First- and second-order clustering transitions for a system with infinite-range attractive interaction

We consider a Hamiltonian system made of N classical particles moving in two dimensions, coupled via an infinite-range interaction gauged by a parameter A. This system shows a low energy phase with most of the particles trapped in a unique cluster. At higher energy it exhibits a transition towards a homogenous phase. For sufficiently strong coupling A, an intermediate phase characterized by two clusters appears. Depending on the value of A, the observed transitions can be either second or first order in the canonical ensemble. In the latter case, microcanonical results differ dramatically from canonical ones. However, a canonical analysis, extended to metastable and unstable states, is able to describe the microcanonical equilibrium phase. In particular, a microcanonical negative specific heat regime is observed in the proximity of the transition whenever it is canonically discontinuous. In this regime, microcanonically stable states are shown to correspond to saddles of the Helmholtz free energy, located inside the spinodal region.