# Estimate the difference between two sets of atoms

I've been working on amorphous structures derived from a crystalline one (using MD) containing $N$ atoms. I want to prove that these structures are different and to quantify their "differentness". One could think of simply displaying the structures, but:

• I actually have 20 of them and I don't want to show them 2 by 2 (poor information density)
• This does not quantify these differences (the "differentness" metric)

I thought about calculating the RMS of the corresponding atom pairs distances using two different structures, but:

• I have to associate pairs so that the RMS is minimized
• A single atom cannot form two pairs
• A rotation and a shift (of unknown magnitude) should be allowed due to the periodic and symmetric nature of the reference cell to help minimizing the RMS

I've been working on this for a week with mitigated success: the algorithm is slow and incomplete, the execution time is too slow but it kinda works. But the algorithm(/method?) is too sensitive to outliers: when I compare 3 structures, 2 close from each other (visually) and one more different, I get almost the same RMS (due to outliers).

I think now that the RMS might not be a good idea after all, what figure/graph would you use to quantify the differences between these structures?

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 What are you trying to do??? Why do you want to compute this??? For physical objects, if you have two crystals shifted by half a unit cell, they are essentially the same object, even though the RMS of distances isn't 0. If you have two boxes filled with nitrogen gas, you want to see whether they are at equilibrium, and whether they have equal temperatures. If you have $N$ abstract particles and you're comparing them, you want to ask this in math.SE or stats.SE or cs.SE. There are a number of statistical distances that answer this problem. The best answer depends on why you want to do this. – Peter Shor Mar 16 at 15:37 I have two amorphous structures derived from a crystal. I want to be able to tell that these are different. The problem is that showing the two structures does not work well. If I had crystals shifted by half a unit cell my algorithm returns 0 since it chooses the atom pairs to minimize the RMS (and by the way choose rotation angles and a shift to do that efficiently). I've been on it for a week... – Gael Mar 16 at 15:48 If you don't get any answers in a few days, I would ask this question in cs.se or cstheory.se. It's similar to some problems computer scientists have looked at, and it seems that computing the RMS (or whatever metric you want) is one of the biggest bottlenecks. But stats.SE might be another good place to ask it, if you don't get any satisfactory answers from cs. – Peter Shor Mar 16 at 15:58 Also, I think in this case the better stackexchange etiquette would be to revise your question to include the motivation in the question, rather than just leaving it in the comments. – Peter Shor Mar 16 at 16:00 Thanks Peter. I wanted to keep it short but I'll update my question... Actually, I've been to maths.SE then stats.SE then here as I was building my question. But double posting is not well seen here and I thought I would begin with physics.SE. I did not think about cs.SE though. and I should. – Gael Mar 16 at 16:14
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