Anxiety and the Equation: Understanding Boltzmann's Entropy

by Eric Johnson

A man and his the anxiety-plagued nineteenth-century physicist who contributed significantly to our understanding of the second law of thermodynamics.Ludwig Boltzmann's grave in Vienna's Central Cemetery bears a cryptic S = k log W. This equation was Boltzmann's great discovery, and it contributed significantly to our understanding of the second law of thermodynamics. In Anxiety and the Equation, Eric Johnson tells the story of a man and his the anxiety-plagued nineteenth-century physicist who did his most important work as he struggled with mental illness.

Johnson explains that “S” in Boltzmann's equation refers to entropy, and that entropy is the central quantity in the second law of thermodynamics. The second law is always on, running in the background of our lives, providing a way to differentiate between past and future. We know that the future will be a state of higher entropy than the past, and we have Boltzmann to thank for discovering the equation that underlies that fundamental trend. Johnson, accessibly and engagingly, reassembles Boltzmann's equation from its various components and presents episodes from Boltzmann's life—beginning at the end, with “Boltzmann Kills Himself” and “Boltzmann Is Buried (Not Once, But Twice).” Johnson explains the second law in simple terms, introduces key concepts through thought experiments, and explores Boltzmann's work. He argues that Boltzmann, diagnosed by his contemporaries as neurasthenic, suffered from an anxiety disorder. He was, says Johnson, a man of reason who suffered from irrational concerns about his work, worrying especially about opposition from the scientific establishment of the day.

Johnson's clear and concise explanations will acquaint the nonspecialist reader with such seemingly esoteric concepts as microstates, macrostates, fluctuations, the distribution of energy, log functions, and equilibrium. He describes Boltzmann's relationships with other scientists, including Max Planck and Henri Poincaré, and, finally, imagines “an alternative ending,” in which Boltzmann lived on and died of natural causes.

Genres: Science, Physics, Nonfiction, History, Biography

173 pages, Kindle Edition

Published October 23, 2018

About the author

Eric Johnson teaches chemistry at Mount St. Joseph University. His most redeeming quality (albeit self-reported) is that he explains things clearly, which would be more impressive if he could explain a greater number of things. Two topics about which he can speak with some authority are entropy, which guides the fate of everything, and Ludwig Boltzmann, the physicist who discovered the reason why. These topics are explored in Johnson's first book: Anxiety and the Equation.

Photo courtesy of Mount St. Joseph University/Don Denney Photography.

Reviews

[John Mauro · Rating 5 out of 5]

Although the concept of atoms dates back to the ancient Greek philosophers, most notably Democritus (ca. 460-370 B.C.), the Austrian physicist Ludwig Boltzmann (1844-1906) was the first scientist to develop the physics connecting atomic theory and thermodynamics. In doing so, Boltzmann became the father of a new field of physics known as statistical mechanics, which explains the macroscopic properties of materials in terms of their underlying atomic structure or microstates.

The fundamental postulate of statistical mechanics is that the measurable properties of a system are a direct result of suitable averaging over its various microstates. Since the microstates are not necessarily known, they must be described in terms of their probabilities of occurrence. The macrostate of a system can then be determined based on this set of microstates and their respective probabilities.

Boltzmann’s theory of statistical mechanics faced stiff criticism and outright rejection from many prominent scientists in the late 19th century. At that time, the field of statistics was considered to be an “immoral” branch of mathematics, suitable only for gambling and not for legitimate science. Moreover, the atomic theory of matter itself was highly controversial. For example, in a famous exchange at the conclusion of one of Boltzmann’s lectures, the eminent physicist Ernst Mach rose from his seat and boldly declared, “I do not believe that atoms exist,” flatly rejecting Boltzmann’s entire life’s work.

Indeed, Boltzmann’s ideas proved to be a few decades ahead of their time. Unfortunately, he did not cope well with this rejection from his less enlightened contemporaries. Boltzmann took his own life on September 5, 1906, by hanging himself while on a vacation with his family near Trieste, in what is now northeastern Italy. Boltzmann’s tombstone in Vienna is engraved with his famous equation for entropy (see book cover).

The importance of Boltzmann’s contributions to science cannot be overstated. Boltzmann was the first scientist to establish a rigorous connection between the macroscopic properties of a material and its underlying atomic structure. In founding the field of statistical mechanics, Boltzmann was also the first physicist to understand the importance of probability theory and the role of uncertainty in the formulation of physics. Today, Boltzmann’s impact stretches far beyond statistical mechanics, as his entropy equation is also the basis for the field of information theory, pioneered by Shannon more than a half century later.

In "Anxiety and the Equation," Eric Johnson presents a very well-written biography of Ludwig Boltzmann, intertwined with lucid explanations of his fundamental concepts of entropy and statistical mechanics. This book is a must-read for statistical physicists and anyone interested in the history of science.

[Alexandru Tudorica · Rating 4 out of 5]

When dealing with anxiety isn't as easy as statistical thermodynamics.

