In the first version, we wrote: “[W]e claim that unless some new stability effect is identified, there is no reason to believe in nonlinear Landau damping for, say, gravitational interaction, in any regularity class lower than analytic.” Since then we have identified precisely such an effect (echoes occurring at different frequencies are asymptotically well separated). Exploiting it led to the above-mentioned improvements.
As a corollary, our work now includes new results of stability for homogeneous equilibria of the Vlasov–Poisson equation, such as the stability of certain nonmonotonic distributions in the repulsive case (a longstanding open problem), and stability below the Jeans length in the attractive case.
Another referee expressed a reservation about our use of nonconventional functional spaces. While this may be the case for our “working norm,” it is not so for the naïve norm appearing in our assumptions and conclusions, already used by others. Passing from one norm to another is done by means of Theorem 4.20.
The paper was entirely rewritten to incorporate these improvements, and carefully proofread. To prevent any further increase in the paper’s length, we have cut all expository passages and comments which were not strictly related to our main result; most of the remaining remarks are intended simply to explain the results and methods.
A final comment about length: we are open to discussion regarding adjustments to the organization of the paper, and we note that the modular presentation of the tools used in our work probably makes it possible for some referees to work as a team, thereby hopefully alleviating their task.
We very much hope that this paper will satisfy the experts and remain
Yours truly,
Clément Mouhot & Cédric Villani
* * *
THIRTY-NINE
Saint-Rémy-lès-Cheuvreuse
January 7, 2010
Reading email first thing in the morning, as soon as you’ve gotten out of bed, is a sort of intellectual drug injection. Not too much of a jolt, just enough to get you started.
Among the new messages today, Laurent Desvillettes sends unhappy news: our mutual friend Carlo Cercignani has died.
Carlo’s name is inseparable from that of Ludwig Boltzmann. Carlo devoted his professional life to Boltzmann, to Boltzmann’s theories, to his celebrated equation and its many applications. He wrote three of the standard works on Boltzmann, one of them the first research monograph I ever read.
Carlo’s mathematical interests were nevertheless extraordinarily diverse. Boltzmann allowed him to explore a great many topics, some of them having no obvious connection to the equation Carlo so cherished.
And that wasn’t the half of this universal man, polyglot and polymath, highly cultured, who refused to limit himself to the sciences: his works include a play for the theater, a collection of poems, and translations from Homer.
My first important result, or at least the first one I was really proud of, concerned what is known as Cercignani’s conjecture. I was twenty-four years old and raring to go when Giuseppe Toscani invited me to spend two weeks at the University of Pavia in October 1997. An idea had just occurred to Giuseppe about how to tackle this famous conjecture, and he suggested that I give it a go during my brief stay. After a few hours I could see that his crude frontal assault had no chance of success … but I did notice an interesting calculation, something that had the ring of truth about it, the hint of a remarkable new identity. It was all I needed: the mathematical rocket was ready to be launched.
I began by showing that Cercignani’s conjecture regarding entropy production in Boltzmann’s equation could be reduced to an estimate of entropy production in a problem in plasma physics that, as it happened, I had already studied with Laurent. And then I mixed in a bit of information theory, something that has always fascinated me. An incredible combination of circumstances that never would have come about had Giuseppe’s misguided intuition not struck him at precisely the moment I turned up!
Working together, we had almost cracked the case by the time I had to leave. Later the same month the opportunity presented itself to announce our results to an audience of the leading experts on the Boltzmann equation at a conference in Toulouse. It was here that Carlo, like many others, first came to know about me. I can still hear the excitement in his voice as he urged me on: “Cédric, prove my conjecture!”
At twenty-four, it was one of my first published articles. Five years later, in my twenty-fourth article, I returned to the problem, only now with more experience and more technique, and succeeded finally in proving Carlo’s brilliant conjecture. He was so proud of me.
The Boltzmann equation still has quite a few loose ends, however, and Carlo was counting on me to tie up some of the most important—and the most maddening—ones. That was one of my ambitions as well. But then, without warning, I wandered off, first in the direction of optimal transport and geometry, then of the Vlasov equation and Landau damping.
I still have it in mind to come back to Boltzmann, just not now. But even if one day my dream comes true, I shall never know the joy of telling Carlo that I’ve tamed his favorite monster, the one he loved before all others.
* * *
Cercignani’s conjecture concerns the relationship between entropy and entropy production in a gas. Let’s simplify, and leave to one side the spatially inhomogeneous character of the gas, so that the only thing that matters is the velocity distribution. Therefore, let f(v) be a velocity distribution in a gas away from equilibrium: since this distribution is not equal to the Gaussian γ(v), the entropy is not as high as it might be. Boltzmann’s equation predicts that the entropy will increase. Will it increase by a lot or only by very little?
Cercignani’s conjecture suggests that the instantaneous increase in entropy is at least proportional to the difference between the entropy of the Gaussian and the entropy of the distribution that we’re interested in:
The conjecture has implications for figuring out how fast the distribution converges to equilibrium—a fundamental question connected with Boltzmann’s fascinating discovery of irreversibility.
