  In this paper, we analyze the dynamics of an $N$ particles system evolving
according the gradient flow of an energy functional. The particle system is a
consistent approximation of the Lagrangian formulation of a one parameter
family of non-local drift-diffusion equations in one spatial dimension. We
shall prove the global in time existence of the trajectories of the particles
(under a sufficient condition on the initial distribution) and give two blow-up
criteria. All these results are consequences of the competition between the
discrete entropy and the discrete interaction energy. They are also consistent
with the continuous setting, that in turn is a one dimension reformulation of
the parabolic-elliptic Keller-Segel in high dimensions.
