r},t)=\rho({\bf r},t)S_{ij}\left( \mathbf{r},t\right) $. The polarization ${\bf P}$ is directly proportional to the collective velocity of the active particles, while $S_{ij}$ is the conventional nematic order parameter tensor familiar from liquid crystal physics. For a uniaxial system in two dimensions, $Q_{ij}$ is a symmetric traceless tensor with only two independent components and can be written in terms of a unit vector ${\bf \hat{n}}$ as $Q_{ij}=Q(\hat{n}_i\hat{n}_j-\frac12\delta_{ij})$, where $Q=\rho S$; $S$ is the magnitude of the order parameter and the director ${\bf \hat{n}}$ identifies the direction of spontaneously broken symmetry in the nematic state. For simplicity most of the discussion below refers to the case where the active particles are modeled as long thin rods with repulsive interactions.

Active Nematic Hydrodynamics {#subsec:active_nematic}
----------------------------

We first construct the dynamical equations of the system including single-particle convection terms induced by self propulsion, but assuming that self-propulsion does not modify the interaction between two rods. The hydrodynamic equations then take the form [@Baskaran2008]

$$\begin{gathered}
\partial _{t}\rho +v_{0}\nabla \cdot \bm{\tau }=D\nabla ^{2}\rho
\label{1.1}\\
\partial _{t}\bm{\tau }+D_r\bm{\tau }+v_{0}\nabla \cdot \mathbf{Q}+
\frac{v_{0}}{2}\nabla \rho =D_\tau\nabla ^{2}\bm{\tau }  \label{1.2}\end{gathered}$$

$$\begin{aligned}
\partial _{t}Q_{ij}-&D_r\left[ \alpha\left( \rho \right) -\beta
{\bf Q}{\bf :}{\bf Q}\right] Q_{ij}+v_{0}F_{ij} = D_{b}\nabla ^{2}Q_{ij}\notag\\
&+D_{s}\partial _{k}\left( \partial _{i}Q_{kj}+\partial
_{j}Q_{ik}-\delta _{ij}\partial 