Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 3 / d / Solution

The result is a fluid Eulerian perturbation generated by translating the entire equilibrium phase-space distribution function through ${\xi }_x$:Here $\Delta$ follows the displaced flux tube. Magnetic flux freezing carries the tube and its particles across the equilibrium pressure gradient. To leading order the tube changes neither its field strength nor its length by enough to alter the particle phase-space distribution function: these changes are smaller by plasma beta or $L_p/R$. Parallel streaming remains within the same displaced tube, on which the original equilibrium phase-space distribution function was uniform. It therefore cannot erase the displacement-generated contrast between adjacent tubes. This is why the term $v_{\parallel }\delta b_x{\partial }_x{f}_{0\alpha }$ in c exactly cancels the apparent parallel relaxation of $-{\xi }_x{\partial }_x{f}_{0\alpha }$. No collisional restoration of a Maxwell-Boltzmann distribution is needed.