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In Figs. 6, 7 and 8, we plot the normalized energy Λ as a function of the central density 1 /λ for the first three branches n = 1 , 2 , 3 in different dimensions of space d = 1 , 2 , 3. 

Let us consider the asymptotic behaviors of the energy (we only describe the first branch n = 1) and compare with the Newtonian model (see, e.g., [29]): 

• In d = 1: for the ordinary Newtonian model, the se- ries of equilibria is parameterized by α , which is a mea- sure of the central density. When α → + ∞ , the distri- bution tends to a Dirac peak ρ = Mδ ( x ) and the energy Λ → 0. When α → 0, the distribution is homogeneous and the energy Λ → −∞ . For the modified Newtonian model, the series of equilibria is parameterized by the central density 1 /λ . When 1 /λ → + ∞ , the distribution tends to a Dirac peak ρ = Mδ ( x ) and Λ → Λ (1) max = 1 / 6 (see Appendix D). When λ = 1 the distribution is ho- mogeneous and Λ = Λ ∗ c = − 1 / (2 η ∗ $_{c}$) ≃ − 0 . 0506606. When 1 /λ → 0, the distribution tends to a Dirac peak ρ = M 2 ( δ ( x − R ) + δ ( x + R )) concentrated at the box and Λ → Λ (1) $_{max}$. In d = 2: for the ordinary Newtonian model, the se- 

• 

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ries of equilibria is parameterized by α . When α → + ∞ , the distribution tends to a Dirac peak ρ = Mδ ( r ) and the energy Λ → + ∞ . When α → 0, the distribution is homogeneous and the energy Λ → −∞ . For the modified Newtonian model, the series of equilibria is parameter- ized by 1 /λ . When 1 /λ → + ∞ , the distribution tends to a Dirac peak ρ = Mδ ( r ) and Λ → + ∞ . When λ = 1 the distribution is homogeneous and Λ = Λ ∗ c = − 1 /η ∗ c ≃ − 0 . 13622121. When 1 /λ → 0, the distribution is con- centrated at the boundary and we numerically find that Λ → 0 . 1. In d = 3: for the ordinary Newtonian model, the 

• series of equilibria is parameterized by α . When α → + ∞ , the distribution tends to the singular isothermal sphere ρ$_{s}$ ( r ) = 1 / (2 πGβmr $^{2}$) with energy Λ$_{s}$ = 1 / 4. The curve Λ( α ) undergoes damped oscillations around this value. When α → 0, the distribution is homogeneous and the energy Λ → −∞ . For the modified Newtonian model, the series of equilibria is parameterized by 1 /λ . When 1 /λ → + ∞ , the distribution is concentrated at the center and we numerically find that Λ → − 0 . 38 ... (the value is different from the Newtonian result Λ$_{s}$ = 1 / 4 due to different boundary conditions). The curve Λ( λ ) undergoes damped oscillations around this value. When λ = 1 the distribution is homogeneous and Λ = Λ ∗ c = − 3 / (2 η ∗ $_{c}$) ≃ − 0 . 22287452. When λ → 0, the distribution is concentrated at the boundary and we numerically find that Λ → 0 . 05. 

## E. The entropy and the free energy 

Finally, we relate the central density 1 /λ to the entropy S and to the free energy F . Using Eqs. (4), (10) and (11), the entropy is given by 

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