Multi-mode Pom-Pom Model

Summary

Multi-mode PomPom Model based on [3]:

\[\begin{split}\boldsymbol \sigma &= 3 \sum_{i=1}^n G_i \lambda_i^2(t) \boldsymbol S_i (t),\\ \boldsymbol S_i &= \dfrac{\boldsymbol A_i } {\mathrm{Tr} \boldsymbol A_i}\\ \dfrac {\mathrm D \boldsymbol A_i} {\mathrm D t} &= \boldsymbol \kappa \cdot \boldsymbol A_i + \boldsymbol A_i\cdot \boldsymbol \kappa ^T - \dfrac {1} {\tau_{\mathrm b, i}} (\boldsymbol A_i - \boldsymbol I), \\ \dfrac {\mathrm D \lambda_i} {\mathrm D t} &= \lambda_i (\boldsymbol \kappa : \boldsymbol S_i) - \dfrac {1} {\tau_{\mathrm s, i}} (\lambda_i - 1) \exp\left( \nu^* (\lambda_i - 1) \right),\end{split}\]
where, for each mode \(i\):
  • \(G_i\): weight of mode \(i\)

  • \(\tau_{\mathrm b, i}\): backbone orientation relaxation time of mode \(i\)

  • \(\tau_{\mathrm s, i}\): backbone stretch relaxation time of mode \(i\)

  • \(\nu_i^* = \dfrac{2}{q_i - 1}\)

  • \(q_i\): the number of dangling arms of each mode

  • Parameters
    • q_i \(\equiv q_i\): the number of dangling arms of each mode

    • ratio_i \(\equiv \dfrac{\tau_{\mathrm b, i}}{\tau_{\mathrm s, i}}\):

    the ratio of orientation to stretch relaxation times of each mode