Field of Science

The case for dumb kindness

On June 22, 1941, Nazi Germany attacked the Soviet Union in a typhoon of steel and firepower without precedent in history. In spite of telltale signs and repeated warnings, Joseph Stalin who had indulged in wishful thinking was caught completely off guard. He was so stunned that he became almost catatonic, shutting himself in his dacha, not even coming out to make a formal announcement. It was days later that he regained his composure and spoke to the nation from the heart, awakening a decrepit albeit enormous war machine that would change the fate of tens of millions forever. By this time, the German juggernaut had advanced almost to the doors of Moscow, and the Soviet Union threw everything that it had to stop Hitler from breaking down the door and bringing the whole rotten structure on the Russian people’s heads, as the Führer had boasted of doing.
Among the multitudes of citizens and soldiers mobilized was a shortsighted, overweight Jewish journalist named Vasily Grossman. Grossman had been declared unfit for regular duty because of his physical shortcomings, but he somehow squeezed himself all the way to the front through connections. During the next four years, he became one of the most celebrated war correspondents of all time, witnessing human conflict whose sheer brutality beggared belief. To pass the time in this most unreal of landscapes, Grossman had a single novel to keep him company – War and Peace. It was to prove to be a prophetic choice.
Not only was Grossman present during the siege and eventual victory at Stalingrad – a single battle in which more Soviet soldiers and citizens died than American soldiers during all of World War 2 – but he was also part of the Soviet advance into the occupied territories in which the Nazis had waged a racial war of extermination that would almost annihilate an entire race of people. While forward-deployed units of Nazi Einsatzgruppen killed more than a million Jews in Ukraine, Lithuania and other countries, this “holocaust by bullets” was only a precursor to the horror of Auschwitz and Treblinka. Grossman became the first journalist to enter Treblinka and describe what words could scarcely bring themselves to describe. Most of all, the Holocaust hit home for him in a devastatingly personal way – Grossman’s own mother was murdered by the Nazis in the village of Berdychiv; the prewar Jewish population of this small town numbering more than 40,000 was completely annihilated. This singular episode shaped Grossman’s worldview for the rest of his life.
Over the next ten years Grossman who had seen Stalin’s 1937 purges and the postwar takeover of Europe became witness to his own country’s descent into oppression, conquest and genocidal aspirations. The words that proclaimed liberty and brotherhood during the fight against the Nazis started sounding hollow. In 1960 he put the finishing touches to what was the culmination of his career and thinking – Life and Fate, a 900-page magnum opus that was on par with some of the greatest fiction of all time. Today Life and Fate stands shoulder to shoulder with the great novels. And similar to the great novels, it takes in the entire world and nothing seems to be missing from its pages. Love, hatred, war, peace, childhood, motherhood, jealousy, bravery, cowardice, introspection, economics, politics, science, philosophy…everything is contained in its universe. More importantly, like the great works of literature, like Shakespeare and Dante, Dickens and Hemingway, like Grossman’s compatriots Tolstoy and Dostoevsky, the themes in Life and Fate are timeless, transcending nationality, race, gender and even its wartime setting. It will be relevant two hundred years from now when men and women will still be fighting and killing and discussing and loving. The novel speaks to human beings struggling with common problems across the gulf of time. And it speaks doggedly against the identity politics that riddles our discourse so widely.
Like War and Peace, Life and Fate straddles almost a hundred and fifty characters spread over a variety of times and locations, from the quiet warmth of a matriarch’s dwelling to the absolute nihilism of an extermination camp to several battle locations on the front spread around Stalingrad. Here we encounter characters whose views of life have been forced to be stripped down to their bare bones because of the sheer bleak brutality around them and forced minimalism of their existence. While there are hundreds of major and minor characters, a few key ones stand out. Broadly speaking, the characters fan out from the person of Alexandra Vladimirovna, a factory worker and steely matriarch who had lived in Stalingrad before moving out because of the war, and her two daughters Lyudmila and Yevgenia. The action also centers on Yevgenia’s old husband Krymov who has been an important party official and her new lover Novikov who is a tank commander. Meanwhile, Lyudmila lives with her husband Victor Shtrum, who in many ways speaks for the conscience of the various other characters in the novel. At least in one sense the most interesting person is Mikhail Mostovskoy, a friend of the family who has ended up in a German concentration camp.
