  • Reach out to students from other states and countries who are interested in your topic but don’t have your book (yet). 
  • Have fun. 
  • Increase book sales!
Next I discuss the blog for IPMB.
I am coauthor with Russell Hobbie of the textbook Intermediate Physics for Medicine and Biology (5th edition, Springer, 2015) My blog can be found at The blog began in 2007. I post once a week, every Friday morning, with over 600 posts so far. I also share the weekly posts on the book’s Facebook page. I use the blogger software, which is free and easy to learn;
After that, I describe my different types of posts.
  • Useful for Instructors: Posts that will be especially helpful to faculty teaching from your book, such as sample syllabi, information about prerequisites, and links. 
  • Book Reviews: Reviews of books that are related to mine. 
  • Obituaries: Stories of famous scientists who have died recently. 
  • New Homework Problems: I often post new homework problems that instructors can use in class or on exams. 
  • My Own Research: Stories from my own research, to serve as examples of how to apply the material in the textbook. 
  • Lots of Math: Some of my posts are very mathematical, and I warn the reader. 
  • Personal Favorites: About 10% of my posts I list as personal favorites. These are particularly interesting, especially well written, or sometimes autobiographical.
Finally, I provide a sample post. I chose one of my favorites about craps, published on August 10, 2018.

A big thank you to my graduate student Dilmini Wijesinghe, who helped me design the poster. She’ll be at the meeting too, presenting another poster about biomechanics and mechanotransduction. But that’s another story.

Friday, October 4, 2019

Spiral MRI

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss a type of magnetic resonance imaging called echo-planar imaging.
In EPI the echoes are not created using π pulses. Instead, they are created by dephasing the spins at different positions along the x axis using a Gx gradient, and then reversing that gradient to rephase the spins, as shown in Fig. 18.32. Whenever the integral of Gx(t) is zero, the spins are all in phase and the signal appears. A large negative Gy pulse sets the initial value of ky to be negative; small positive Gy pulses (“blips”) then increase the value of ky for each successive kx readout. Echo-planar imaging requires strong gradients—at least five times those for normal studies—so that the data can be acquired quickly. Moreover, the rise and fall-times of these pulses are short, which induces large voltages in the coils. Eddy currents are also induced in the patient, and it is necessary to keep these below the threshold for neural activation. These problems can be reduced by using sinusoidally-varying gradient currents. The engineering problems are discussed in Schmitt et al. (1998); in Vlaardingerbroek and den Boer (2004); and in Bernstein et al. (2004).
Echo-Planar Imaging: Theory, Technique and Application, edited by Schmitt, Stehling, and Turner, superimposed on Intermediate Physics for Medicine and Biology.
Echo-Planar Imaging:
Theory, Technique and Application
edited by Schmitt, Stehling, and Turner.
To learn more about “sinusoidally-varying gradient currents,” I consulted the first of the three references, Echo-Planar Imaging: Theory, Technique and Application, edited by Franz Schmitt, Michael Stehling, and Robert Turner (Springer, 1998). In his chapter on the “Theory of Echo-Planar Imaging,” Mark Cohen discusses a spiral echo-planar pulse sequence in which the gradient fields have the unusual form Gx = Go t sin(ωt) and Gy = Go t cos(ωt).

Below I show the pulse sequence, which you can compare with the echo-planar imaging sequence in Fig. 18.32 of IPMB if you have the book by your side (don’t you always keep IPMB by your side?). The top two curves are the conventional slice selection sequence: a gradient Gz (red) in the z direction is applied during a radiofrequency π/2 pulse Bx (black), which rotates the spins into the x-y plane. The unconventional readout gradient Gx (blue) varies as an increasing sine wave. It produces a gradient echo at times corresponding approximately to the extrema of the Gx curve (excluding the first small positive peak). The phase encoding gradient Gy (green), an increasing cosine wave, is approximately zero at the echo times, but will shift the phase and therefore impact the amplitude of the echo.
A pulse sequence for spiral echo-planar imaging, based on Fig. 14 of “Theory of Echo-Planar Imaging,” by Mark Cohen in Echo-Planar Imaging: Theory, Technique and Application, edited by Schmitt, Stehling, and Turner.
A pulse sequence for spiral echo-planar imaging,
based on Fig. 14 of “Theory of Echo-Planar Imaging,”
by Mark Cohen.

If you look at the output in terms of spatial frequencies (kx, ky), you find that the echos correspond to points along an Archimedean spiral.

The spiral echo-planar imaging technique as viewed in frequency space, based on Fig. 13 of “Theory of Echo-Planar Imaging,” by Mark Cohen, in Echo-Planar Imaging: Theory, Technique and Application, edited by Schmitt, Stehling, and Turner.
The spiral echo-planar imaging technique as viewed in frequency space,
based on Fig. 13 of “Theory of Echo-Planar Imaging,” by Mark Cohen.

