Further Developments of Sinai's Ideas: The Boltzmann-Sinai Hypothesis

Abstract

In 1963 Ya. G. Sinai formulated a modern version of Boltzmann's ergodic hypothesis, what we now call the ``Boltzmann-Sinai Ergodic Hypothesis'': The billiard system of $N$ ($N\ge 2$) hard balls of unit mass moving on the flat torus $\mathbb{T}^\nu=\mathbb{R}^\nu/\mathbb{Z}^\nu$ ($\nu\ge 2$) is ergodic after we make the standard reductions by fixing the values of trivial invariant quantities. It took fifty years and the efforts of several people, including Sinai himself, until this conjecture was finally proved. In this short survey we provide a quick review of the closing part of this process, by showing how Sinai's original ideas developed further between 2000 and 2013, eventually leading the proof of the conjecture.