Random-matrix theory of parametric correlations in the spectra of disordered metals and chaotic billiards

We discuss a Dyson's Brownian-motion model for the evolution of an ensemble of Hermitian matrices as a function of an external fictitious time to find the distribution of energy eigenvalues for one-dimensional Brownian motion of N classical particles merged in a thermal viscous fluid and interacting through a logarithmic Coulomb potential.

A Fokker-Planck equation for the evolution of the distribution function is solved to yield the correlation of level densities at different energies and different parameter values. An approximate solution is obtained by asymptotic expansion and an exact solution by mapping onto a free-fermion model passing the Fokker-Planck equation onto a Schrödinger equation in the Gaussian unitary ensemble.