Now, three mathematicians finally provide such a result. Their work not only represents a significant advancement in Hilbert’s plan, but also involves issues concerning the irreversible nature of time.
“It’s a beautiful piece,” said Gregory Falkovich, a physicist at the Weizmann Academy of Sciences. “Tour.”
Under medium
Consider a gas with very particles distributed. Physicists can model it in a number of ways.
At the microscopic level, gas consists of single molecules that act like billiards, moving through space according to Isaac Newton’s 350-year-old law of motion. This model of gas behavior is called a hard ball particle system.
Now zoom in a little bit. On this new “Media” scale, your field of vision covers too many molecules that can’t be tracked individually. Instead, you will model the gas using equations developed in the late 19th century by physicists James Clerk Maxwell and Ludwig Boltzmann. It is called the Boltzmann equation, which describes the possible behavior of gas molecules, telling you that you can expect particles that move at different velocities at different locations. This gas model allows physicists to study how air moves on small scales, for example, how it flows around the space shuttle.
Shrink again, you can no longer say that the gas is made up of individual particles. It acts like a continuous substance. To model this macro behavior – how dense the gas is and how fast it moves anywhere in space – you need another set of equations, called the Navier-Stokes equation.
Physicists believe that these three different models of gas behavior are compatible. They are just different shots of understanding the same thing. But mathematicians who hope to contribute to Hilbert’s sixth problem want to rigorously prove this. They need to prove that Newton’s single particle model produced Bolzman’s statistical description, and Bolzman’s equation produced the Navier-Stokes equation.
Mathematicians succeeded in the second step, proving that macroscopic models can be obtained from gases with mesoscopic content under various circumstances. But they cannot solve the first step, making the logical chain incomplete.
Now it has changed. In a series of papers, mathematicians Yu Deng, Zaher Hani and Xiao Ma demonstrated the harder microscopic to moderate mirroring steps in gases in one of the environments, which is the first time the chain is completed. Yan Guo of Brown University said the result and the technology that made it possible was a “paradigm shift.”
Declaration of Independence
Bowlzmann can already show that Newton’s law of motion would cause his mesoscopic equations, as long as one key assumption is correct: the particles in the gas move more or less independently of each other. That is, it must be very rare for a pair of molecules to collide with each other multiple times.
But Balzmann was unable to confirm that this assumption was correct. “Of course, the fact that he couldn’t do proves this,” said Sergio Simonella of the University of Sapinza, Rome. “There was no structure, there was no tools at the time.”
After all, there are many ways to bump into particles and rotting particles. “You only get a huge explosion in the possible directions they could go,” Levermore said, “It’s a “nightmare” that actually proves that scenes involving many memories are as rare as Boltzmann needs them.
In 1975, a mathematician named Oscar Lanford managed to prove this, but only for a short period of time. (The exact time depends on the initial state of the gas, but it is smaller than the blink of an Simonella.) Then it proves to be ruptured. Until most particles have a chance or even a collision, Langford can no longer guarantee that memories will be rare.
