J Lalic

D Fougeron

E Madsen

E R A Joly

J Christiansen

L Flores Sanz De Acedo

M Menouni

M Standke

P Barrillon

P Rymaszewski

L Flores

Sanz De

Acedo M Menouni

T Strebler

Single event effects on the RD53B pixel chip digital logic and on-chip CDR

Keywords:

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

RD53 is a common design framework of final pixel readout chips for the ATLAS and CMS phase 2 upgrades developed in 65 nm CMOS technology. This framework aims at providing a fault tolerant system which meets stringent requirements of the HL-LHC environment and provides reliable functionality in its extreme radiation field. Besides having tolerance to immense total ionizing dose of 1 Grad, this field demands high redundancy to Single Event Effects (SEEs). Critical parts of its digital design are protected with different mitigation schemes to ensure reliable operation. Several heavy-ions and proton beam tests were performed to measure the efficiency of these protection techniques and to estimate a soft error rate in the final detector. During beam testing it has been seen that Single Event Transients (SET) can impact high-speed data link stability by causing several μs long dropouts which motivated more detailed studies. The on-chip Clock Data Recovery (CDR) circuit was characterized by capturing heavy-ion-induced error signatures in its high-speed clock which drives the output data link.

RD53B design for soft error mitigation

The RD53B chip [START_REF] Garcia-Sciveres | RD53B Manual[END_REF] is the pre-production chip which will be followed by the final RD53C production chip with submission planned in 2022. The pixel chips for ATLAS and CMS are two instances of a common design with different analog front-ends and pixel matrix sizes. However, all SEE results are valid for both chips since they have the same digital design and analog blocks in the periphery including the on-chip CDR. Design robustness for soft error mitigation depends on the target level of reliability on one side and acceptable power and area overheads on the other. RD53 objectives are minimization of the upset rate for critical information and signals, and the capability of immediate self-recovering in case of critical data corruption. This implies that the configuration memory, state machines, critical event data, look-up tables are all SEE robust. The on-chip CDR is SET hardened with triplicated clock divider, a bang-bang phase detector and an optimized voltage-controlled oscillator (VCO) [START_REF] Moustakas | A Clock and Data Recovery Circuit for the ALTAS/CMS HL-LHC Pixel Front End Chip in 65 nm CMOS Technology[END_REF]. As a means of having a small pixel size and staying within power dissipation constraints, the chip has different protection schemes for its pixel and global configuration memory which prevents soft errors at two different levels of efficiency [START_REF] Menouni | Single event effects testing of the RD53B chip, Submitted to JPCS[END_REF]. In case chip operation is critically affected by SEEs, a Clear command is implemented for fast clear of data buffers and state machines.

SEE sensitivity of the Aurora serialized data

During beam campaigns, the 1.28 Gbit/s readout links showed susceptibility to SEEs with temporary loss of synchronization with the DAQ receiver causing loss of multiple events. However, this behavior is not observed if the high-frequency clock driving the data link output is provided externally bypassing the on-chip CDR block.

Chip default operation mode and CDR-Bypass

Figure 1 shows clock generation inside the analog chip bottom and a simplified data flow. During chip default operation mode, the CDR block recovers 160 MHz reference clock from the received 160 Mbit/s input command stream and generates a 1.28 GHz clock. This high-frequency clock drives up to 4 serializers, each providing a 1.28 Gbit/s output data stream on a corresponding data lane via a CML output driver. While characterizing performance of the pixel digital logic, when exposed to particle beam, it is beneficial to decouple its soft error rate from the CDR SET sensitivity. Thus, the chip features the possibility of enabling a so-called CDR-Bypass mode in which both reference and serializer clocks are provided externally by the DAQ system.

Hit loss induced by the SET sensitivity of the high-speed data link

Performance of the digital chain for hit processing is evaluated by injecting digital hit pulses while irradiating device at a given flux and recording pixel matrix response. Hit data is acquired through the 1.28 Gbit/s serial link via Aurora Xilinx IP receiver on a Kintex7-based DAQ [START_REF] Daas | BDAQ53, a versatile pixel detector readout and test system for the ATLAS and CMS HL-LHC upgrades[END_REF] board.

Measurements with both heavy-ions and protons show that if the chip operates in the CDR-Bypass modes, its data stream has high signal integrity with no stability issues induced by SETs. However, the same test sequence with on-chip CDR enabled, leads to loss of the synchronization state with the DAQ receiver due to clock cycle slipping. To measure hit loss caused by the SET sensitivity of the serializer clock, receiver volatility to phase shifts of the input stream is eliminated. This is achieved by disabling its sync state machine, thus all received bit-slipped headers are ignored without instantiating Aurora lane re-synchronization. In figure 2a, data bit-slip rate induced by 480 MeV protons is shown for multiple iterations of 110 s long digital injection. If the proton beam is run at the at flux of 1.5 × 10 9 protons cm 2 ×s , the average bit-slip rate is 3.13 s -1 . With half the flux density, the bit-slip rate is 6.78 s -1 . An example of an occupancy map when the digital injection is repeated 100 times per pixel is shown in figure 2b (RD53-ATLAS has 384 × 400 pixels). In the 480 MeV proton beam environment, hit loss induced by the data bit-slips can be up to 0.2%. This motivated further studies to determine the exact signature shape and duration of induced transients in the clock signal.

