Approaching ‘Entropy’

Science popularization has its legends and heroes, just like any other field, though I’ve heard no plans as yet to open a Hall of Fame. Should that day come, one of the first inductees would undoubtedly be the astronomer and documentarian Carl Sagan -- and there’s a passage in Eric Johnson’s Anxiety and the Equation: Understanding Boltzmann’s Entropy (MIT Press) that someone will want to read out during the ceremony:

Sagan helped us to feel at home in the universe -- as if we might actually belong here, not trapped on some rapidly spinning rock, along with a bunch of mostly crazy strangers, destined to go around and around an unremarkable star amid a mostly empty space. He was a reassuring tour guide in a universe where we play a seemingly inconsequential role. He offered us awe and wonder when we might otherwise have succumbed to a sense of insignificance. The cosmos seemed a more welcoming place because he shared it with us.

This tribute appears near the opening of the eighth chapter, by which point the reader should sense the irony. For Johnson, an associate professor of chemistry at Mount St. Joseph University in Cincinnati, is writing about the universe at another extreme of scale (atomic and molecular) as understood by a profoundly melancholy scientist, the Austrian physicist Ludwig Boltzmann (1844-1906), who, when we are introduced to him on the first page, is already dead by his own hand.

The challenge for the author, then, is to elicit and reward our interest despite a full and fair warning that the cosmos is not going to seem “a more welcoming place because he shared it with us.” (Not even a little.)

In fact Johnson succeeds at this marvelously through a combination of dry humor and well-turned pivots between the narrative and expositive chapters. He keeps the lay reader’s eyes on the prize of understanding (at least very generally) the formula inscribed on the monument at Boltzmann’s grave: S = k. log W.

Citing Stephen Hawking’s quip that each equation in a book cuts its sales in half, Johnson writes that his own brief history of entropy is therefore probably doomed not to break into the best-seller lists. For one thing, it can scarcely avoid a short chapter on logarithms. (“Math courses,” he writes, “like many other valuable experiences, are generally wasted on the young.”) But popularization is just pedagogy by different means, and Johnson knows how to build an introductory lecture. Starting out with two atoms behaving in ways readily graspable by ordinary intuition (“a surprisingly poor guide throughout much of modern physics”), he builds up a series of well-designed and patiently explained diagrams and graphs that give the lay reader a feel for how the movement of one octillion gas particles can be understood statistically.

With the motion of any given particle being unpredictable and the tracking of them all at once impossible (one octillion is the numeral 1 followed by 27 zeroes: a figure larger than the number of stars in the universe), it might seem as if it would be hopeless to try to say anything about their movement over time. Nonetheless, a spontaneous yet predictable long-term tendency exists for the particles to reach equilibrium in their diffusion and temperature -- with that tendency also being known as entropy.

The latter concept was not Boltzmann’s contribution, as such. Rather, he framed entropy as a result of probabilities over time, rather than something increasing inexorably within a system. Entropy à la Boltzmann could fluctuate. And while not as surreal as some more recent developments (time-traveling subatomic particles and the like) randomly fluctuating entropy was strange enough to make the new physics of Boltzmann’s final years a scene of much dispute and consternation.

Not that the strain of controversy made him hang himself, though it clearly did not help. “It was not his first attempt at suicide,” Johnson writes, “but it was his most successful.” Anxiety and the Equation presents Boltzmann as the archetypal disheveled genius on the verge of a nervous breakdown. Gifted in the classroom as well as in his research, he sounds more genial than what his colleagues regarded as quite appropriate for an eminent scientist and professor -- but also prone to crippling bouts of depression and to brooding over slights, real and imagined. In later years, Johnson says, Boltzmann forgot “the many accolades that he had accumulated over the course of his career and felt compelled instead to insert his own personal struggles into his scientific writing. If he were alive today, he would be that aggrieved person who clicks Send late at night rather than first sleeping on the matter.”

The book is neither a full biography nor an intro to statistical mechanics. Its virtue comes from selecting just enough of the life and the work to elucidate each other for readers who will remember Boltzmann, if at all, only as a name mentioned in connection with Albert Einstein or Max Planck -- younger colleagues who built on his work. Johnson's depiction is both clear-eyed about his faults and evocative of the suffering Boltzmann endured. It is a piece of scientific popularization as well as a remarkably humane book.