It’s hard to keep track of all the characters, but one of the most remarkable things is how even some of the minor, intermittent players leave an indelible memory because of their pronunciations and ideas. There are some extraordinarily poignant moments, such as when Lyudmila’s son Tolya is wounded on the front and she hurries to visit him in the hospital, only to find that he has died shortly before. She asks to be escorted to his grave and spends a moment of hauntingly beautiful, ethereal and yet earthly tragedy mourning at his side, covering him with his shawl so that he won’t be cold. It takes her several minutes to realize the bare truth of Tolya’s non-existence:
“The water of life, the water that had gushed over the ice and brought Tolya back from the darkness, had disappeared; the world created by the mother’s despair, the world that for a moment had broken its fetters and become reality, was no more.”
Perhaps there is no story more emotionally devastating in the book than the story of Sofya Levinton, a Jewish friend of Lyudmila’s who has the misfortune of being snared by the Nazis and put on a cattle train to Auschwitz. On the train Sofya runs into David, a six or seven year-old boy who also shared the misfortune of being cut off from his mother and put in a ghetto with his grandmother. When his grandmother died of disease, the woman she had entrusted David to was too busy trying to save herself. Like two atomic particles randomly bumping into each other by accident, David and Sofya bump into each other on the train. They have no one else, so they have each other. They accompany each other into the camp, into the dressing room, and finally into the gas chamber where there is no light, no life, no meaning. As the Zyklon B starts hissing from the openings above, David clings to the unmarried, childless Sofya:
“Sofya Levinton felt the boy’s body subside in her hands. Once again she had fallen behind him. In mineshafts where the air becomes poisoned, it is always the little creatures, the birds and mice, that die first. This boy, with his slight, bird-like body, had left before her.
‘I’ve become a mother,’ she thought.
That was her last thought.”
In another German concentration camp, Mikhail Mostovskoy has philosophical disputes with a few prisoners who are trying to shake his confidence in communism and are also trying to organize an escape. Mostovskoy is a true believer and is keeping the flame burning bright. But reality is not so easy. The denouement comes when he is called to the office of the camp commandant. His name is Liss. Liss is interested in certain documents which a dissident named Ikkonikov has thrust into Mostovskoy’s hands, right before refusing to help build a gas chamber and being executed as a result. But that is not Liss’s main concern, and he is not here to punish Mostovskoy. Instead he does something worse than provide an easy death: he brings the hammer down on Mostovskoy’s entire worldview when he tells him how similar Nazism and Stalinism are, how they are built on the backs of oppressed and murdered people, how true believers in both ideologies should ideally stand shoulder to shoulder with each other, how this whole war is therefore an unnecessary farce. Mostovskoy is shaken, and his loss of faith very much mirrors Grossman’s own by the time he wrote the book: with its murder and suppression of all dissent, complete control of people’s lives and total disregard for individual freedom, were fascism and communism that different?
But if Mostovskoy had any lingering doubts about whether his faith in collective action has been built on a house of cards, it collapses completely when he reads Ikkonikov’s pamphlets and hears him speaking from the grave. It’s strange: Ikkonikov is a minor character who appears perhaps in four or five pages of the volume, and the transcript of his documents occupies not more than ten pages in a book numbering almost a thousand pages, and yet in many ways his pamphlet is the single-most important part of the book, communicating as it does the overwhelming significance of individual kindness and action in the face of utter, unending conflict. Individual kindness is the only thing that remains when all humanity has been stripped away from both oppressor and oppressed; when every trace of nationality, race, gender and political views has been obliterated by sheer terror and murder, this kindness is the only elemental thing connecting all human beings simply because they are human beings and nothing else, it is this kindness, this dumb, senseless kindness, that will keep propelling humanity onwards when all else is lost. It is this kindness that goes by the name of ‘good’. As Ikkonikov says,
“Good is to be found neither in the sermons of religious teachers and prophets, nor in the teachings of sociologists and popular leaders, nor in the ethical systems of philosophers… And yet ordinary people bear love in their hearts, are naturally full of love and pity for any living thing. At the end of the day’s work they prefer the warmth of the hearth to a bonfire in the public square.
