What do you think?
Rate this book
272 pages, Paperback
First published August 1, 2012
One of the fundamental and initially very controversial theories of classical physics is Boltzmann’s kinetic theory of gases. Instead of tracking the individual motion of billions of individual atoms it studies the evolution of the probability that a particle occupies a certain position and has a certain velocity. The equilibrium probability distributions are well known for more than a hundred years, but to understand whether and how fast convergence to equilibrium occurs has been very difficult. Villani (in collaboration with Desvillettes) obtained the first result on the convergence rate for initial data not close to equilibrium. Later in joint work with his collaborator Mouhot he rigorously established the so-called non-linear Landau damping for the kinetic equations of plasma physics, settling a long-standing debate.
Imagine you’re walking through the woods on a peaceful summer’s afternoon. You pause at the edge of a pond. Everything is perfectly calm, not the slightest breeze. Suddenly the surface of the pond becomes agitated, as though seized by convulsions; a few moments later, it is sucked down into a roaring whirlpool. And then, a few moments after that, everything is calm once more. Still not a breath of air, not even a ripple on the surface from a fish swimming beneath it. So what happened? The Scheffer–Shnirelman paradox, surely the most astonishing result in all of fluid mechanics, proves that such a monstrosity is possible, at least in the mathematical world. It is not based on an exotic model of quantum probabilities or dark energy or anything of that sort. It rests on the incompressible Euler equations, the oldest of all partial differential equations, used by mathematicians and physicists everywhere to describe a perfectly incompressible fluid without any internal friction. It has been more than two hundred fifty years since Euler derived his fundamental equations, and yet not all of their mysteries have been penetrated. Indeed, they are still considered to mark out one of the most treacherous regions of the mathematical world. When the Clay Mathematics Institute set seven “millennium problems” in 2000, offering $1 million apiece for their solution, it did not hesitate to include the regularity of solutions to the Navier–Stokes equations. It was very careful, however, to avoid any mention of Euler’s equations—a far greater and more terrifying beast. And yet at first glance Euler’s equations seem so simple, so innocent, utterly devoid of guile or cunning. No need to model variations in density or to grapple with the enigmas of viscosity. One has only to write down the classical laws of conservation: conservation of mass, quantity of motion, and energy. But then … suddenly, in 1993, Scheffer showed that Euler’s equations in the plane are consistent with the spontaneous creation of energy! Energy created from nothing! No one has ever seen such bizarre behavior in fluids in the natural world! All the more reason, then, to suspect that Euler’s equations hold still more surprises in store for us. Big surprises. Scheffer’s proof is a stunning feat of mathematical virtuosity, as obscure as it is difficult. I doubt that anyone other than its author has read it carefully from beginning to end, and I am certain that no one could reconstruct its reasoning, unaided, in every detail. There was more to come. Four years later, in 1997, the Russian-born mathematician Alexander Shnirelman, renowned for his originality, presented a new proof of this staggering result. Shortly afterward Shnirelman proposed a physically realistic criterion for solutions to Euler’s equations that would prohibit pathological phenomena of the sort Scheffer had discovered. Alas! A few years ago, two brilliant young mathematicians, Camillo De Lellis, an Italian, and László Székelyhidi, a Hungarian, proved a general and still more shocking theorem that showed, among other things, that Shnirelman’s criterion was powerless to resolve the paradox. Additionally, using the techniques of convex integration, they were able to develop a new method for producing these “wild” solutions, an elegant procedure that grew out of earlier research by a number of mathematicians, including Vladimir Šverák, Stefan Müller, and Bernd Kirchheim. Thanks to De Lellis and Székelyhidi, we now realize that even less is known about Euler’s equations than we thought. And what we thought we knew wasn’t much to begin with.
Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets this word) seems to playa role. In a famous passage, Poincare describes how he discovered Fuchsian functions in such a moment. About such states, Gauss is said to have remarked as follows: "Procreare jucundum (to conceive is a pleasure)"; he added, however, "sed parturire molestum (but to give birth is painful)." Unlike sexual pleasure, this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do so at will, unless perhaps by dogged work which it seems to reward with its appearance.
Knuth continually worked to improve the original model. The version numbers he assigned to his program are approximations of π, ever more precisely estimated as the program was gradually perfected: after version 3.14 came version 3.141, then 3.1415, and so on. The current version is 3.1415926; according to the terms of Knuth’s will, it will change to π immediately following his death, thus fixing TEX for all eternity.