Laurent Desvillettes worked on Cercignani’s conjecture in the early 1990s, and after him Eric Carlen and Maria Carvalho. They obtained partial results that opened up completely new perspectives, but they were still a long way from proving it. Cercignani himself, with the help of the Russian mathematician Sasha Bobylev, had shown that the conjecture was overly optimistic, that it couldn’t be true … unless perhaps if one were to assume extremely strong collisions, interactions harder than those of so-called hard spheres, with the cross-section increasing at least proportionally to the relative velocity—“very hard spheres,” in the jargon of the kinetic theory of gases.
But in 1997 Giuseppe Toscani and I demonstrated the existence of a bound that is “almost” as good:
where ε is as small as one likes, under certain quite restrictive conditions governing collisions.
In 2003, I showed that this result holds true for all interactions satisfying certain reasonable assumptions; and, more strikingly still, I managed to show that the conjecture is true if the high-velocity collisions are of the very hard sphere type. The key identity, discovered with Toscani six years earlier, was the following:
If (St)t≥0 is the semigroup associated with the Fokker–Planck equation, , and then
This identity plays a key role in the representation formula
where F(v, v*) = f(v)f(v*) and G(v, v*) is the average of all the products f(v′)f(v′*) when (v′, v′*) describes all pairs of postcollision velocities compatible with the precollision velocities (v, v*). This formula lies at the heart of the solution of Cercignani’s conjecture.
Carlo Cercignani
Theorem (Villani, 2003). Let S(f) = − ∫ f log f denote the Boltzmann entropy associated with a velocity distribution f = f(v). Let B be a Boltzmann collision kernel satisfying for some constant KB > 0, and denote by the associated entropy production functional,
Let f = f(v) be a probab
ility distribution on with zero mean and unit temperature. Then
where
FORTY
Paris
February 16, 2010
Late afternoon in my spacious office at the Institut Henri Poincaré. I had the handsome blackboard enlarged and got rid of a few pieces of furniture to make more room. I’ve given a lot of thought to how I want to redecorate.
First, the bulky air conditioning unit has got to go—it’s normal to be hot in the summer!
Against the wall, a large display cabinet will hold some personal items and a few of the jewels from the Institute’s collection of geometric models.
To the left of my desk I plan to install the bust of a rather stern-looking Henri Poincaré that his grandson, François Poincaré, generously donated to the IHP.
And behind me a large space is reserved for a portrait of Catherine Ribeiro! I’ve already selected the image, found it on the Internet: Catherine with her arms spread wide in a vast gesture signifying struggle, peace, strength, and hope. Arms spread wide like the rebel in Goya’s El tres de Mayo, kneeling before Napoleon’s firing squad, or like Miyazaki’s Nausicaä before the soldiers of the royal house of Pejite. An image of strength, but also of abandonment and vulnerability. This idea appeals to me: you can’t expect to go forward if you’re not prepared to expose yourself to chance, risk, even danger. The iconic image of the pasionaria, the hopelessly passionate, vulnerable artist, which occurs also in Baudoin’s magnificent Salade Niçoise—I need it to watch over me, I need to confront it myself every day, face to face, in the person of Catherine.
Today, a day like every other, filled with appointments, discussions, meetings. This morning, a long telephone conversation with the chairman of my board of directors, the CEO of an insurance company who is deeply committed to enlisting the private sector in the service of scientific research. And this afternoon, a photo session to illustrate an interview I’ve given to a popular science magazine. Nothing very disagreeable about any of this. It’s been more than six months since my life took a fascinating turn: new people to meet, new things to learn, new ideas to talk about.
The photographer was unpacking his equipment in my office, setting up the tripod and reflector, when the telephone rang. Absent-mindedly, I picked up the receiver.
“Allô, oui.”
“Hello, is this Cédric Villani?”
“Yes.”
“This is Lázló Lovász calling from Budapest.”
For a moment my heart stopped. Lovász is president of the International Mathematical Union—and, by virtue of his office, chair of the Fields Medal Committee. This, by the way, is all I know about the committee; apart from him, I haven’t the faintest idea who’s on it.
“Hello, Professor Lovász, how are you doing?”
“Good, I’m fine. I have news—good news for you.”
“Oh, really?”
It was just like in a movie … I knew that these were the same words that Wendelin Werner had heard four years ago. But could it really be, so early in the year?
“Yes, I’m glad to tell you that you have won a Fields Medal.”
“Oh, this is unbelievable! This is one of the most beautiful days in my life. What should I say?”
“I think you should just be glad and accept it.”
Ever since Grigori Perelman refused the Fields Medal, the committee can’t help but be uneasy: What if someone else should refuse it? But I’m hardly on Perelman’s level, and I have no qualms about saying yes.
Lovász hastened to add that the laureates were being notified of the committee’s decision earlier than usual to ensure that the formal announcement will come from the IMU, and not through a leak.