Yes, as well as this terrible Good with a capital ‘G’, there is everyday human kindness. The kindness of an old woman carrying a piece of bread to a prisoner, the kindness of a soldier allowing a wounded enemy to drink from his water-flask, the kindness of youth towards age, the kindness of a peasant hiding an old Jew in his loft. The kindness of a prison guard who risks his own liberty to pass on letters written by a prisoner not to his ideological comrades, but to his wife and mother.
The private kindness of one individual towards another; a petty, thoughtless kindness; an unwitnessed kindness. Something we could call senseless kindness. A kindness outside any system of social or religious good.
But if we think about it, we realize that this private, senseless, incidental kindness is in fact eternal. It is extended to everything living, even to a mouse, even to a bent branch that a man straightens as he walks by.
Even at the most terrible times, through all the mad acts carried out in the name of Universal Good and the glory of States, times when people were tossed about like branches in the wind, filling ditches and gullies like stones in an avalanche – even then this senseless, pathetic kindness remained scattered throughout life like atom…
This kindness, this stupid kindness, is what is most truly human in a human being. It is what sets man apart, the highest achievement of his soul. No, it says, life is not evil!”
And who promotes this kindness? Not religion with its conditional acceptance and demands to conform. Not the state which also imposes its own demands for conformity. Not even capitalism which makes kindness conditional on the invisible hand of selfish actions. In fact no system of organization can impose this kindness, no matter how much it speaks of it in glowing terms. It can only come about when all systems of organization have been obliterated, when humanity’s bare existence compels its members to recognize a quality in each other that is completely independent of every group identification, every kind of “ism”.
And who spoke of this kindness? Not the religious prophets who sought salvation in the one true God and heaven, not the commissars whose mind-numbing bureaucratic machinations threatened to grind every human particle of unique identity into the featureless dust of one level playing field, not even the scientific rationalists whose discoveries can only describe, not prescribe. No, to describe senseless, stupid, all-encompassing kindness one must look to the great poets and writers, not the philosophers. And through everyday characters and conversations, nobody demonstrates the timeless nature of individual kindness as well as Chekhov:
“Chekhov said: let’s put God – and all these grand progressive ideas – to one side. Let’s begin with man; let’s be kind and attentive to the individual man – whether he’s a bishop, a peasant, an industrial magnate, a convict in the Sakhalin Islands or a waiter in a restaurant. Let’s begin with respect, compassion and love for the individual – or we’ll never get anywhere.”
If you haven’t already, dear reader, I cannot exhort you enough to read Chekhov. Read his plays, read especially his short stories, read anything by him. Throughout Life and Fate the nature of indivisible, immutable bonds between human beings – whether it is a commander and his aide, an aging communist and her son-in-law, and of course the more common and enduring sets of relationships between sons and mothers, daughters and fathers – stand above and beyond the basic essentials of the narrative.
Another character, in a completely different set of circumstances on the Stalingrad front:
“Human groupings have one main purpose: to assert everyone’s right to be different, to be special, to think, feel and live in his or her own way. People join together in order to win or defend this right. But this is where a terrible, fateful error is born: the belief that these groupings in the name of a race, a God, a party or a State are the very purpose of life and not simply a means to an end. No! The only true and lasting meaning of the struggle for life lies in the individual, in his modest peculiarities and in his right to these peculiarities.”
If that is not a soaring counterpoint to and a damning indictment of the identity politics that has completely taken over our discourse today, I do not know what is.
