  The density of the transmission eigenvalues of Pb nano-contacts has been
estimated recently in mechanically controllable break-junction experiments.
Motivated by these experimental analyses, here we study the evolution of the
density of the transmission eigenvalues with the disorder strength and the
number of channels supported by the ballistic constriction of a quantum point
contact in the framework of the Dorokhov-Mello-Pereyra-Kumar equation. We find
that the transmission density evolves rapidly into the density in the diffusive
metallic regime as the number of channels $N_c$ of the constriction increase.
Therefore, the transmission density distribution for a few $N_c$ channels comes
close to the known bimodal density distribution in the metallic limit. This is
in agreement with the experimental statistical-studies in Pb nano-contacts. For
the two analyzed cases, we show that the experimental densities are seen to be
well described by the corresponding theoretical results.
