behavior [@Glenn]. The microscopic density of segment orientation and position of $n$ wormlike polymer chains is defined as $\hat{\rho} \left({\bf r},{\bf u} \right)\equiv\sum_{j=1}^{n}\int_{0}^{L_{C}}\mathrm{d}s \; \delta \left({ \bf r}-  {\bf r}_{j}\left(s \right) \right) \delta \left({ \bf u}-  {\bf u}_{j}\left(s \right) \right)$. The arc length of the polymer is defined by $s\in \left[0,L_{C} \right]$ where $L_{C}$ is the contour length of the polymer. The configuration of the chain is represented by ${\bf r}_{j}\left(s \right)$, therefore, ${\bf u}_{j}\left(s \right) =\mathrm{d}{\bf r}_{j}/\mathrm{d}s$ is the tangent vector to the polymer at $s$. The canonical partition function of a system consists of main-chain nematic wormlike polymers is ${\cal Z}_{C}\left(n,V,T \right)={\cal Z}_{0} \int {\cal D}\rho \int {\cal D} w\; \exp \left(-H\left[ \rho,w\right] \right)$, where ${\cal Z}_{0}$ is the partition function for an ideal gas of $n$ non-interacting wormlike chains which can be regarded as configurational entropy and $H$ is the effective Hamiltonian

$$\begin{split}
 H\left[ \rho,w\right]&=-i \int \mathrm{d}{\bf r} \int \mathrm{d}{\bf u}\; w \left({\bf r},{\bf u} \right) \rho \left({\bf r},{\bf u} \right) -n \ln Q\left[ i w\right] \\ 
 & \hspace{-1cm} +\frac{\beta}{2} \int \mathrm{d}{\bf r} \int \mathrm{d}{\bf u}\int \mathrm{d}{\bf u^{\prime}}\; \rho \left({\bf r},{\bf u} \right) v \left({\bf u},{\bf u^{\prime}} \right) \rho \left({\bf r},{\bf u^{\prime}} \right)\label{eq:Hamiltonian}
 \end{split}$$

The first term 