When word of Grossman’s magnum opus got out the KGB stormed his apartment. They considered the novel so dangerous that they confiscated not only the manuscript but also the typewriter ribbons which were used to craft the novel. This level of paranoia could only exist in the Soviet Union. Why they did this is clear after reading it. Not only does Life and Fate show, through devastatingly understated examples of indelible characters who gradually become disillusioned, the hollow nature of the Soviet system’s promises and its similarity with the fascism that its patriotic adherents thought they were fighting, but it also demonstrated through the character of physicist Victor Shtrum, the anti-Semitism that while not as fatal as that in Nazi Germany, was slowly but surely brewing in the country’s corridors and the hearts and minds of its people. Even before the war ended it was clear that the Germans’ campaign of Jewish cleansing in Ukraine and parts of Russia could not have been carried out without the complicity of local populations who held grudges against Jews for decades. Grossman’s personal motivation because of his mother’s murder brought to his depiction of the Soviet Union’s initially “benign” and then increasingly oppressive anti-Semitism particularly strident and urgent force. The party line in the country refused to have writers like Grossman single out Jewish victims of the Holocaust because they knew that doing so would shine a mirror into their own faces. The combination of Grossman’s expose of the Soviets as being little different from the Nazis and anti-Semites to boot sealed his novel’s fate.
When Grossman asked when his book might see the light of day, a high-ranking party official named Suslov said there was no question of the volume being published for another two hundred years; by announcing such a draconian sentence on Grossman’s work, he inadvertently announced the novel’s incendiary nature. Grossman died in 1964 without seeing his book smuggled out and translated by Robert Chandler, a sad and lonely man in a Moscow apartment battling stomach cancer.
But his act of defiance, expressed in this profound book as an assertion of the fundamental nature of the individual and a rejection of collectivism of all kinds, spoke to the ages, escaped the fetters of its two hundred-year oppressors and brought about the collapse of the Soviet Union. And it could well bring about the collapse of the systems we take so much pride in because we fail to see how they are turning us into inchoate groups. So let us now practice thoughtless, stupid, unwitnessed kindness. It’s the one constant in life and fate.
First published on 3 Quarks Daily

The Fermi-Pasta-Ulam-Tsingou Problem: A Foray Into The Beautifully Simple And The Simply Beautiful

In November 1918, a 17-year-student from Rome sat for the entrance examination of the Scuola Normale Superiore in Pisa, Italy’s most prestigious science institution. Students applying to the institute had to write an essay on a topic that the examiners picked. The topics were usually quite general, so the students had considerable leeway. Most students wrote about well-known subjects that they had already learnt about in high school. But this student was different. The title of the topic he had been given was “Characteristics of Sound”, and instead of stating basic facts about sound, he “set forth the partial differential equation of a vibrating rod and solved it using Fourier analysis, finding the eigenvalues and eigenfrequencies. The entire essay continued on this level which would have been creditable for a doctoral examination.” The man writing these words was the 17-year-old’s future student, friend and Nobel laureate, Emilio Segre. The student was Enrico Fermi. The examiner was so startled by the originality and sophistication of Fermi’s analysis that he broke precedent and invited the boy to meet him in his office, partly to make sure that the essay had not been plagiarized. After convincing himself that Enrico had done the work himself, the examiner congratulated him and predicted that he would become an important scientist.
Twenty five years later Fermi was indeed an important scientist, so important in fact that J. Robert Oppenheimer had created an entire division called F-Division under his name at Los Alamos, New Mexico to harness his unique talents for the Manhattan Project. By that time the Italian emigre was the world’s foremost nuclear physicist as well as perhaps the only universalist in physics – in the words of a recent admiring biographer, “the last man who knew everything”. He had led the creation of the world’s first nuclear reactor in a squash court at the University of Chicago in 1942 and had won a Nobel Prize in 1938 for his work on using neutrons to breed new elements, laying the foundations of the atomic age.
The purpose of F-division was to use Fermi’s unprecedented joint abilities in both experimental and theoretical physics to solve problems that stumped others. To Fermi other scientists would take their problems in all branches of physics, many of them current or future Nobel laureates. They would take advantage of Fermi’s startlingly simple approach to problem-solving, where he would first qualitatively estimate the parameters and solution and then plug in complicated mathematics only when necessary to drive relentlessly toward the solution. He had many nicknames including “The Roadroller”, but the one that stuck was “The Pope” because his judgement on any physics problem was often infallible and the last word.
