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Danny-Boy

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I have a somewhat strange/vague question and hoped that someone here could point me in the right direction to find a solution.

I have a system comprising of a number of identical particles (i.e., I can't differentiate between them) in 2-D space. I have numerous observations of their spatial distribution at discrete points in time. Is there any way of deducing from these observations any information at all about the dynamics of the system? (I suspect I'm barking up the wrong tree, but I read that Feynman path integrals take into account all possible paths a particle can take, but if I cannot differentiate one particle from another presumably such an approach is impossible.)

Apologies if this inappropriate for this forum!