_{l}Q_{kl}\right)\notag\\
 &\label{1.3}
 \end{aligned}$$

where ${\bf Q}{\bf :}{\bf Q}=Q_{kl}Q_{kl}$, $D_r$ is the rotational diffusion rate, and $F_{ij}=\left( \partial _{i}\tau _{j}+\partial _{j}\tau _{i}-\delta
_{ij}\nabla \cdot \tau \right)$. All terms proportional to $v_0$ arise from one-particle convection due to self-propulsion and are the only consequence of activity in this simple model. The repulsive interactions among the particles generate the cubic homogeneous term (with $\beta>0$) on the right hand side of Eq.  and a change in sign of $\alpha(\rho)\sim \rho-\rho_c$ at a critical density $\rho_c$, controlling the transition between the isotropic and the nematic states.  [^1] Interactions also give density-dependent corrections to the various diffusion coefficients for density ($D$), polarization ($D_\tau$), splay ($D_s$) and bend ($D_b$) deformations of the nematic alignment tensor. We will ignore all such corrections in the following [^2]. Due to the fact that the interactions are purely nematic, the polarization $\bm \tau $ decays on short time scales $\sim D_r^{-1}$ for all strengths of activity. At long time, a hydrodynamic description can then be obtained by neglecting $\partial_t\bm\tau$ in Eq. , and using Eq.  to eliminate $\bm\tau$ from the other equations, with the result (to leading order in gradients)

$$\begin{gathered}
\partial _{t}\rho =D\nabla ^{2}\rho+{\cal D}_Q\bm\nabla\bm\nabla{\bf :}{\bf Q}
\label{2.1}\end{gathered}$$

$$\begin{aligned}
\partial 