It’s not a textbook, but there’s “something you can learn from it”
Ludwig Boltzmann was an excruciatingly anxious person and also one of the best scientific minds of his generation. Boltzmann’s revolutionary work on entropy paved the way for Einstein’s quantum revolution of the early 20th century, and yet he still spent hours, late into the night, cramming to teach the most basic introductory physics lectures.
In Anxiety and the Equation: Understanding Boltzmann’s Entropy, Eric Johnson offers no easy definitions. His book, delightfully, presents as many contradictions as it does answers. Half history, half science, the book’s form pushes against the too staid, too unwieldy genres of literary biography and popular physics: “This book is not a textbook,” Johnson writes in the introduction, “but there’s a chance that you learn something from it. It’s not really a biography, though it has a lot to say about a man named Ludwig Boltzmann.”
Was Boltzmann the stereotypical white-coated, absent-minded scientist: unkempt hair with a well-kept brain? Not quite: Though his mind was exceptional, it was disorganized; his writing was circuitous, untrimmed, and unfocused, like a “bulldozer.” And though Boltzmann’s beard was messy, most of his life was well in order—happy family, good quiet job, admiring students, a love of good beer.
That is, until a prestigious job offer came Boltzmann’s way. Then, “the most badly bungled job negotiation in this history of the academy,” full of unbearably awkward letters and a rescinded invitation to one of top universities in Europe, “forever changed his life.” Boltzmann quickly fell into a downward spiral of anxiety, until finally, at age 62, Boltzmann killed himself.
But to tell you that Boltzmann took his own life does not give away the end of the book, merely the book’s beginning. Johnson’s first chapter starts: “It was an inelegant death. Hanging there. He was a fat man.” Likewise, to tell you that Boltzmann invented the modern concept of entropy does not spoil the scientific revelation: Boltzmann’s famous entropic law, the second law of thermodynamics, S=klogW, adorns the book’s cover.
Johnson is not a psychologist or a historian—though he is a chemist—Johnson is first of all a teacher, an explainer. This is a book of revelation not through biographical plot but through the careful discovery of causes: How did Boltzmann go from a bumbling yet happy professor at an obscure Austrian university to that self-defeating spiral; how do we understand the symbols and numbers that describe entropy, especially for those of us many years removed from high school math?
Johnson writes pedagogically, but not pedantically. He is charming and colloquial. He uses (maybe a few too many) parenthetical asides. The book barely tops 150 pages, with short chapters switching easily between history and science; it’s a book that can be read lightly and in a weekend. Psychology buffs or math enthusiasts may be disappointed by the simplified, even rushed, treatment of important ideas like bipolar disorder or factorials. But in reworking and relearning quite simple concepts, Johnson imparts considerably more knowledge than one might expect, both about human behavior and about math.
The statistical concepts that Johnson explains in this book start deceptively simple—two particles of gas in an empty room—and finish with pleasurable complexity—numbers larger than can be written in simple forms. Likewise the questions he begins with seem so obvious as to be ridiculous: Why is the temperature of the room constant from one side to another, why does ice melt on a warm day, or why don’t puddles refreeze on warm days? But if probabilities describe the universe: why not? Maybe one room out of many will have a very weird temperature distribution. What mathematical principles prove that those strange scenarios won’t come to pass?
Methodically, and endearingly, Boltzmann teaches. Starting with that sparsely populated room of two particles—or two bedbugs, his other frequent, but gross, example—Johnson steps toward increasingly large numbers. He shows that nature tends, statistically, toward the even distribution of energy: It’s much, much more likely for a room’s atoms to have relatively equal kinetic energies, resulting in an even temperature, than for there to be an uneven distribution of kinetic energy and for one side of the room to be much hotter than the other.
Johnson illustrates these principles literally, with helpful charts and graphs. He’ll include a few equations along the way despite Stephen Hawking’s warning “that each equation cuts a book’s sales in half”—Hawking’s A Brief History of Time “included just one,” according to Johnson, “and that equation was E=mc^2 (which maybe should have been considered a gimme).” But instead of obfuscating, the equations come alive—for those who forgot what “logarithm” even means, the term transforms from obtuse math word into a magical prediction for how the universe behaves.
Similarly, the book’s psychology is simultaneously learned and casual. In chapters like “Postmortem Psychiatry” and “A Case for Anxiety,” Johnson discusses Boltzmann’s discernible mental symptoms. “Conventional wisdom asserts that he suffered from bipolar disorder,” Johnson summarizes. But Johnson disputes the diagnosis. “Was he depressed? Yes. Was he excitable, irritable, and maybe even hypomanic? Fair enough. But each of these symptoms was likely not so much a cause as it was an effect. The underlying cause of instability was likely anxiety. Boltzmann was a nineteenth-century victim of a twenty-first century disease.”
Nestled in the middle of the book are two short, revealing chapters. The first, “The Night Before,” describes Boltzmann’s paralyzing anxiety: Here he is, one of the brightest physicists in the world, spending hours preparing for a first-year lecture. “It was nonsense,” Johnson writes, “Rather than enjoying the success that he had earned over the course of his career, he was worrying about the tasks that he had already performed a thousand times before.” The image of the old man, sweating, frantic, hunched over his notes, is pathetic.
Then Johnson suggests a beautifully wrought addendum: “The Next Day.” After an awkward start the professor “settled in” to his role. He transforms into a virtuoso, a teacher par excellence. It’s not a stretch to hear Johnson describing himself, and his own book, when he praises Boltzmann’s “remarkably casual, even modern” tone or, more broadly, any teacher at the top of their game: “He was methodical in his approach, attentive to every detail and to the needs of his students. He anticipated potential misunderstandings and welcomed their questions. He imparted meaning to the mathematics. These symbols refer to something real.”
Boltzmann never actually came up with the equation on this book’s cover, S=klogW. One of the students of Boltzmann’s writings, Max Planck, distilled dozens of Boltzmann’s inelegant pages into that elegant formula. But if nothing else, Boltzmann, like Johnson, is an explainer, a teacher. Besides Planck, Boltzmann’s student Paul Ehrenfest would become a major figure in quantum mechanics and his student Lise Meitner would help discover nuclear fission. After the lecture, Johnson imagines Boltzmann, as he “surveyed the now empty room, admired what he had managed to somehow create on those blackboards.” Just as importantly, what he had sparked in the minds of his students.