Now, three mathematicians have finally provided such a result. Their work not only represents great progress in the Hilbert program, but also draws questions about the irreversible nature of the time.
“It’s a good job,” he said Gregory Falkovichphysical at the Weizmann Institute of Science. “A tour de force.”
Under the mesoscope
Consider a gas whose particles are widespread. There are many ways in which a physique could model it.
On a microscopic level, the gas is composed of individual molecules that act as a billiard balls, which move in space according to the laws of the 350 -year movement of Isaac Newton. This model of gas behavior is called hard -ball particles.
Now enlarge a little. At this new “Mesoscopic” scale, your visual field contains too many molecules to keep track of individuals. Instead, you will modulate the gas using an equation that the physicists James Clerk Maxwell and Ludwig Boltzmann developed at the end of the nineteenth century. Called the equation of Boltzmann, it describes the probable behavior of the gas molecules, telling you how many particles you can expect to find in different positions that move at different speeds. This gas model allows physicists to study how the air moves on small stairs, for example how it could flow around a space shuttle.
Zoom again and it is no longer possible to say that the gas consists of individual particles. It behaves like a continuous substance. To model this macroscopic-quiet behavior, the gas and speed with which you are moving at any time in the space-must be needed for another series of equations, called the Navier-Stakes equations.
Physics consider these three different models of gas behavior compatible; They are simply different lenses to understand the same thing. But the mathematicians who hoped to contribute to the sixth Hilbert problem wanted to prove it rigorously. They needed to demonstrate that the model of individual Newton particles gives rise to the statistical description of Boltzmann and that the Boltzmann equation in turn gives rise to the Navier-Stakes equations.
Mathematics have had some success with the second step, showing that it is possible to derive a macroscopic model of a gas from a mesoscopic in various contexts. But they were unable to solve the first step, leaving the logic chain incomplete.
Now it has changed. In a series of articles, mathematics Yu Deng, Zaher HaniAND Xiao but He demonstrated the most hard microscopic passage For a gas in one of these settings, complete the chain For the first time. The result and techniques that made it possible are “movements of the paradigm”, he said Yan Guo of Brown University.
Declaration of independence
Boltzmann could already demonstrate that the laws of Newton’s movement give rise to his mesoscopic equation, as long as a crucial prerequisite is true: that the particles in the gas move more or less independently of each other. That is, it must be very rare that a particular couple of molecules collided with each other several times.
But Boltzmann was unable to definitively demonstrate that this hypothesis was true. “What could not do, of course, is to demonstrate the theorems about it,” he said Sergio Simonella of the University of Sapienza in Rome. “At the time there was no structure, there were no tools.”
After all, there are many ways in which a collection of particles could clash and remember. “Get only this huge explosion of possible directions in which they can go,” said Levermore, making it a “nightmare” to actually demonstrate that the scenarios involving as many memories are as rare as Boltzmann needed to be.
In 1975, a mathematician named Oscar Lanford managed to demonstrate thisBut only for extremely short periods of time. (The exact time depends on the initial state of the gas, but is lower than the beating of an eye, according to Simonella.) So the test broke; Before most particles had the opportunity to clash once, Lanford could no longer guarantee that memories would remain a rare event.