  The classical Toda flow is a well-known integrable Hamiltonian system that
diagonalizes matrices. By keeping track of the distribution of entries and
precise scattering asymptotics, one can exhibit matrix models for log-gases on
the real line. These types of scattering asymptotics date back to fundamental
work of Moser.
  More precisely, using the classical Toda flow acting on symmetric real
tridiagonal matrices, we give a "symplectic" proof of the fact that the
Dumitriu-Edelman tridiagonal model has a spectrum following the Gaussian
$\beta$-ensemble.
