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## III. THE MODIFIED NEWTONIAN MODEL 

In this section, we discuss phase transitions that ap- pear in the modified Newtonian model. 

We consider a system of particles interacting via a mean field potential Φ( r , t ) that is solution of the modi- fied Poisson equation 

where ρ = M/V is the average density (conserved quan- tity). At statistical equilibrium, the density is given by the Boltzmann distribution 

We have used the notations of astrophysics (where G is the constant of gravity and S$_{d}$ the surface of a unit sphere in d dimensions) in order to make the connection with ordinary self-gravitating systems where Eq. (43) is re- placed by the Poisson equation ΔΦ = S$_{d}$Gρ . However, this model can have application in other contexts as ex- plained below. We assume that the system is confined in a finite domain (box) and we impose the Neumann boundary conditions 

where n is a unit vector normal to the boundary of the box (the explicit expression of the potential in d = 1 is given in Appendix B). This model admits spatially ho- mogeneous solutions ( ρ = ρ and Φ = 0) at any tempera- ture. It also admits spatially inhomogeneous solutions at sufficiently low temperatures. We shall study this model in arbitrary dimensions of space d with explicit compu- tations for d = 1 , 2 , 3. This model has different physical applications: 

(i) It describes self-gravitating systems in a cosmolog- ical setting [12]. Due to the expansion of the universe, when we work in the comoving frame, the Poisson equa- tion takes the form of Eq. (43) where the potential is produced by the deviation between the actual density ρ ( r , t ) and the mean density ρ . In cosmology, we must also account for the scale factor a ( t ) but if we consider timescales that are short with respect to the Hubble time H − 1 = a/ ˙ a , we can ignore this time dependence. This model has been studied by Valageas [41] in d = 1 with pe- riodic boundary conditions. In that context, the relevant ensemble is the MCE since the system is isolated. 

(ii) By a proper reinterpretation of the parameters, the field equation (43) describes the relation between the concentration of the chemical and the density of bacteria in the Keller-Segel model (39). In that case, the most physical dimension is d = 2 and the boundary conditions are of the form (45). Furthermore, the relevant ensemble 

## A. Physical motivation of the model 

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is the CE since the KS model has a canonical structure. This model has been studied by applied mathematicians, starting with J¨ ager & Luckhaus [11], but they have not performed the type of study that we are developing in this paper. 

In view of these different applications, we shall study this model in the microcanonical and canonical ensembles in any dimension of space. 

In the modified Newtonian model, the statistical equi- librium state is given by the Boltzmann distribution (44) coupled to the modified Poisson equation (43). We look for spherically symmetric solutions because, for non ro- tating systems, entropy maxima (or minima of free en- ergy) are spherically symmetric. Introducing the cen- tral density ρ$_{0}$ = ρ (0), the central potential Φ$_{0}$ = Φ(0), the new field ψ = βm (Φ − Φ$_{0}$) and the scaled distance ξ = ( S$_{d}$Gβmρ$_{0}$ ) 1 / $^{2}$r , the Boltzmann distribution (44) can be rewritten 

Substituting this relation in the modified Poisson equa- tion (43), we obtain the modified Emden equation 

where λ = ρ/ρ$_{0}$ plays the role of the inverse central den- sity. Since Φ $^{′}$(0) = 0 for a spherically symmetric system, the boundary conditions at the origin are 

The ordinary Emden equation [53] is recovered for λ = 0, i.e. for very large central densities with respect to the average density. The function e − ψ ( ξ ) is plotted in Figs. 1 and 2 for different values of λ and different dimensions of space d . It presents an infinity of oscillations. For d = 1, the oscillations are undamped and their period is given by Eq. (E11). For d ≥ 2, the oscillations are damped and the function ψ ( ξ ) tends to the asymptotic value − ln λ for ξ → + ∞ . We assume that the system is enclosed in a spheri- 

cal box of radius R . The normalized box radius α = ( S$_{d}$Gβmρ$_{0}$ ) 1 / $^{2}$R is determined by the boundary condi- tion Φ $^{′}$( R ) = 0 that becomes 

For a given value of λ , we need to integrate the modi- fied Emden equation (47)-(48) until the point ξ = α such that ψ $^{′}$( α ) = 0. Since the function ψ ( ξ ) presents an in- finite number of oscillations, this determines an infinity of solutions α$_{1}$ ( λ ), α$_{2}$ ( λ ),... that will correspond to dif- ferent branches in the following diagrams. Once α$_{n}$ ( λ ) 

## B. The modified Emden equation 

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