Non-dimensionalization
Non-dimensionalization¶
As with any computer simulation of a physical system, dimensionful quantities are represented numerically by non-dimensionalized counterparts, so it’s important to rescale quantities consistently.
Within open-Qmin, all energy densities are scaled in units of the magnitude of the first Landau-de Gennes bulk free energy coefficient:
This applies to the LdG bulk free energy coefficients themselves:
You may choose to provide a value of \(\tilde A\) besides \(-1\) using the --phaseConstantA or -a flag; indeed the default value is \(-0.172\). However, before beginning computation, open-Qmin rescales the entered \(B\) and \(C\) values by \(|A|\), so -a <A_value> -b <B_value> -c <C_value> is equivalent to -a -1 -b <B_value/|A_value|> -c <C_value/|A_value|>. No such automatic rescaling is performed for other values such as the elastic constants.
The elastic constants, which have dimensions of \( (\text{energy density}) \times (\text{length})^2 \), are non-dimensionalized using \(A\) and the lattice spacing \(\Delta x\):
Thus, in the one-elastic-constant approximation, for a nematic material with a given \(L_1\) and \(|A|\), the chosen value for \(\tilde L_1\) sets the dimensionful length that corresponds to the lattice spacing:
Generally, the dimensionful \(\Delta x\) should be slightly smaller than the material’s defect core size; larger values make it challenging to resolve defect cores in a finite difference approach.
For external fields, the dimensionful products \(\mu_0 |\bf H|^2 \) and \(\varepsilon_0 |\bf E|^2\) have units of energy density, so the non-dimensionalized versions of these products must be given in units of \(|A|\). One option is to set \(\mu_0=1\) and thus define your non-dimensionalized magnetic field as \(\tilde {\bf H} = {\bf H} \sqrt{\mu_0 / |A|} \), (and likewise for \(\tilde {\bf E} = {\bf E}\sqrt{\varepsilon_0/|A|}\)).