I am building a computational model of ellipsoidal cell network formation and I would like to use a particle order parameter to study my model's behavior. I have come across this article Steinhardt, Nelson, Ronchetti, PRB (1983), that was apparently the one that started everything, but the math is quite exotic for me. I have found that the formula:
$$ S = 1/3<cos^2θ - 1> $$
is the order parameter for a 3-D system. This is convenient as S = 0 means no order, while S = 1 means complete alignment. A colleague at the univ has mentioned that for a 2-D particle system the previous equation for S becomes:
$$ S = cos2θ $$
but I don't think this can be right, as in the following example: 2 particles have a 90 degree angle between them, so they are completely unaligned.
cos2θ = cos2*90 = -1 which would imply they are completely aligned, which is wrong. Also, the range is [-1:1] instead of 0 < S < 1.
I would really appreciate it, if anyone could point me to the right direction.