# Order parameters

An order parameter is some observable physical quantity that is able to distinguish between two distinct phases. The choice of order parameter is not necessarily unique.

## Solid-liquid transition

Possible choices:

• Fourier transform of the density
• Shear modulus

## Isotropic-nematic transition

The uniaxial order parameter is zero for an isotropic fluid and one for a perfectly aligned system. First one calculates a director vector [1]

${\displaystyle Q_{\alpha \beta }={\frac {1}{N}}\sum _{j=1}^{N}\left({\frac {3}{2}}{\hat {e}}_{j\alpha }{\hat {e}}_{j\beta }-{\frac {1}{2}}\delta _{\alpha \beta }\right),~~~~~\alpha ,\beta =x,y,z,}$

where ${\displaystyle Q}$ is a second rank tensor, ${\displaystyle {\hat {e}}_{j}}$ is a unit vector along the molecular long axis, and ${\displaystyle \delta _{\alpha \beta }}$ is the Kronecker delta. Diagonalisation of this tensor gives three eigenvalues ${\displaystyle \lambda _{+}}$, ${\displaystyle \lambda _{0}}$ and ${\displaystyle \lambda _{-}}$, and ${\displaystyle n}$ is the eigenvector associated with the largest eigenvalue (${\displaystyle \lambda _{+}}$). From this director vector the nematic order parameter is calculated from [2]

${\displaystyle S_{2}={\frac {d\langle \cos ^{2}\theta \rangle -1}{d-1}}}$

where d is the dimensionality of the system.

i.e. in three dimensions [3]

${\displaystyle S_{2}=\lambda _{+}=\langle P_{2}(n\cdot e)\rangle =\langle P_{2}(\cos \theta )\rangle =\langle {\frac {3}{2}}\cos ^{2}\theta -{\frac {1}{2}}\rangle }$

where ${\displaystyle S_{2}}$ is known as the uniaxial order parameter. Here ${\displaystyle P_{2}}$ is the second order Legendre polynomial, ${\displaystyle \theta }$ is the angle between a molecular axes and the director ${\displaystyle n}$, and the angle brackets indicate an ensemble average.