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This may be more of a mathematics question but it comes about in a physics context. I have a box of area $64cm \times 32cm$ and it is partitioned into two parts using a divider which can slide along one axis if hit by a ball.There are $N_{L}$ balls on the left side and $N_{R}$ on the right, all balls have radius $R=4$.

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My question is can I find out the probability of my slider being in a particular interval.

Originally I partitioned the box into an $8 \times 4$ grid and calculated the number of $N$ balls into $n$ boxes. However I want a continuous model and am a bit confused about how exactly I could go about this. I feel like I will need to integrate of the area available for any particular ball from $x=8$ to $x=56$, where x is the position of the slider.

Any help is greatly appreciated!

Sam
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  • The cursor can move between the positions determined by the most compact arrangement of balls on each side. My guess is that it is being determined by the column arrangement $4|4|4|...$ and the pyramidal arrangement $4|3|4|3|...$. (I guess you have only 1 layer, since you didn't provide a height for the box). But I do not understand the probability question, since you say nothing about ball movement ? – zwim Apr 22 '17 at 05:48
  • Respectfully, I don't think your problem is well posed. If you give me a deterministic model for how the balls interact with each other, the slider, and the wall and tell me the initial conditions, then I can tell you the position of the slider exactly for all points in time. And as the balls have an explicit volume which is not many orders of magnitude less than the dimensions of the box, it doesn't make sense to consider the problem statistical mechanically in the thermodynamic limit $N_L, N_R \to \infty$. – eepperly16 Apr 22 '17 at 06:04
  • No you're right its not very well written I'm terrible at explaining questions, however I was thinking that if I let $(x_i,y_i)$ be the position for the center of mass of the ith ball then I can calculate the microstates accessible which is proportional to the positional area of the phase-space $$ \int dx_1 dy_1 \int dx_2 dy_2... $$ – Sam Apr 22 '17 at 06:25

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