We consider the dynamics of a system of particles with logarithmic attractive interaction, on the torus, at inverse temperature beta. We show phase transitions on the stability and uniqueness of the uniform distribution. Investigating the mean-field convergence of the system by the modulated free energy method,  we deduce that  uniform-in-time convergence is not always true.

This is joint work with Antonin Chodron de Courcel and Matthew Rosenzweig.