emergent physics of active polar fluids.

Since the goal of this presentation is to highlight the mechanisms responsible for emergent structures, we simplify the equations by setting all of the equilibrium-like diffusion coefficients to be equal, i.e., $D=D_\tau=D_b=D_{0}$, with the exception of the splay relaxation constants $D_{s}$. In addition, we assume $\lambda_i=\lambda $ for all $i$’s and $\gamma_1=\gamma_2=\gamma$. Finally, we measure time in units of $1/D_r$ and lengths in units of $
\sqrt{D_{0}/D_r}$. The hydrodynamic equations then simplify to (in nondimensional form)

$$\begin{gathered}
\partial _{t}\rho +\overline{v}\bm\nabla \cdot \bm\tau=\nabla ^{2}\rho +
\overline{D}_Q\bm\nabla \bm\nabla :\mathbf{Q}  \label{4.1}\\
%
\partial _{t}\bm{\tau }+\left(1+\gamma{\bf Q}{\bf :}{\bf Q}\right)\bm\tau-\gamma\bm{\tau \cdot Q}
+\lambda \bm{\tau }\cdot \nabla \bm{\tau } =-\overline{v}
\bm\nabla \cdot \mathbf{Q}-\frac{\overline{v}}{2}\bm\nabla \rho
+\lambda \left( \bm{\tau }\nabla \cdot \bm{\tau }+\frac12\nabla \tau
^{2}\right) +\nabla ^{2}\bm{\tau }  \label{4.2} \\
%
\partial _{t}Q_{ij}-\left( \alpha-\beta{\bf Q}{\bf :}{\bf Q}  \right) Q_{ij}+\overline{v} F_{ij}+\lambda G_{ij} =
\overline{D}_\rho( \partial _{i}\partial _{j}-\frac12 \delta _{ij}\nabla ^{2}) \rho
+\overline{D}_{S}\partial _{k}\left( \partial _{i}Q_{kj}+\partial
_{j}Q_{ik}-\delta _{ij}\partial _{l}Q_{kl}\right)
+\nabla ^{2}Q_{ij}
\label{4.3}\end{gathered}$$

\[4\]

with $\overline{D}_Q={\cal D}_Q/D_0$, $\overline{D}_\rho={\cal D}_\rho/D_0$ and $\overline{D}_s=D_s/D_0$. Finally, we assume $\alpha=\frac{\rho }{\rho _{c}}-1 $ and $
\beta $ independent of $\rho$. The effect of activity is assumed to affect the mean field phase transition only through the dependence of the critical 