Let us sum up all the hints. The magnetic field is so strong (as compared to the thermal energy) that perpendicular-to-the-magnetic-field motion is not excited, and hence, the motion of electrons is one-dimensional. The shock moves slowly, therefore the adiabatic invariant is conserved, except when phase trajectory is crossing a separatrix, and adding (deducting) thereby the surface area of the newly added (deducted) loop in the phase space. Initially, the phase trajectory is a rectangle, constrained between the wall and the shock; it becomes shorter and higher in time until the momentum has reached a high-enough value to overcome the shock’s barrier. At this point, a new loop of space trajectory is added to the existing loop, so that the new trajectory is a union of two rectangles. Further, the adiabatic invariant is conserved until the end of the process when the space trajectory has again taken the shape of a single rectangle. The pressure on the wall can be calculated if we know the probability distribution of the speeds at the end, which is the same as knowing the probability distribution of the surface areas of the space trajectories. So we need to figure out the probability distribution of the surface area of the newly added loop during the separatrix-crossing. This can be done by noticing that the speed excess over the minimally required value when an electron “climbs”, for the first time, onto the shock barrier, is evenly distributed between 0 and 2u (because the electron’s speed is increased by 2u during each period).
This is the last hint. The final deadline for Problem 4 is 31st May, 23:59 GMT.
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