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John Darby
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I assume classical mechanics applies to this problem.

If by the spheres "do not interact with each other" you mean no collisions, then only those spheres initially headed in a direction that hit the circle are counted.

I assume by the spheres "do not interact with each other" you mean they can only elastically collide. You would have to do a simulation stepping thorough time to calculate collisions and the final velocity (vector) of each sphere considering oblique elastic collisions. As an approximation you can treat the particles as points of equal mass that only change direction (not speed) after an elastic collision. Particles can both enter and exit the circle. If you remove the wall, particles can escape the box, but if the wall remains particles scatter off the wall. Over a long time (steady state), if the wall is removed and there are no collisions outside the wall (no reflection of particles back into the box) all particles eventually escape the box. Over a long time (steady state), if the wall is not removed, the fraction of particles inside the box is equal to the ratio of the circle area to the box area.

There may be a good answer from statistical mechanics, but I am not an expert in that area. Perhaps others can address this as @Carlos Gauss suggests in an earlier comment.

I assume classical mechanics applies to this problem.

If by the spheres "do not interact with each other" you mean no collisions, then only those spheres initially headed in a direction that hit the circle are counted.

I assume by the spheres "do not interact with each other" you mean they can only elastically collide. You would have to do a simulation stepping thorough time to calculate collisions and the final velocity (vector) of each sphere considering oblique elastic collisions. Particles can both enter and exit the circle. If you remove the wall, particles can escape the box, but if the wall remains particles scatter off the wall. Over a long time (steady state), if the wall is removed and there are no collisions outside the wall (no reflection of particles back into the box) all particles eventually escape the box. Over a long time (steady state), if the wall is not removed, the fraction of particles inside the box is equal to the ratio of the circle area to the box area.

There may be a good answer from statistical mechanics, but I am not an expert in that area. Perhaps others can address this as @Carlos Gauss suggests in an earlier comment.

I assume classical mechanics applies to this problem.

If by the spheres "do not interact with each other" you mean no collisions, then only those spheres initially headed in a direction that hit the circle are counted.

I assume by the spheres "do not interact with each other" you mean they can only elastically collide. You would have to do a simulation stepping thorough time to calculate collisions and the final velocity (vector) of each sphere considering oblique elastic collisions. As an approximation you can treat the particles as points of equal mass that only change direction (not speed) after an elastic collision. Particles can both enter and exit the circle. If you remove the wall, particles can escape the box, but if the wall remains particles scatter off the wall. Over a long time (steady state), if the wall is removed and there are no collisions outside the wall (no reflection of particles back into the box) all particles eventually escape the box. Over a long time (steady state), if the wall is not removed, the fraction of particles inside the box is equal to the ratio of the circle area to the box area.

There may be a good answer from statistical mechanics, but I am not an expert in that area. Perhaps others can address this as @Carlos Gauss suggests in an earlier comment.

Source Link
John Darby
  • 9.5k
  • 2
  • 15
  • 36

I assume classical mechanics applies to this problem.

If by the spheres "do not interact with each other" you mean no collisions, then only those spheres initially headed in a direction that hit the circle are counted.

I assume by the spheres "do not interact with each other" you mean they can only elastically collide. You would have to do a simulation stepping thorough time to calculate collisions and the final velocity (vector) of each sphere considering oblique elastic collisions. Particles can both enter and exit the circle. If you remove the wall, particles can escape the box, but if the wall remains particles scatter off the wall. Over a long time (steady state), if the wall is removed and there are no collisions outside the wall (no reflection of particles back into the box) all particles eventually escape the box. Over a long time (steady state), if the wall is not removed, the fraction of particles inside the box is equal to the ratio of the circle area to the box area.

There may be a good answer from statistical mechanics, but I am not an expert in that area. Perhaps others can address this as @Carlos Gauss suggests in an earlier comment.