29 For the modified Newtonian model (43), the potential can be written where u ( r , r $^{′}$) is the Green function of the Poisson equa- tion with Neumann boundary conditions. Comparing Eq. (C1) with Eq. (7), we find that the external potential is for any perturbation δρ that conserves mass at first or- der, i.e. ∫ δρ d r = 0. This stability criterion is equivalent to the stability criterion (20) but it is simpler because it is expressed in terms of the density instead of the distri- bution function. Remark: From the stability criteria (A12) and (A19), we clearly see that canonical stability implies micro- canonical stability (but not the converse). Indeed, since the last term in Eq. (A12) is negative, it is clear that if inequality (A19) is satisfied then inequality (A12) is automatically satisfied. In general, this is not reciprocal and we may have ensembles inequivalence. However, if we consider a spatially homogeneous system for which Φ is uniform, the last term in Eq. (A12) vanishes (since the mass is conserved) and the stability criteria (A12) and (A19) coincide. Therefore, for spatially homoge- neous systems, we have ensembles equivalence. Using Eq. (C2), the potential energy (8) can be written Interchanging the dummy variables r and r ′ and using the symmetry u ( r $^{′}$, r ) = u ( r , r $^{′}$), we get Finally, using Eq. (7), we obtain canonical ensemble iff the second order variations of free energy (A15) are positive definite Appendix C: External potential for the modified Newtonian model Φ( r ) = ∫ ( ρ − ρ )( r $^{′}$) u ( r , r $^{′}$) d r $^{′}$, (C1) V ( r ) = − ρ ∫ u ( r , r $^{′}$) d r $^{′}$. (C2) W = 1 2 ∫ ρ Φ d r − 1 2 ρ ∫ ρ ( r ) u ( r , r $^{′}$) d r d r $^{′}$. (C3) W = 1 2 ∫ ρ Φ d r − 1 2 ρ ∫ ρ ( r $^{′}$) u ( r , r $^{′}$) d r d r $^{′}$. (C4) 1 2 ∫ δρδ Φ d r + k$_{B}$T m ∫ ( δρ ) 2 2 ρ d r ≥ 0 , (A19) W = 1 2 ∫ ( ρ − ρ )Φ d r + 1 2 ρ ∫ V ( r ) d r . (C5) V ( x ) = − ρG ( x 2 + R $^{2}$) . (C6) 1 2 ρ ∫ R − R V ( x ) dx = − 4 3 Gρ $^{2}$R$^{3}$. (C7) Φ( x ) = ∫ R − R ρ ( x $^{′}$) u ( x, x $^{′}$) dx ′ (B1) u ( x, x $^{′}$) = − G k$_{0}$ 1 sinh(2 k$_{0}$R ) × (cosh [ k$_{0}$ (2 R − | x − x $^{′}$| )] + cosh [ k$_{0}$ ( x + x $^{′}$)]) . (B2) u ( | x − x $^{′}$| ) = − G k$_{0}$ e − k 0 | x − x $^{$^{′}$|}$. (B3) Therefore, the potential energy is given by Eq. (53) up to an unimportant additive constant 1 $_{2}$ρ ∫ V ( r ) d r = − 1 $_{2}$ρ 2 ∫ u ( r , r $^{′}$) d r d r $^{′}$. In d = 1, according to Eq. (B4), the potential of inter- In d = 1, according to Eq. (B4), the potential of inter- action is u = G | x − x $^{′}$| . Therefore, the external potential is explicitly given by In this Appendix, we limit ourselves to the case d = 1 although the results can be easily generalized to any dimension. Using standard methods, we can obtain the Green function associated with the screened Poisson equation (64) in a box with Neumann boundary condi- tions. Then, we find that the potential is explicitly given by with In an infinite domain ( R → + ∞ ), we obtain The additive constant in the energy is Let us consider the modified Newtonian model (43) in d = 1. At T = 0, the density profile is a Dirac peak ρ = Mδ ( x ) and the energy (56) is Similarly, for the modified Newtonian model (43) with Neumann boundary conditions, the potential is explicitly given by Appendix B: Explicit expressions of the potential Appendix D: The minimum energy E = − 1 4 G ∫ R − R ( d Φ dx ) 2 dx. (D1) Φ $^{′}$( x ) = 2 G ∫ x 0 ρ ( x $^{′}$) dx ′ − 2 Gρx, (D2) Φ( x ) = G ∫ R − R ( ρ − ρ )( x $^{′}$) | x − x $^{′}$| dx $^{′}$, (B4) For a symmetric density profile, the modified Poisson equation can be integrated into and this expression remains valid in an infinite domain ( R → + ∞ ).