 In 1906, a 62-year-old Ludwig Boltzmann was vacationing with his family at the Bay of Duino in Italy. On the evening of September 5th, while his wife and daughter were out swimming, he took a length of cord, tied it to a curtain rod, and hanged himself. No one knows exactly why the famous physicist from Vienna committed suicide. It is true that he was in poor health and suffered bouts of depression. It is also true that he faced withering criticism from scientists who thought his life's work was all wrong. Boltzmann even fled one university appointment to get away from a detractor. What was Boltzmann's daring idea that drew such relentless attack? It was his attempt to connect the invisible microscopic world with the observable macroscopic one. He sought to explain how the properties of atoms can produce the things we see and measure every day, like the temperature of a gas or the thickness of a liquid. Today, it's hard to believe Boltzmann was controversial. Supported by later findings, his work has become an essential component of mathematical physics. It not only serves as a basis to understand thermodynamics and fluids, but also to probe the nature of entropy, information, and even time. In the 19th century, the idea of atoms was not entirely accepted, even though the concept dates to ancient Greece. Chemists liked atoms because they provided a framework to understand chemical reactions. Physicists, however, doubted that atoms were truly the physical entities we know them to be today. Boltzmann challenged many of his contemporaries when he used atomic theory to explain the nature of gases. He saw gases as composed of particles bouncing around, like in a kettle boiling water and creating vapor and steam. All the little balls flying around in different directions and speeds can be described by Isaac Newton's laws of motion, including his famous second law, the force on an object equals its mass times its acceleration. Newton's equations accurately predict the movement of objects from billiard balls to planets and other heavenly bodies. And they should also apply to gases, if atoms are real and can bounce around. With Newton's laws, you can figure out how each particle is moving and then add up the results, deriving, for instance, the temperature of the gas. Practically speaking, that's a tall order. A liter of gas contains an astronomical number of molecules, about one followed by 22 zeros. Boltzmann figured out a better way, use probability and statistics. He came up with a mathematical framework to describe the motions of all the particles as a system by looking at the distributions of positions and velocities of typical And with this framework, Boltzmann devised the kinetic theory of gases and created a new field of physics called statistical mechanics. This all sounds straightforward. After all, in school today we learn the temperature is a measure of the average kinetic energy. What's more, Boltzmann's kinetic theory provided a microscopic interpretation of the second law of thermodynamics, which emerged out of studies of heat engines and the transport of energy. Broadly speaking, the second law deals with phenomena like why you can't unscramble an egg or unmix the cream from your coffee. It states that the evolution of an isolated system with a huge number of particles is irreversible. You cannot watch eggs unscramble themselves or the coffee go back to black. This directional flow of time in which entropy or disorder always increases stands in contrast to Newton's laws, which do not have a preferred temporal direction. Newton's equations work equally well if time flows forward or backward. So how do Newton's equations, which do not depend on time, lead to a theory of gases in which time flows one way? While the dynamics of the microscopic particles are time-invariant, Boltzmann asserted that the system as a whole is not. Imagine a box divided in half by a wall with a hole in its middle and that at the start most of the particles are on one side. After a time, both sides of the box will have roughly the same number of particles. You will not get a situation in which the particles are clustered like this or this. Boltzmann saw entropy as related to the number of microscopic states in thermal equilibrium that cannot be distinguished by macroscopic measurement. All these microscopic arrangements, for instance, look alike, even if the particles end up in different places. Mathematically, Boltzmann described entropy this way. The entropy S of a system is equal to a constant K times the natural logarithm of the number of microstates, W, that look alike macroscopically. As the gas in the box evens out, the more microstates there are, which means that S, the entropy, only gets bigger. In this way, entropy never decreases, which means time moves in one direction. Or consider the microscopic perspective. In a system packed with many particles moving around, the particles will collide, frequently. For many chaotic collisions, information about the particles' kinetic histories is lost. As a result, retracing their past is not possible, and exact reversals cannot happen. In this way, the past remains in the past. Boltzmann's statistical mechanics ultimately changed the way physicists viewed thermodynamics. And for