Fermi’s love for semi-quantitative, order-of-magnitude estimates gave him an unusual oeuvre. He loved working out the most rigorous physics theories as much as doing back-of-the-envelope calculations designed to test ideas; the latter approach led to the famous set of problems called ‘Fermi problems‘. Simplicity and semi-quantitative approaches to problems are the hallmark of models, and Fermi inevitably became one of the first modelers. Simple models such as the quintessential “spherical cow in a vacuum” are the lifeblood of physics, and some of the most interesting insights have come from using such simplicity to build toward complexity. Interestingly, the problem that the 17-year-old Enrico had solved in 1918 would inspire him in a completely novel way many years later. It would be the perfect example of finding complexity in simplicity and would herald the beginnings of at least two new, groundbreaking fields.
Los Alamos was an unprecedented exercise in bringing a century’s worth of physics, chemistry and engineering to bear on problems of fearsome complexity. Scientists quickly realized that the standard tools of pen and paper that they had been using for centuries would be insufficient, and so for help they turned to some of the first computers in history. At that time the word “computer” meant two different things. One meaning was women who calculated. The other meaning was machines which calculated. Women who were then excluded from most of the highest echelons of science were employed in large numbers to perform repetitive calculations on complicated physics problems. Many of these problems at Los Alamos were related to the tortuous flow of neutrons and shock waves from an exploding nuclear weapon. Helping the female computers were some of the earliest punched card calculators manufactured by IBM. Although they didn’t know it yet, these dedicated women working on those primitive calculators became history’s first pioneering programmers. They were the forerunners of the women who worked at NASA two decades later on the space program.
Fermi had always been interested in these computers as a way to speed up calculations or to find new ways to do them. At Los Alamos a few other far-seeing physicists and mathematicians had realized their utility, among them the youthful Richard Feynman who was put in charge of a computing division. But perhaps the biggest computing pioneer at the secret lab was Fermi’s friend, the dazzling Johnny von Neumann, widely regarded as the world’s foremost mathematician and polymath and fastest thinker. Von Neumann who had been recruited by Oppenheimer as a consultant because of his deep knowledge of shock waves and hydrodynamics had become interested in computers after learning that a new calculating machine called ENIAC was being built at the University of Pennsylvania by engineers J. Presper Eckert, John Mauchly, Herman Goldstine and others. Von Neumann realized the great potential of what we today call the shared program concept, a system of programming that contains both the instructions for doing something and the process itself in the same location, both coded in the same syntax.
Fermi was a good friend of von Neumann’s, but his best friend was Stanislaw Ulam, a mathematician of stunning versatility and simplicity who had been part of the famous Lwów School of mathematics in Poland. Ulam belonged to the romantic generation of Central European mathematics, a time during the early twentieth century when mathematicians had marathon sessions fueled by coffee in Lwów, Vienna and Warsaw’s famous cafes, where they scribbled on the marble tables and argued mathematics and philosophy late into the night. Ulam had come to the United States in the 1930s; by then von Neumann had already been firmly ensconced at Princeton’s Institute for Advanced Study with a select group of mathematicians and physicists including Einstein. Ulam had started his career in the most rarefied parts of mathematics including set theory; he later joked that during the war he had to stoop to the level of manipulating actual numbers instead of merely abstract symbols. After the war started Ulam had wanted to help with the war effort. One day he got a call from Johnny, asking him to a move to a secret location in New Mexico. At Los Alamos Ulam worked closely with von Neumann and Fermi and met the volatile Hungarian physicist Edward Teller with whom he began a fractious, consequential working relationship.
