Now, three mathematicians have finally provided such a result. Their work not only represents a major advance in Hilbert’s program, but also taps into questions about the irreversible nature of time.
“It’s a beautiful work,” said Gregory Falkovich, a physicist at the Weizmann Institute of Science. “A tour de force.”
Under the Mesoscope
Consider a gas whose particles are very spread out. There are many ways a physicist might model it.
At a microscopic level, the gas is composed of individual molecules that act like billiard balls, moving through space according to Isaac Newton’s 350-year-old laws of motion. This model of the gas’s behavior is called the hard-sphere particle system.
Now zoom out a bit. At this new “mesoscopic” scale, your field of vision encompasses too many molecules to individually track. Instead, you’ll model the gas using an equation that the physicists James Clerk Maxwell and Ludwig Boltzmann developed in the late 19th century. Called the Boltzmann equation, it describes the likely behavior of the gas’s molecules, telling you how many particles you can expect to find at different locations moving at different speeds. This model of the gas lets physicists study how air moves at small scales—for instance, how it might flow around a space shuttle.
“What mathematicians do to physicists is they wake us up.” Gregory Falkovich
Zoom out again, and you can no longer tell that the gas is made up of individual particles. It acts like one continuous substance. To model this macroscopic behavior—how dense the gas is and how fast it’s moving at any point in space—you’ll need yet another set of equations, called the Navier-Stokes equations.
Physicists view these three different models of the gas’s behavior as compatible; they’re simply different lenses for understanding the same thing. But mathematicians hoping to contribute to Hilbert’s sixth problem wanted to prove that rigorously. They needed to show that Newton’s model of individual particles gives rise to Boltzmann’s statistical description, and that Boltzmann’s equation in turn gives rise to the Navier-Stokes equations.
Mathematicians have had some success with the second step, proving that it’s possible to derive a macroscopic model of a gas from a mesoscopic one in various settings. But they couldn’t resolve the first step, leaving the chain of logic incomplete.
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