0
$\begingroup$

If I have a differential equation for the order parameter of a uniaxial nematic, like

$\frac{\partial S_{\alpha \beta}}{\partial t} = f(S_{\alpha \beta})$,

for

$S_{\alpha \beta} = S(n_{\alpha}n_{\beta} - \frac{1}{3}\delta_{\alpha \beta})$ ,

is it possible to convert this to a differential equation for $n_{\alpha}$? Meaning, can I perform some transformation to get an equation of the form

$\frac{\partial n_{\alpha}}{\partial t} = g(n_{\alpha})$.

If not, is there a way to insert angles $\phi$ and $\theta$ into a 3-D $S_{\alpha \beta}$, where $\phi$ is in one direction, and $\alpha$ the perpendicular direction (imagine essentially the angles of spherical coordinates)? These angles are the angles that define the director $n_{\alpha}$, as a unit vector of constant magnitude 1.

$\endgroup$

1 Answer 1

0
$\begingroup$

For future physicists wondering the same; you can in fact do so. You need only to dot in a power of the director on your equation! The tensor $\mathsf{S}$ dotted with $n$ results in a power of $n$, so all works out in the end.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.