a, Curvature of ballistic current profile versus l MR /W. The analytic solution (blue curve) is based on de Jong and Molenkamp2 and the red curve is a Monte Carlo billiard ball simulation result. The two methods agree perfectly until l MR exceeds the channel length used in the billiard ball simulation, beyond which the solutions begin to deviate. b, Curvature κ of E y , as in Fig. 4b, calculated by Boltzmann simulation (see Methods), as a function of l ee /W and l MR /W for W/R c = 1.3. Curvature is calculated over the centre of the channel. Green lines divide the panel into flow regimes as in Fig. 4b. c, Curvature κ of j x , extracted from the same simulation as a. For j x , the curvature in the ballistic regime is essentially constant at κ ≈ 0.31 and so the curvature of j x is less discriminating between the hydrodynamic and ballistic regimes than the curvature of E y , which becomes negative. In the other regimes, the curvatures of j x and E y are very similar, and the differences between them diminish as each of the length scales becomes much smaller than W. In the hydrodynamic regime the curvature saturates on the maximal possible value for a strictly parabolic profile, and in the porous regime it follows the length scale \({D}_{

u }=\frac{1}{2}\sqrt{{l}_{{\rm{MR}}}{l}_{{\rm{ee}}}}\) as expected.