“It’s very important that you keep it perfectly secret,” Lovász went on. “You can tell your family, but that is all. None of your colleagues should know.”
I shall therefore have to keep quiet for … six months. That’s a hell of a long time! In six months (and three days) the news will be broadcast on television and radio throughout the entire world. Until then I must do everything in my power to protect this highly confidential information. I’ll have to prepare myself psychologically for a marathon, not a sprint.
In the meantime, speculation about the medal winners will be rampant. But my lips shall remain sealed. As my colleague from Lyon, Michelle Schatzman, once observed, “Those who know do not speak, those who speak do not know.”
Before Lovász’s call, I put my chances of winning the medal at 40%. Now they’ve suddenly shot up to 99%! But still not to 100%—the possibility that it may have been a hoax can’t be ruled out entirely. Landau himself once played a trick with a friend on a fellow physicist whom they despised. They sent him a phony telegram from the Royal Swedish Academy: “Congratulations, you’ve won the Nobel Prize, etc.” The bastards …
So don’t get too excited yet, Cédric. How do you know that was really Lovász on the other end? Wait until a letter arrives, formally confirming the award, before you break out the champagne!
Ah, the secret, yes—but what about the photographer in my office!?
Apparently nothing registered, he must not understand English. Let’s hope not. Our session finally got under way. A picture of me posing with the mathematical physics trophy I brought home from Prague, another one of me in front of the Institute …
“That’s good, I think I’ve got what we need. One thing I wanted to ask you—in the article it says that you might win a prize or something?”
“You mean the Fields Medal? That was just speculation on the interviewer’s part. The announcement won’t be made for quite a while, the congress doesn’t take place until August.”
“I see. Think you’ll win?”
“Ohhh, I don’t know, it’s awfully hard to predict … no one can really say!”
* * *
In the years after 1918, harmony needed to be restored among the war-ravaged nations of Europe, where the Treaty of Versailles weighed heavily on Germany and the other Central Powers. What was true for society was also true for science: institutions had to be rebuilt.
In France, the mathematician and politician Émile Borel drew up plans for the Institut Henri Poincaré. In Canada, the mathematician John Charles Fields, an influential member of the recently founded International Mathematical Union, had the idea of creating an award that would serve both to recognize important work, as the Nobel Prize did, and to encourage talented younger mathematicians. It was to be embodied in the form of a medal and accompanied by a modest sum of money.
Fields donated the necessary funds, commissioned the medal’s reliefs from a Canadian sculptor, and composed its inscriptions in Latin, a common language chosen to reflect the universality of mathematics.
On the obverse of the medal, Archimedes is shown in right-facing profile together with the inscription TRANSIRE SUUM PECTUS MUNDOQUE POTIRI (Rise above oneself and grasp the world).
On the reverse, laurels frame an illustration of a theorem by Archimedes on the calculation of volumes of spheres and cylinders, with the inscription CONGREGATI EX TOTO ORBE MATHEMATICI OB SCRIPTA INSIGNIA TRIBUERE (Mathematicians gathered from all over the world have paid tribute to a remarkable work).
On the rim, the name of the laureate and the year of the award.
And the medal itself: solid gold.
Fields did not want the prize to be named after anyone, but upon his death in 1932 it was obvious to all that it should be called the Fields Medal. It was awarded for the first time four years later, in 1936, and then every four years from 1950 onward at the International Congress of Mathematicians, the grand meeting of the mathematical world. Its location changes from one occasion to the next, with as many as five thousand men and women taking part.
In keeping with Fields’s wish that the prize should serve to stimulate future achievement, it is awarded to mathematicians under the age of forty. In 2006, the age-counting rule was clarified: eligible candidates must not yet be forty years old on the first day of
January of the year in which the congress takes place. The number of laureates may vary between two and four, as the jury appointed by the Executive Committee of the IMU sees fit to decide.
A strict embargo on the announcement of the jury’s decision, combined with careful media coordination, assures Fields Medal winners of unrivaled publicity within the mathematical community and even beyond. The medals are usually presented by the head of state in the country where the congress is held. From there the news spreads throughout the world at once.
FORTY-ONE
RER B, Paris
May 6, 2010
In the world of Paris rapid transit, each of the RER lines is remarkable in its own way. In the case of the RER B, the line I take to go to work, it would not be an exaggeration to say that it breaks down every day, and that most days it is packed until midnight or one in the morning. To be fair, it also has its virtues: it assures its passengers of regular physical exercise by making them change trains frequently, and it improves their mental agility by keeping them in suspense as to exactly when a train will reach its destination and where it will stop along the way.
But this morning, on my way home from a conference in Cairo, it is very, very early and the train is nearly empty.
The outbound flight, in the company of the cutest girl you could ever hope to meet, couldn’t have been more delightful. We watched a film together on my computer, sharing earphones like brother and sister (always fly economy class, by the way, the girls are statistically cuter).
Birth of a Theorem: A Mathematical Adventure Page 17
