An increasing number of low carrier density materials exhibit a surprisingly large transport mean free path due to inefficient momentum relaxation. Consequently, charge transport in these systems is markedly non-ohmic but rather ballistic or hydrodynamic, features which can be explored by driving current through narrow channels. Using a kinetic equation approach we theoretically investigate how a non-quantizing magnetic field discerns ballistic and hydrodynamic transport, in particular in the spatial dependence of the transverse electric field, $E_y$: We find that $E_y$ is locally enhanced when the flow exhibits a sharp directional anisotropy in the non-equilibrium density. As a consequence, at weak magnetic fields, the curvature of $E_y$ has opposite signs in the ballistic and hydrodynamic regimes. Moreover, we find a robust signature of the onset of non-local correlations in the form of distinctive peaks of the transverse field, which are accessible by local measurements. Our results demonstrate that a purely hydrodynamic approach is insufficient in the Gurzhi regime once a magnetic field is introduced.
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