Fermi, Ulam and von Neumann all worked on the intricate calculations involving neutron and thermal diffusion in nuclear weapons and they witnessed the first successful test of an atomic weapon on July 16th, 1945. All three of them realized the importance of computers, although only von Neumann’s mind was creative and far-reaching enough to imagine arcane and highly significant applications of these as yet primitive machines – weather control and prediction, hydrogen bombs and self-replicating automata, entities which would come to play a prominent role in both biology and science fiction. After the war ended, computers became even more important in the early 1950s. Von Neumann and his engineers spearheaded the construction of a pioneering computer in Princeton. After the computer achieved success in doing hydrogen bomb calculations at night and artificial life calculations during the day, it was shut down because the project was considered too applied by the pure mathematicians. But copies started springing up at other places, including one at Los Alamos. Partly in deference to the destructive weapons whose workings would be modeled on it, the thousand ton Los Alamos machine was jokingly christened MANIAC, for Mathematical Analyzer Numerical Integrator and Computer. It was based on the basic plan proposed by von Neumann which is still the most common plan used for computers worldwide – the von Neumann architecture.
After the war, Enrico Fermi had moved to the University of Chicago which he had turned into the foremost center of physics research in the country. Among his colleagues and students there were T. D. Lee, Edward Teller and Subrahmanyan Chandrasekhar. But the Cold War imposed on his duties, and the patriotic Fermi started making periodic visits to Los Alamos after President Truman announced in 1951 that he was asking the United States Atomic Energy Commission to resume work on the hydrogen bomb as a top priority. Ulam joined him there. By that time Edward Teller had been single-mindedly pushing for the construction of a hydrogen bomb for several years. Teller’s initial design was highly flawed and would have turned into a dud. Working with pencil and paper, Fermi, Ulam and von Neumann all confirmed the pessimistic outlook for Teller’s design, but in 1951, Ulam had a revolutionary insight into how a feasible thermonuclear weapon could be made. Teller honed this insight into a practical design which was tested in November 1952, and the thermonuclear age was born. Since then, the vast majority of thermonuclear weapons in the world’s nuclear arsenals have been based on some variant of the Teller-Ulam design.
By this time Fermi had acutely recognized the importance of computers, to such an extent in fact that in the preceding years he had taught himself how to code. Work on the thermonuclear brought Fermi and Ulam together, and in 1955 Fermi proposed a novel project to Ulam. To help with the project Fermi recruited a visiting physicist named John Pasta who had worked as a beat cop in New York City during the Depression. With the MANIAC ready and standing by, Fermi was especially interested in problems where highly repetitive calculations on complex systems could take advantage of the power of computing. Such calculations would be almost impossible in terms of time to perform by hand. As Ulam recalled later,
“Fermi held many discussions with me on the kind of future problems which could be studied through the use of such machines. We decided to try a selection of problems for heuristic work where in the absence of closed analytic solutions experimental work on a computing machine might perhaps contribute to the understanding of properties of solutions. This could be particularly fruitful for problems involving the asymptotic, long time or “in the large” behavior of non-linear physical systems…Fermi expressed often a belief that future fundamental theories in physics may involve non-linear operators and equations, and that it would be useful to attempt practice in the mathematics needed for the understanding of non-linear systems. The plan was then to start with the possibly simplest such physical model and to study the results of the calculation of its long-term behavior.”
Fermi and Ulam had caught the bull by its horns. Crudely speaking, linear systems are systems where the response is proportional to the input. Non-linear systems are ones where the response can vary disproportionately. Linear systems are the ones which many physicists study in textbooks and as students. Non-linear systems include almost everything encountered in the real world. In fact, the word “non-linear” is highly misleading, and Ulam nailed the incongruity best: “To say that a system is non-linear is to say that most animals are non-elephants.” Non-linear systems are the rule rather than the exception, and by 1955 physics wasn’t really well-equipped to handle this ubiquity. Fermi and Ulam astutely realized that the MANIAC was ideally placed to attempt a solution to non-linear problems. But what kind of problem would be complex enough to attempt by computer, yet simple enough to provide insights into the workings of a physical system? Enter Fermi’s youthful fascination with vibrating rods and strings.
The simple harmonic oscillator is an entity which physics students encounter in their first or second year of college. Its distinguishing characteristic is that the force applied to it is proportional to the displacement. But as students are taught, this is an approximation. Real oscillators – real pendulums, real vibrating rods and strings in the real world – are not simple. The force applied results in a complicated function of the displacement. Fermi and Ulam set up a system consisting of a string attached to one end. They considered four models; one where the force is proportional to the displacement, one where the force is proportional to the square of the displacement, one where it’s proportional to the cube, and one where the displacement varies in a discontinuous way with the force, going from broken to linear and back. In reality the string was modeled as a series of 64 points all connected through these different forces. The four graphs from the original paper are shown below, with force on the x-axis and displacement on the y-axis and the dotted line indicating the linear case.
