_{t}Q_{ij}-&D_r\left[ \alpha\left( \rho \right) -\beta
{\bf Q}{\bf :}{\bf Q}\right] Q_{ij} = D_{b}\nabla ^{2}Q_{ij}\notag\\
&+D_{s}\partial _{k}\left( \partial _{i}Q_{kj}+\partial
_{j}Q_{ik}-\delta _{ij}\partial _{l}Q_{kl}\right)\notag\\
&+{\cal D}_\rho( \partial _{i}\partial _{j}-\frac12\delta _{ij}\nabla
^{2}) \rho
 &\label{2.2}
 \end{aligned}$$

\[2\]

This procedure yields qualitative new terms proportional to ${\cal D}_Q$ and ${\cal D}_\rho$ that vanish in equilibrium. The term proportional to ${\cal D}_Q$ is the curvature induced density flux that has been discussed extensively by Ramaswamy and collaborators [@Ramaswamy2003] and shown to be responsible for giant number fluctuations in the ordered state of active nematic. The diffusive coupling proportional to ${\cal D}_\rho$ describes similar physics but has not been considered in earlier description of active nematic fluids. In addition, activity also yields corrections to the various diffusion coefficients. We have, however, implicitly neglected those here by retaining the same notation for these quantities as in Eqs. - to highlight the difference between these corrections that do not change the dynamics qualitatively and the new terms proportional to ${\cal D}_Q$ and ${\cal D}_\rho$. Although obtained here by considering a system of self-propelled particles, Eqs.  and have the same structure as the hydrodynamic equations of an active nematic, consisting of a collection of *apolar* active particles (shakers) with apolar interactions. This 