[John · Rating 4 out of 5]

Short science books, starting with Galileo's Sidereus Nuncius, have often been our best books on technical topics. Brevity is not appreciated as much at should be particularly in academia. Anxiety and the Equation is a nice addition to the short canon. It contains one of the most lucid explanations of microstates and macrostates I've run across. This is entropy for fifth graders and that's a compliment.

[McKinley Paul · Rating 3 out of 5]

Pretty good. The biographical sections were so-so, focusing a bit too much on retroactive psychoanalysis for my taste. There were still interesting tidbits about Boltzmann’s life and significance, but were sometimes a little slow to get through. Luckily they are short sections alternating between biography and science. Fortunatrly, the sections explaining entropy were quite simple and lucid. Very good overall. Johnson does a great job relating microstates to macrostates and introducing the concept of multiplicity that is pretty much entropy. No one can deny that an intuitive, simple, and thorough explanation of S=klog(W) was given. This itself is interesting, takes up half the book and is worthwhile. However, more discussion of the second law, as well as the importance of entropy could have been given.

[Chanda Prescod-weinstein]

I’m reviewing this one for Physics World so I won’t rate it here.

[David Coorey · Rating 5 out of 5]

If you've ever found entropy to be a topic surrounded by mystery and confusion, then "Anxiety and the Equation" is the perfect antidote. Eric Johnson reduces the complexity down to a handful of easily understood and entertaining examples, encapsulated in short and sharp chapters. The maths presented is very layman-accessible, including a wonderful chapter explaining why entropy is defined by k*log(W), rather than just k*W. The book is also packed with historical facts about Boltzmann and his peers, some of which provide a tragic insight into his psyche.

In short, I found the book to be the clearest explanation I've read on an otherwise very confusing branch of science. In fact, the beauty of the book was to show that this topic need not be confusing after all :).

[Matthew Vallely · Rating 3 out of 5]

I enjoyed this book’s elementary explanations of Boltzmann’s life, personal relationships, and general demeanor. It was a challenge to see the truth that this man, although looked on as a hero in some regards, was more so the antagonist in his own life.

I also enjoyed the author’s surface level versions of micro states and macro states. The analogies used and the simple figures that fit onto one page are a good starting point at making a complex concept like entropy seem a lot more simple. The surface level explanation of entropy as a whole was underwhelming to me, as I was hoping for a more mathematical and technical approach. Still, the authors descriptions were a nice entry level into Boltzmann’s work for the average educated individual, but with my background and previous knowledge of these concepts, I finished the book still wanting a bit more. I did learn some neat facts about his life as well as tangential contributions from other contemporary physicists like Brown, Bohr, Mach, and Einstein and it offered a neat perspective how all of their timelines overlapped and built on each other.

[Mathew Mehrian · Rating 5 out of 5]

Great book. I really liked the biographical direction that was taken in this book. It is not a full biography of Boltzmann but more so about his mental state with respect to his work and colleagues interwoven with explanations of how to understand his equation. Great for those who are interested to in scientist bios and those interested in a better understanding of entropy both mathematically and conceptually regardless of math skills.

[Bookish Hedgehog · Rating 4 out of 5]

This is a sweet little book. Johnson approaches Boltzmann's life with the care and detail of a novelist, not a biographer. On the other hand, this brief touch with fiction allows him to use fancy thought experiments and metaphors to engage the reader.

I wish there was more elaboration on the applications of entropy and what the k means, but given the size of the book, it's definitely worth a read!

[Pavle · Rating 5 out of 5]

Read it because of a quote from a statistical mechanics textbook: "Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest [Boltzmann's student], carrying on the work, died similarly in 1933. Now it's our turn to study statistical mechanics.", the reality is more complicated. The best of pop-science, clear explanation of somewhat unintuitive science, and a good depiction of a complicated person.

[Joshua · Rating 4 out of 5]

I quite enjoyed this presentation of statistical mechanics and entropy. The historical accounting of Boltzmann’s life and the scientific climate at the birth of quantum physics was also well done. But I did not care for the numerous distracting (often self-deprecating, for some reason) interjections from the author. I also liked that it was a quick read.

[Hannah Kersemeier · Rating 4 out of 5]

Good book about science and mental health. Does a good job of breaking down complicated chemistry topics and talking about the anxiety that comes in academia. Don’t really love or agree with the authors philosophy at the end tho

[Peter Morfe · Rating 3 out of 5]

This is a short little book about Ludwig Boltzmann and the equation written on his grave stone. For me, it served as a good introduction to the story of Boltzmann's life. The physical/mathematical aspects were very brief for my taste and not completely convincing.

[Jacob · Rating 5 out of 5]

Well written, funny, and engaging while being extremely informative. Written for someone with a high school to post-doc grasp of mathematics and physics looking to fill in the history of Boltzmann and S = k log(W) in a light and relaxing read. Looking forward to rereading it regularly.

[Indudhar Andanur · Rating 4 out of 5]

If there is no table in the chapter, that’s most likely to be a section on biography. Skim those chapters/sections, what you are left with is a book highly readable on Entropy.

[William T · Rating 5 out of 5]

I am sure that there are a few things that readers could quibble about, but I had fun and found this a rewarding quick read.

[Roy Kenagy]

RK NOOK

[MS · Rating 5 out of 5]

5 entropic starzzz