# 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