Here’s what the physicists expected: the case for a linear oscillator, familiar to physics students, is simple. The string shows a single sinusoidal node that remains constant. The expectation was that when the force became non-linear, higher frequencies corresponding to two, three and more sinusoidal modes would be excited (these are called harmonics or overtones). The global expectation was that adding a non-linear force to the system would lead to an equal distribution or “thermalization” of the energy, leading to all modes being excited and the higher modes being heavily so.
What was seen was something that was completely unexpected and startling, even to the “last man who knew everything.” When the quadratic force was applied, the system did indeed transition to the two and three-mode system, but the system then suddenly did something very different.
“Starting in one problem with a quadratic force and a pure sine wave as the initial position of the string, we indeed observe initially a gradual increase of energy in the higher modes as predicted. Mode 2 starts increasing first, followed by mode 3 and so on. Later on, however, this gradual sharing of energy among successive modes ceases. Instead, it is one or the other mode that predominates. For example, mode 2 decides, as it were, to increase rather rapidly at the cost of all other modes and becomes predominant. At one time, it has more energy than all the others put together! Then mode 3 undertakes this role.”
Fermi and Ulam could not resist adding an exclamation point even in the staid language of scientific publication. Part of the discovery was in fact accidental; the computer had been left running overnight, giving it enough time to go through many more cycles. The word “decides” is also interesting; it’s as if the system seems to have a life of its own and starts dancing of its own volition between one or two lower modes; Ulam thought that the system was playing a game of musical chairs. Finally it comes back to mode 1, as if it were linear, and then continues this periodic behavior. An important way to describe this behavior is to say that instead of the initial expectation of equal distribution of energy among the different modes, the system seems to periodically concentrate most or all of its energy in one or a very small number of modes. The following graph for the quadratic case makes this feature clear: on the y-axis is energy while on the x-axis is the number of cycles ranging into the thousands (as an aside, this very large number of cycles is partly why it would be impossible to solve this problem using pen and paper in reasonable time). As is readily seen, the height of modes 2 and 3 is much larger than the higher modes.
The actual shapes of the string corresponding to this asymmetric energy exchange are even more striking, indicating how the lower modes are disproportionately excited. The large numbers here again correspond to the number of cycles.
The graphs for the cubic and broken displacement case are similar but even more complex, leading to higher modes being excited more often but the energy still concentrated into the lower modes. Needless to say, these results were profoundly unexpected and fascinating. The physicists did not quite know what to make of them, and Ulam found them “truly amazing”. Fermi told him that he thought they had made a “little discovery”.
The 1955 paper contains an odd footnote: “We thank Ms. Mary Tsingou for efficient coding of the problems and for running the computations on the Los Alamos MANIAC machine.” Mary Tsingou was the underappreciated character in the story. She was a Greek immigrant whose family barely escaped Italy before Mussolini took over. With bachelor’s and master’s degrees in mathematics from Wisconsin and Michigan, in 1955 she was a “computer” at Los Alamos, just like many other women. Her programming of the computer was crucial and non-trivial, but she was acknowledged in the work and not in the writing. She worked later with von Neumann on diffusion problems, was the first FORTRAN programmer, and even did some calculations for Ronald Reagan’s infamous “Star Wars” program. As of 2020, Mary Tsingou is still alive and 92 and living in Los Alamos. The Fermi-Pasta-Ulam problem should be called the Fermi-Pasta-Ulam-Tsingou problem.
