  One special interest for the industrial development of Hall thruster is
characterizing the anomalous cross-field electron transport observed after the
channel exit. Since the ionization efficiency is more than 90%, the neutral
atom density in that domain is so low that the electron collisions cannot
explain the high electron flux observed experimentally. Indeed this is 100
times higher than the collisional transport. In Hall thruster geometry, as ions
are not magnetized the electric and magnetic field configuration creates a huge
difference in drift velocity between electrons and ions, which generates
electron cyclotron drift instability or $\vec E \times \vec B$ electron drift
instability. Here we are focusing on collision-less chaotic transport of
electrons by those unstable modes generated by $\vec E \times \vec B$ drift
instability. We found that in presence of these electrostatic modes electron
dynamics become chaotic. They gain energy from the background waves which
increases electron temperature along perpendicular direction by a significant
amount, $T_{\rm perp}/T_{\rm parallel}\sim 4$, and a significant amount of
crossfield electron transport is observed along the axial direction.
