2 with point ( k$_{0}$ )$_{m}$ R ≃ 11 . 8, where R is the system size. If k$_{0}$ < ( k$_{0}$ )$_{c}$, the system presents canonical and micro- canonical second order phase transitions. In that case, the ensembles are equivalent. If ( k$_{0}$ )$_{c}$ < k$_{0}$ < ( k$_{0}$ )$_{m}$, the system presents a canonical first order phase transi- tion and a microcanonical second order phase transition. In that case, there exists a region of negative specific heats in MCE and the ensembles are inequivalent. If k$_{0}$ > ( k$_{0}$ )$_{m}$, the system presents canonical and micro- canonical first order phase transitions. In d = 2 and d = 3, the phase transitions are similar to those reported for the modified Newtonian model. In Sec. V, we study the dynamical stability of the homogeneous phase and analytically determine the critical point ( E ∗ $_{c}$, T ∗ c ) that marks the onset of instability of the homogeneous branch and the starting point of the bifurcated inhomogeneous branch. Direct numerical simulations associated with these phase transitions will be reported in a forthcom- ing paper. where ρ ( r , t ) = ∫ f ( r , v , t ) d v is the spatial density. In- troducing the mean field potential Finally, it may be noted that the phase transitions reported in this paper share analogies (but also differ- ences) with phase transitions observed in the Hamil- tonian mean field (HMF) model [32–36], the spherical mass shell (SMS) model [37], the Blume-Emery-Griffiths (BEG) model [38], the infinite-range attactive interaction (IRAI) model [39], the self-gravitating Fermi gas (SGF) model [28], the self-gravitating ring (SGR) model [40] and the one-dimensional static cosmology (OSC) model [41]. the energy can also be written S = − k$_{B}$ ∫ f m ln f m d r d v , (4) M = ∫ ρ d r , (5) E = ∫ f v 2 2 d r d v + 1 2 ∫ ρ ( r , t ) u ( r , r $^{′}$) ρ ( r $^{′}$, t ) d r d r ′ + ∫ ρV d r , (6) Φ( r ) = ∫ u ( r , r $^{′}$) ρ ( r $^{′}$) d r ′ + V ( r ) , (7) E = 1 2 ∫ fv 2 d r d v + 1 2 ∫ ρ (Φ + V ) d r . (8) δS − 1 T δE − αδM = 0 . (9) f = Ae − βm ( v 2 2 +Φ ) , (10) ρ = A $^{′}$e − m Φ k B T . (11) δ $^{2}$J = − ∫ ( δf ) 2 2 mf d r d v − 1 2 β ∫ δρδ Φ d r ≤ 0 , (12) m d r i dt = ∂H ∂ v i , m d v i dt = − ∂H ∂ r i , (1) H = ∑ i 1 2 mv 2 i + m 2 ∑ i<j u ( r $_{i}$, r $_{j}$) + m ∑ i V ( r $_{i}$) . (2) We shall be interested in global and local entropy max- ima. Let us first determine the critical points of entropy at fixed mass and energy which cancel the first order vari- ations. Introducing Lagrange multipliers, they satisfy The variations are straightforward to evaluate and we obtain the mean field Maxwell-Boltzmann distribution where β = 1 /k$_{B}$T and Φ( r ) is given by Eq. (7). Integrat- ing over the velocity, the find that the density is given by the mean field Boltzmann distribution We consider an isolated system of N particles in inter- action described by the Hamiltonian equations where We assume that the particles interact through a binary potential u ( r , r $^{′}$) that is symmetric with respect to the interchange of r and r $^{′}$, and that they also evolve in a fixed external potential V ( r ). Since the system is iso- lated, with strict conservation of energy and mass, the proper statistical ensemble is the microcanonical ensem- ble [9]. In this paper, we shall use a mean field approach [63]. In the microcanonical ensemble, the statistical equi- librium state is obtained by maximizing the Boltzmann entropy at fixed mass and energy. We thus have to solve the maximization problem This critical point is a (local) entropy maximum at fixed mass and energy iff for all perturbations δf that conserve mass and energy at first order. In Appendix A, we provide an equivalent but simpler condition of stability in the microcanonical ensemble [see inequality (A12)]. The time evolution of the distribution function f ( r , v , t ) is governed by a kinetic equation of the form II. KINETIC MODELS AND STATISTICAL EQUILIBRIUM STATES 1. Isolated systems ∂f ∂t + v · ∂f ∂ v − ∇ Φ · ∂f ∂ r = ( ∂f ∂t ) coll , (13) max f { S [ f ] | E [ f ] = E, M [ f ] = M } , (3)