Fermi’s sense of having made a “little discovery” has to be one of the great understatements of 20th century physics. The results that he, Ulam, Pasta and Tsingou obtained went beyond harmonic systems and the MANIAC. Until then there had been two revolutions in 20th century physics that changed our view of the universe – the theory of relativity and quantum mechanics. The third revolution was quieter and started with French mathematician Henri Poincare who studied non-linear problems at the beginning of the century. It kicked into high gear in the 1960s and 70s but still evolved under the radar, partly because it spanned several different fields and did not have the flashy reputation that the then-popular fields of cosmology and particle physics had. The field went by several names, including “non-linear dynamics”, but the one we are most familiar with is chaos theory.
As James Gleick who gets the credit for popularizing the field in his 1987 book says, “Where chaos begins, classical science stops.” Classical science was the science of pen and pencil and linear systems. Chaos was the science of computers and non-linear systems. Fermi, Ulam, Pasta and Tsingou’s 1955 paper left little reverberations, but in hindsight it is seminal and signals the beginning of studies of chaotic systems in their most essential form. Not only did it bring non-linear physics which also happens to be the physics of real world problems to the forefront, but it signaled a new way of doing science by computer, a paradigm that is the forerunner of modeling and simulation in fields as varied as climatology, ecology, chemistry and nuclear studies. Gleick does not mention the report in his book, and he begins the story of chaos with Edward Lorenz’s famous meteorology experiment in 1963 where Lorenz discovered the basic characteristic of chaotic systems – acute sensitivity to initial conditions. His work led to the iconic figure of the Lorenz attractor where a system seems to hover in a complicated and yet simple way around one or two basins of attraction. But the 1955 Los Alamos work got there first. Fermi and his colleagues certainly demonstrated the pull of physical systems toward certain favored behavior, but the graphs also showed how dramatically the behavior would change if the coefficients for the quadratic and other non-linear terms were changed. The paper is beautiful. It is beautiful because it is simple.
It is also beautiful because it points to another, potentially profound ramification of the universe that could extend from the non-living to the living. The behavior that the system demonstrated was non-ergodic or quasiergodic. In simple terms, an ergodic system is one which visits all its states given enough time. A non-ergodic system is one which will gravitate toward certain states at the expense of others. This was certainly something Fermi and the others observed. Another system that as far as we know is non-ergodic is biological evolution. It is non-ergodic because of historical contingency which plays a crucial role in natural selection. At least on earth, we know that the human species evolved only once, and so did many other species. In fact the world of butterflies, bats, humans and whales bears some eerie resemblances to the chaotic world of pendulums and vibrating strings. Just like these seemingly simple systems, biological systems demonstrate a bewitching mix of the simple and the complex. Evolution seems to descend on the same body plans for instance, fashioning bilateral symmetry and aerodynamic shapes from the same abstract designs, but it does not produce the final product twice. Given enough time, would evolution be ergodic and visit the same state multiple times? We don’t know the answer to this question, and finding life elsewhere in the universe would certainly shed light on the problem, but the Fermi-Pasta-Ulam-Tsingou problem points to the non-ergodic behavior exhibited by complex systems that arise from simple rules. Biological evolution with its own simple rules of random variation, natural selection and neutral drift may well be a Fermi-Pasta-Ulam-Tsingou problem waiting to be unraveled.
The Los Alamos report was written in 1955, but Enrico Fermi was not one of the actual co-authors because he had tragically died in November 1954, the untimely consequence of stomach cancer. He was still at the height of his powers and would have likely made many other important discoveries compounding his reputation as one of history’s greatest physicists. When he was in the hospital Stan Ulam paid him a visit and came out shaken and in tears, partly because his friend seemed so composed. He later remembered the words Crito said in Plato’s account of the death of Socrates: “That now was the death one of the wisest men known.” Just three years later Ulam’s best friend Johnny von Neumann also passed into history. Von Neumann had already started thinking about applying computers to weather control, but in spite of the great work done by his friends in 1955, he did not realize that chaos might play havoc with the prediction of a system as sensitive to initial conditions as the global climate. It took only seven years before Lorenz found that out. Ulam himself died in 1984 after a long and productive career in physics and mathematics. Just like their vibrating strings, Fermi, Ulam and von Neumann had ascended to the non-ergodic, higher modes of the metaphysical universe.
First posted on 3 Quarks Daily.