  Using the Fokker-Planck equation describing the evolution of the transmission
eigenvalues for Dyson's Brownian motion ensemble, we calculate the
magnetoconductance of a ballistic chaotic dot in in the crossover regime from
the orthogonal to the unitary symmetry. The correlation functions of the
transmission eigenvalues are expressed in terms of quaternion determinants for
arbitrary number $N$ of scattering channels. The corresponding average,
variance and autocorrelation function of the magnetoconductance are given as a
function of the Brownian motion time $t$. A microscopic derivation of this
$S$-Brownian motion approach is discussed and $t$ is related to the applied
flux. This exactly solvable random matrix model yields the right expression for
the suppression of the weak localization corrections in the large $N$-limit and
for small applied fluxes. An appropriate rescaling of $t$ could extend its
validity to larger magnetic fluxes for the averages, but not for the
correlation functions.
