  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.