Characterization of heavy-ion-induced transients in the 1.28 GHz clock

To address the problem of high-speed data link dropouts in heavy-ion and proton beam environments, transients in the clock signal driving this output are characterized. The experiment was carried out at the GANIL irradiation facility in France, using a Xenon beam with a LET of 27 MeV×cm 2 mg .

Irradiation was done in the air at room temperature using the setup depicted in figure 3.

Setup with the 10.24 Gb/s signal oversampling

The high-frequency on-chip CDR output clock used for serialization of Aurora data is first divided by two inside the chip analog bottom (SER_CLK/2) and routed to the CML output driver resulting in a 640 MHz frequency clock signal available for monitoring. Monitoring is done on an oscilloscope which triggers which triggers in case of a phase shift larger than 390.625 ps with respect to reference clock. The reference clock itself is provided by the DAQ board located outside the beam area. To acquire better timing resolution, an FPGA-based setup was developed using the KC705 Xilinx Evaluation Board to monitor the signal in parallel. This system is using a Xilinx Gigabit Transciever (GTX) [START_REF]Xilinx, UG476, 7 Series FPGAs GTX/GTH Transceivers User Guide[END_REF] to oversample the clock under test with a 10.24 Gbit/s sampling rate, providing 64 bits deserialized clock data at a 160 MHz word frequency. To enable oversampling inside the transceiver, it is necessary to lock its internal CDR to the reference clock. The best sampling performance is obtained by choosing Low Power Mode (LPM) equalizer and disabling its auto-adaptive filtering. This system incorporates a triggering mechanism on the clock pulse width deviation bigger than 97.65 ps, which is the resolution of the circuit.

Results

(a) (b) All transients in this group have a duration of less than 18 μs and a clock phase shift of up to 5 ns.

However, the second group of events shows that ion strikes can cause clock amplitude and biasing deviation. This further leads to a cumulative phase shift in the range from 50 to 700 clock cycles, as seen in figure 4b. All events in this group are longer than 20 μs which is the duration of saved waveforms. Event duration distribution is shown in figure 5 for two values of Xe beam flux and

Conclusion

By evaluating the RD53B design robustness against SEEs, it has been observed that heavy-ions and protons can induce high-speed data link dropouts limiting the reliability of the system. The susceptibility of the 1.28 GHz clock driving the data output to transient faults is successfully estimated by capturing its phase shifts with an oscilloscope and the proposed high-speed signal sampling inside the FPGA. Two groups of different SET signatures were detected. In cases where driver biasing and clock frequency are not affected, the phase shift is up to a few ns long. However, ions can cause clock signal deviation and lead to phase shifts of several hundred clock cycles. These shifts further lead to data link dropout and result in hit losses. The different nature of these events indicates that except for the CDR block itself, its biasing circuit may also be SET sensitive. Further assessment of the chip's biasing chain is being carried out with two-photon absorption pulsed laser testing. This study has already pinpointed sensitivity in the chip's bandgap as the cause of the long upsets. When finalized, the results will be documented in a separate publication.

Figure 1 :

 1 Figure 1: Simplified data flow with the blocks for clock recovery and encoded data serialization.

Figure 2 :

 2 Figure 2: (a) Bit-slip error rate per digital injection. (b) Hit loss in the occupancy map caused by data bit-slips.

Figure 3 :

 3 Figure 3: Setup for measuring clock SET sensitivity.

Figure 4 :

 4 Figure 4: Accumulated phase difference of the SER_CLK/2's rising edge from its ideal position measured over 12800 clock cycles. (a) Example of a transient with 2 ns phase shift and duration of less than 18 μs. (b) Example of a transient with large accumulated phase shift.

Figure 5 :

 5 Figure 5: Distribution of transient duration at two different values of Xe flux. The red signal is an example of a deviated 640 MHz clock under test.


Charles Elbar
email: charles.elbar@sorbonne-universite.fr

Marco Mason

Benoît Perthame
email: benoit.perthame@sorbonne-universite.fr

Jakub Skrzeczkowski
email: jakub.skrzeczkowski@student.uw.edu.pl

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux. limits in parabolic equations" no.