Spiral echo-planar imaging has some drawbacks. Data in k-space is not collected over a uniform array, so you need to interpolate onto a square grid before performing a numerical two-dimensional inverse Fourier transform to produce the image. Moreover, you get blurring from chemical shift and susceptibility artifacts. The good news is that you eliminate the rapid turning on and off of gradient pulses, which reduces eddy currents that can cause their own image distortions and possibly neural stimulation. So, spiral imaging has advantages, but the pulse sequence sure looks weird.

Echo-planar imaging in general, and spiral imaging in particular, are very fast. In his chapter on “Spiral Echo-Planar Imaging,” Craig Meyer discusses his philosophy about using EPI.
Spiral scanning is a promising alternative to traditional EPI. The properties of spiral scanning stem from the circularly symmetric nature of the technique. Among the attractive properties of spiral scanning are its efficiency and its good behavior in the presence of flowing material; the most unattractive property is uncorrected inhomogeneity leads to image blurring. Spiral image reconstruction can be performed rapidly using gridding, and there are a number of techniques for compensating for inhomogeneity. There are good techniques for generating efficient spiral gradient waveforms. Among the growing number of applications of spiral scanning are cardiac imaging, angiography, abdominal tumor imaging, functional imaging, and fluoroscopy.

Spiral scanning is a promising technique, but at the present it is still not in routine clinical use. There are many theoretical reasons why spiral scanning may be advantageous for a number of clinical problems, and initial volunteer and clinical studies have yielded very promising results for a number of applications. Still, until spiral scanning is established in routine clinical use, some caution is warranted about proclaiming it to be the answer for any particular question.

Friday, September 27, 2019

The Cauchy Distribution

In an appendix of Intermediate Physics for Medicine and Biology, Russ Hobbie and I analyze the Gaussian probability distribution
An equation for the Gaussian probability distribution.
It has the classic bell shape, centered at mean x with a width determined by the standard deviation σ.

Other distributions have a similar shape. One example is the Cauchy distribution
An equation for the Cauchy probability distribution.
where the distribution is centered at x and has a half-width at half-maximum γ. I initially thought the Cauchy distribution would be as well behaved as any other probability distribution, but it’s not. It has no mean and no standard deviation!

Rather than thinking abstractly about this issue, I prefer to calculate and watch how things fall apart. So, I wrote a simple computer program to generate N random samples using either the Gaussian or the Cauchy distribution. Below is a histogram for each case (N = 1000; Gaussian, x = 0, σ = 1; Cauchy, x = 0, γ = 1).

Histograms for 1000 random samples obtained using the Cauchy (left) and Gaussian (right) probability distribution.

Those samples out on the wings of the Cauchy distribution are what screw things up. The probability falls off so slowly that there is a significant chance of having a random sample that is huge. The histograms shown above are plotted from −20 to 20, but one of the thousand Cauchy samples was about −2400. I’d need to plot the histogram over a range more than one hundred times wider to capture that bin in the histogram. Seven of the samples had a magnitude over one hundred. By contrast, the largest sample from the Gaussian was about 4.6.

What do these few giant samples do to the mean? The average of the thousand samples shown above obtained from the Cauchy distribution is −1.28, which is bigger than the half-width at half-max. The average of the thousand samples obtained from the Gaussian distribution is −0.021, which is much smaller than the standard deviation.

Even more interesting is how the mean varies with N. I tried a bunch of cases, summarized in the figure below.
A plot of the mean versus sample size, for data drawb from the Gassian and Cauchy probability distribution.

There’s a lot of scatter, but the means for the Gaussian data appear to get smaller (closer to the expected value of zero) as N gets larger, The red line is not a fit, but merely drawn by eye. I included it to show how the means fall off with N. It has a slope of −½, implying that the means decay roughly as 1/√N. In contrast, the means for the Cauchy data are large (on the order of one) and don’t fall off with N. No matter how many samples you collect, your mean doesn’t approach the expected value of zero. Some oddball sample comes along and skews the average.

If you calculate the standard deviations for these cases, the problem is even worse. For data generated using the Cauchy distribution, the standard deviation grows with N. For N over a million, the standard deviation is usually over a thousand (remember, the half-width at half-max is one), and for my N = 5,000,000 case the standard deviation was over 600,000. Oddballs dominate the standard deviation.

I’m sorry if my seat-of-the-pants experimental approach to analyzing the Cauchy distribution seems simplistic, but for me it provides insight. The Cauchy distribution is weird, and I’m glad Russ and I didn’t include an appendix about it in Intermediate Physics for Medicine and Biology.