mathematicians, it left open a huge challenge, enunciated by the influential German mathematician David Hilbert more than a hundred years ago. Hilbert put forth 23 problems in various areas of mathematics, and it's his sixth problem that ties in with Boltzmann's work. Hilbert asked if certain branches of physics can be axiomatized, the way mathematical proofs are. Those are first principles, assumptions we regard as basic truths, like that a number line goes to infinity. Specifically, he noted that, quote, Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua, end quote. Boltzmann's phrasing is formal, but he basically asked, can you mathematically connect the dots from the very small world of atoms to the very large world of gases? Boltzmann's equation would be the intermediate step between atoms and fluids, a bridge between the very small and the very large. With Boltzmann's view, we look at particles in aggregate, with a function that addresses the proportion of particles having a given position and velocity. This is called the probability distribution function, F. The Boltzmann equation expresses the fact that particles evolve from the combined effects of when they move freely, called transport, and when they collide. Transport refers to a particle's displacement, dx, while moving with velocity v during a brief instant, dt. In the absence of collisions, the evolution would be given by this partial differential equation relating the change in the function over time and space. Taking into account collisions between particles, however, dramatically complicates this equation. In general, the probability density function F does not contain enough information to describe the statistics of collisions. However, when there are a huge number of particles and their velocities at any given position are independent of one another, collisions can be taken into account by a collision integral, which in Boltzmann's theory is assumed to depend on the probability functions of both particles taken separately. The integral describes the change in the velocity distribution due to the collisions. The independence among particles results in what is called chaos in statistical physics. To describe the microscopic world using Boltzmann's kinetic theory, we have to check whether this chaos assumption is realistic. It should be a good approximation when the gas has a very low density. Indeed, a small particle obeying Newtonian dynamics should rarely collide with the same particles. In the 1970s, American mathematician Oscar Lanford proved that the Boltzmann equation is a good approximation of the microscopic dynamics when the number of particles tends to infinity. This holds for most initial configurations of the gas, but unfortunately, mathematicians have established it rigorously only for short times between collisions. More recent work by Laura Sarramone and her colleagues has cast new light on chaos. They showed how deviations from the Boltzmann equation are explained by correlations in the sequence of collisions, thereby providing a statistical description of the gas beyond the Boltzmann equation, including the prediction of rare events. Addressing the second part of Hilbert's challenge, connecting Boltzmann's equation to the world we see with our eyes, means asking what happens when we zoom out and look at the gas as a continuous medium. In this case, it is like the air around us, which can be described by a few observables like pressure, temperature, or wind velocity. Mathematically, fluids in our macroscopic world are described by the Navier-Stokes equations, and fluid limits refer to regimes when the collision process is much faster than the transport process so that the gas is locally at thermodynamic equilibrium. In a major achievement in 2004, François Gulsa and Sarramone connected kinetic theory with Navier-Stokes, assuming only that the gas has finite energy and entropy. While mathematicians have succeeded in tying Newtonian dynamics with Boltzmann's kinetic theory and Boltzmann's kinetic theory with macroscopic fluids, Hilbert's challenge to connect the dots from atoms to fluids, from the very small to the very large, has yet to be completely solved. The Boltzmann equation has to be shown to be valid for different collision times and other fluid limits. Still, Sarramone and her colleagues have taken major strides, connecting the microscopic, intermediate, and macroscopic levels to provide a big picture of a gas near thermal equilibrium. In recognition of her accomplishments, Sarramone has received several awards and honors, including the Fermat Prize, the Boucher Memorial Prize, and the Prize of the European Mathematical Society, as well as election to the French and European Academies of Science. In September 2021, she became the first female permanent professor at the Institut de Autitude Scientifique, or IHES, an independent research institute based near Paris. Boltzmann himself probably could not have guessed how his study of particles, entropy, and statistics would become so deeply influential. Today, whenever researchers gather in Vienna, many will take the trip to the city's central cemetery to see his final resting place, to remember the man who inspired generations of thought and research that continue to this day.