During today’s special session on probabilistic methods in partial differential equations, I had the pleasure of hearing Zaher Hani of the University of Michigan speak on his recent work on the wave kinetic equation.

In the early twentieth century physicists began formulating the first principles of quantum mechanics. One of their key realizations is that waves are fundamental physical units in the same way particles are. This prompted scientists to start constructing a wave system analogy to Boltzmann’s theory of statistical mechanics for particles.

For those not familiar, Boltzmann statistics describes the average statistical distribution of non-interacting particles in a large system. Boltzmann’s equation describes how this distribution fluctuates with time. The question facing physicists was whether a similar framework could be derived for waves that satisfy the nonlinear Schrodinger equation. It is very difficult to understand the behaviour of these waves, says Hani, because there are so many possible solutions. But it turns out that there is a wave-analog of Boltzmann’s equation, the wave kinetic equation.

Until recently, though, it was not known how to derive this equation in a mathematically rigorous way. Hani and collaborators showed in 2019 that the wave kinetic equation held for short periods of time that depended on the details of the system, but it was suspected that a more universal bound on the time period should hold. Hani, together with Yu Deng of the University of Southern California, has recently improved the bound for certain types of waves and time domains.

It was definitely interesting to spend half an hour today learning about the comparison between wave statistics and Boltzmann statistics. The work involved not only differential equation techniques, but also Feynman diagrams and number theory—a fascinating illustration of how seemingly distinct areas of mathematics can intersect.

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