lew@ihuxr.UUCP (09/23/83)
Alan's mistake in his simulation set-up was in picking a single point to aim the particles from. An infinitely distant point will generate parallel trajectories at a fixed angle. A closer point generates different angles, but they are a function of where the trajectory strikes the hole. WHERE the particle hits is expressed by the impact parameter, which is the distance of closest approach of the incoming trajectory (projected straight through) to some fixed point, usually chosen at a center of symmetry. To correctly run the simulation, you need to randomly choose (with proper weighting) the impact parameter, and INDEPENDENTLY randomize the angle of incidence. Bill Hughes discussion of the trapezoidal hole problem was correct, but I have a minor quibble. I stated before that reversibility of paths didn't prove, in itself, the symmetry of cross section. Here is a counterexample. Imagine a hole which "refracts" the particles according to Snell's Law. ( sin (r) = n*sin(i) ) On one side, the particles with an angle of incidence greater than arcsin(1/n) will be internally reflected. The total cross section from that side is (normalized by the hole area): integral from 0 to arcsin(1/n) of cos(x) * sin(x) * dx The cosine term expresses the foreshortening of the hole and the sine term expresses the increasing range of angles as the angle of incidence increases. The total cross section is 1/n^2 from this side and 1 from the other side. Something like this situation is realized if the two sides of the box are at different potentials. Particles of a given speed will obey Snell's Law when they change potential. Superficially, this is just like the trapezoidal hole, which creates a shadow region on the wide side. However, in this case, the effective area of the hole for angles near perpendicular is INCREASED from the wide side. This is due to particles glancing off the near side of the hole and going through. The mirror argument is air tight I think. This is because every small bundle of incident angles maps to an equal sized bundle of reflected angles. You just have to break the incident paths into bundles which strike the same surfaces. This bundling is very important. You can't add up rays to make solid angles or to cover areas. It is widely known that normal rays will heat a surface faster than oblique rays, but we can nevertheless form a 1 to 1 correspondence between the normal rays and the oblique rays. Every POINT on the surface is struck by one normal ray and one oblique ray. What about curved surfaces? We can approximate them by a lot of little flat surfaces and take the limit, but the general result is elegantly described by Liouville's Theorem. Note that the total cross section is found by integrating over angles AND area. This is the same as integrating over velocity space and configuration space. (The angles express direction of motion, hence position in velocity space.) Liouvilles Theorem says that (under certain conditions) the phase space density of a distribution of particles is constant. I like the example of constant surface luminosity. When you view a luminous object from a distance, you are sampling a small volume of phase space. Your eye subtends a small area and maps a small solid angle onto a spot on the retina. Constant luminosity per unit solid angle per unit area means that the brightness of the image on your retina is constant as you move away from the object. The 1/r^2 intensity law results because of the reduction in TOTAL solid angle subtended by the object as distance increases. Lew Mammel, Jr. ihuxr!lew