# Order parameter to quantify clustering?

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: while the large $$C$$ ordered phase develops multiple clustered domains

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?

• 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? – lcv Dec 18 '19 at 14:40
• @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. – kevinkayaks Dec 18 '19 at 19:04
• Yes, entropic uncertainty accommodates clustering around multiple centers. Please use "disordered" rather than "unordered". – Cosmas Zachos Dec 18 '19 at 19:37