- #1
birulami
- 155
- 0
Suppose I have N ideal particles in an enclosure, be it a ball or a cube or some other form. The particles shall bounce off the walls of the enclosure and against each other without losing speed. The velocity of each particle [tex]i[/tex] shall be such that it fullfills [tex]|v_i|=\rho[/tex], where [tex]\rho[/tex] is constant, i.e. the speed is always the same, but of course the direction in 3D differs.
Further, on the average, the whole ensemble of points shall not move, i.e.
(*) [tex]\sum_{i=1}^N v_i = 0[/tex]
or at least the sum is very close to zero.
Apart from this, the velocities' directions shall be completely random. What exactly this would mean may need to be further defined.
My questions are:
1) How can I estimate [tex]\frac{1}{N^2}\sum_{i<j} (v_i-v_j)^2[/tex]?
2) Is the "boxed" condition used in the derivation or does (*) contain all we need?
Maybe this is not really a physics question, because the setup is too idealised, but I assume that is still closer to physics than to pure math.
Harald.
Further, on the average, the whole ensemble of points shall not move, i.e.
(*) [tex]\sum_{i=1}^N v_i = 0[/tex]
or at least the sum is very close to zero.
Apart from this, the velocities' directions shall be completely random. What exactly this would mean may need to be further defined.
My questions are:
1) How can I estimate [tex]\frac{1}{N^2}\sum_{i<j} (v_i-v_j)^2[/tex]?
2) Is the "boxed" condition used in the derivation or does (*) contain all we need?
Maybe this is not really a physics question, because the setup is too idealised, but I assume that is still closer to physics than to pure math.
Harald.