Hydrodynamics, superfluidity and giant number fluctuations in a model of self-propelled particles

We derive hydrodynamics of a prototypical one dimensional model, having variable-range hopping, which mimics passive diffusion and ballistic motion of active, or self-propelled, particles. The model has two main ingredients - the hardcore interaction and the competing mechanisms of short and long range hopping. We calculate two density-dependent transport coefficients - the bulk-diffusion coefficient and the conductivity, the ratio of which, despite violation of detailed balance, is connected to number fluctuation by an Einstein relation. In the limit of infinite range hopping, the model exhibits, upon tuning density $\rho$ (or activity), a "superfluid" transition from a finitely conducting state to an infinitely conducting one, characterized by a divergence in conductivity $\chi(\rho) \sim (\rho-\rho_c)^{-1}$ with $\rho_c$ being the critical density. The diverging conductivity greatly increases particle (or vacancy) mobility and induces "giant" number fluctuations in the system.