I have a 1D system containing $N$ particles having positions $\{x_1(t),x_2(t),\dots,x_N(t)\}$ in a box of size $L$ with periodic boundaries. The number of particles is conserved.

The dynamics of the particles are governed by some equations with the interaction strength between particles characterized by some control parameter $C$.

Two phases emerge as $C$ is varied. There is an "unordered" phase, where particles behave independently, then there is an "ordered" phase, where particles behave collectively as a result of the interactions.

Putting the $x_i$ on the horizontal axis and splaying out the $y$ coordinate to facilitate visualization, the small $C$ unordered phase appears like this: enter image description here while the large $C$ ordered phase develops multiple clustered domains enter image description here

I would like to quantify the amount of clustering with some order parameter $\Psi$. Were there only one cluster, the standard deviation of position would do a good job:

$$ \Psi = \sqrt {\langle x^2 \rangle - \langle x \rangle^2}.$$ For the unordered case it would be $L$ and for the ordered case it would be $l$ -- the size of the spatial cluster.

Does anyone have a recommendation of an order parameter to study clustering in the case with many clusters? Can I generalize from the standard deviation somehow? Is there an analogous problem in magnetism I could study?

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    $\begingroup$ Can you specify your system a bit more? What is on the vertical axis in the plot? Particle density? Is total number of particle conserved? $\endgroup$ – lcv Dec 18 '19 at 14:40
  • $\begingroup$ @Icv thanks, I added some clarification. Please let me know if you have more questions. The vertical axis just facilitates visualization as the particles exist on the line (horizontal axis) having some positions $x_i\in[0,L]$. There are periodic boundary conditions and the number of particles is conserved. $\endgroup$ – kevinkayaks Dec 18 '19 at 19:04
  • $\begingroup$ Yes, entropic uncertainty accommodates clustering around multiple centers. Please use "disordered" rather than "unordered". $\endgroup$ – Cosmas Zachos Dec 18 '19 at 19:37

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