3 with where We shall be interested by global and local minima of free energy. Let us first determine the critical points of free energy at fixed mass which cancel the first order varia- tions. Introducing a Lagrange multiplier, they satisfy The variations are straightforward to evaluate and we obtain the mean field Maxwell-Boltzmann distribution (10) and the mean field Boltzmann distribution (11) as in the microcanonical ensemble. This critical point is a (local) minimum of free energy iff for all perturbations δf that conserve mass. F = ∫ 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 + k$_{B}$T ∫ f m ln f m d r d v . (18) Φ( r , t ) = ∫ u ( r , r $^{′}$) ρ ( r $^{′}$, t ) d r ′ + V ( r ) , (14) δF + αT δM = 0 . (19) δ $^{2}$F = 1 2 ∫ δρδ Φ d r + k$_{B}$T m ∫ ( δf ) 2 2 f d r d v ≥ 0 , (20) is the time-dependent mean field potential. The l.h.s. is an advective operator (Vlasov) in phase space. The r.h.s. is a “collision” operator like the Boltzmann operator in the kinetic theory of gases or like the Landau (or Lenard- Balescu) operator in plasma physics or stellar dynamics. The “collision” operator in Eq. (13) takes into account the development of correlations between particles. It can have a more or less complicated form but it satis- fies general properties associated with the first and sec- ond principles of thermodynamics: (i) it conserves mass and energy; (ii) it satisfies an H -theorem for the Boltz- mann entropy (4), i.e. ˙ S ≥ 0 with an equality iff f is the Maxwell-Boltzmann distribution (10). Furthermore, the Maxwell-Boltzmann distribution is dynamically sta- ble iff it is a (local) entropy maximum at fixed mass and energy. These general properties can be checked directly for the Boltzmann equation, for the Landau equation, for the Lenard-Balescu equation and for the BGK operator. Therefore, the kinetic equation (13) is consistent with the maximization problem (3) describing the statistical equilibrium state of the system in MCE. If we neglect the collisions for sufficiently short times, Eq. (13) reduces to the Vlasov equation which can experience a complicated process of collisionless violent relaxation towards a quasi stationary state (QSS) [43]. In the mean field approximation, the evolution of the distribution function f ( r , v , t ) is governed by a kinetic equation of the form We consider a dissipative system of N Brownian par- ticles in interaction described by the Langevin equations coupled to the mean field potential (14). This is called the mean field Kramers equation. The mean field Kramers equation conserves mass and satisfies an H - theorem for the Boltzmann free energy (18), i.e. ˙ F ≤ 0 with an equality iff f is the Maxwell-Boltzmann distribu- tion (10). Furthermore, the Maxwell-Boltzmann distri- bution is dynamically stable iff it is a (local) minimum of free energy at fixed mass. Therefore, the kinetic equation (21) is consistent with the minimization problem (17) de- scribing the statistical equilibrium state of the system in CE. ∂f ∂t + v · ∂f ∂ v − ∇ Φ · ∂f ∂ r = ∂ ∂ v ( D ∂f ∂ v + ξf v ) , (21) m d r i dt = ∂H ∂ v i , (15) d v i dt = − 1 m ∂H ∂ r i − ξ v i $_{+}$√$_{2}$$_{D}$ R $_{i}$( t ) , (16) 2. Dissipative systems in phase space Remark: the critical points in MCE and CE are the same because the variational problems (3) and (17) are equivalent at the level of the first order variations (9) and (19). However, they are not equivalent at the level of the second order variations (12) and (20) because of the dif- ferent class of perturbations to consider. Therefore, we can have ensembles inequivalence [22, 31, 44, 45]. In fact, the condition of canonical stability (17) provides a suf- ficient condition of microcanonical stability (3). Indeed, if inequality (20) is satisfied for all perturbations that conserve mass, then it is a fortiori satisfied for perturba- tions that conserve mass and energy, so that inequality (12) is satisfied. Therefore, canonical stability implies microcanonical stability: where H is the Hamiltonian defined by Eq. (2), − ξ v i is a friction force and R $_{i}$( t ) is a white noise satisfying 〈 R $_{i}$( t ) 〉 = 0 and 〈 R µ i ( t ) R ν $_{j}$( t ) 〉 = δ$_{ij}$δ$_{µν}$δ ( t − t $^{′}$). The diffu- sion coefficient D and the friction coefficient ξ are related to each other according to the Einstein relation ξ = Dβm where β = 1 / ( k$_{B}$T ) is the inverse temperature. Since this system is dissipative, the proper statistical ensemble is the canonical ensemble [9]. In the canonical ensemble, the statistical equilibrium state is obtained by minimiz- ing the Boltzmann free energy F [ f ] = E [ f ] − T S [ f ] at fixed mass. We thus have to solve the minimization prob- lem However, the converse is wrong in case of ensembles in- equivalence. (17) ⇒ (3) . (22) min f { F [ f ] | M [ f ] = M } (17)