Inconsistency of magnetic-moment conservation with entropy increase in collisionless shocks

Collisionless shocks are multiscale objects. Energetic ion distributions are gyrotropic at sufficiently large distances upstream and downstream of the shock transition while at the transition itself the ion dynamics is significantly gyrophase dependent. Magnetic-moment conservation of an ion is widely used as a viable approximation during the shock crossing. It is shown that this approximation is inconsistent with the required entropy increase due to the loss of the gyrophase information.