$\sum_{j \in \partial_i}$ runs over all the particles $j$ in the vicinity of particle $i$, i.e. particles which verify $r_{ij} \leq R_{\theta}$. $R_{\theta}$ thus defines the coupling range of the alignment interaction which is held fixed $R_{\theta}=2\sigma$ (This parameter has to be chosen larger than $\sigma$ in order to guarantee the presence of alignment interactions in the infinitely hard disk limit).

It is useful at this stage to identify the relevant non-dimensional parameters that control the collective behavior of the system. From the Langevin equation - we identify: $$\text{Pe}= \frac{v_0}{\sigma \gamma},$$ the rotational Péclet number, which quantifies the self-propulsion strength of the particles, or, equivalently, their persistence length $l_p=\sigma\text{Pe}$; $$\text{g}=\frac{K}{4 \pi \sigma^2 \gamma},$$ the dimensionless coupling parameter, which compares the coupling strength, i.e. the tendency of the particles to align their velocities, with the angular noise intensity that tends to randomly reorient them; and $$\Gamma= \frac{\mu \epsilon}{\sigma v_0 }\, ,$$ providing a measure of the effective particle stiffness as the self-propulsion is varied. In the present work, we restrict ourselves to the regime where excluded volume interactions always dominate over self-propulsion and the particles can be effectively described as hard disks of diameter $\sigma$ (note that our repulsive potential diverges 