Introduction

We consider the following Vlasov-Cahn-Hilliard equation (VCH in short)

ε 2 ∂ t f ε + εξ.∇ x f ε + εF ε .∇ ξ f ε = ε (t, x)M(ξ ) -f ε , t ≥ 0, x ∈ R d , ξ ∈ R d , ε (t, x) = R d f ε (t, x, ξ) dξ, (1)
with an initial data f ε (0, x, ξ) = f 0 (x, ξ) ≥ 0. The unknown is the function

f ε ≡ f ε (t, x, ξ), t ∈ (0, T ), x ∈ R d , ξ ∈ R d ,
such that, for every infinitesimal volume dx dξ around the point (x, ξ) in the phase space, the quantity f ε (t, x, ξ) dx dξ is the number of particles which have position x and velocity ξ at fixed time t. The small parameter ε > 0 arises from physical dimensions of the system and we are interested in the limit when it tends to 0. Following [START_REF] Takata | A simple kinetic model for the phase transition of the van der Waals fluid[END_REF], the force field F ε (t, x) is decomposed as long-range attractive and short-range repulsive

F ε = F L ε + F S ε , F L ,S ε (t, x) = -∇ L ,S ε (t, x). ( 2
)
We define the convolution in the space variable as f g = R d f (y)g(xy) dy and set

S α,ε (t, x) = 1 α 2 ω S ω S ε ,
where ω S ≥ 0 is a function that may be thought of as a centered Gaussian. We use a double convolution in order to enforce positivity of the corresponding operator as it appears in energy considerations. We assume that ω S satisfies

R d ω S (y) dy = 1, R d yω S (y) dy = 0, R d |y| 2 ω S (y) dy < ∞. (3)
The long-range potential is of the form

L α,ε (t, x) = - 1 α 2 ω L α ω S ω S ε , (4)
where ω L α (x) = 1 α d ω L x α may be thought of as a high temperature Gaussian and ω L is a smooth, nonnegative, symmetric, compactly supported function such that, for some δ > 0,

R d ω L (y) dy = 1, R d yω L (y) dy = 0, R d y i y j ω L dy = δ i, j δ, R d ω L (y)|y| 3 dy < ∞. (5)
The equilibrium distribution M(ξ ) ≥ 0 is a Maxwellian that we normalize as

M(ξ ) := 1 2π D d/2 exp - |ξ | 2 2D , (6)
and we have, for i = 1, . . . , d,

R d M(ξ ) dξ = 1, R d ξ i M(ξ ) dξ = 0, R d ξ 2 i M(ξ ) dξ = D < ∞, (7)
so that D can be interpreted as the diffusion coefficient.

1.1. The macroscopic limit. The right-hand side of Equation ( 1) is a relaxation term that conserves mass but neither momentum nor energy since we aim at using a diffusive scaling. Formally one can guess that

f ε (t, x, ξ) → (t, x)M(ξ ), as ε → 0. ( 8
)
The mass conservation equation on ε is obtained by integrating Equation (1) with respect to ξ against 1,

∂ t ε (t, x) + divJ ε (t, x) = 0, J ε (t, x) = R d ξ ε f ε (t, x, ξ) dξ. (9)
Then, integrating against ξ , we obtain the flux equation

ε 2 ∂ t J ε (t, x) + ∇ x • R d ξ ⊗ ξ f ε (t, x, ξ) dξ -F ε ε = -J ε (t, x). ( 10
)
Combined with [START_REF] Ebenbeck | Cahn-Hilliard-Brinkman systems for tumour growth[END_REF], this flux equation allows us to identify the limit of J ε and to prove that as ε, α → 0, the macroscopic densities tend to a solution of a degenerate nonlocal Cahn-Hilliard equation type. More precisely, we have the Theorem 1 (Limit ε → 0). With the assumptions and notations ( 2)- [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: global existence and weak stability[END_REF], let α = ε. Let f 0 be a non-negative distribution that satisfies [START_REF] Frigeri | On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials[END_REF], [START_REF] Garcke | A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis[END_REF] and let f ε be a solution of [START_REF] Bardos | Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data[END_REF] with initial condition f 0 . Then, we can extract a subsequence (not relabelled) such that ε → in L p t L 1 x strongly for 1 ≤ p < ∞ where solves in the distributional sense the equation

∂ t -D -div( ∇ ) = 0, = -δ [ω S ω S ], (11)
with initial data 0 = R d f 0 (x, ξ) dξ .

In fact, [START_REF] Takata | A simple kinetic model for the phase transition of the van der Waals fluid[END_REF] obtains formally a more complete description which we cannot prove at the moment (see Sect. 4).

Remark 2. • Writing formally

= div( ∇ log( )), this term can be added to the potential so as to obtain a kind of Cahn-Hilliard equation.

• Different scaling between α and ε can be considered, α constant is also possible • Uniqueness can be proved in the class of uniformly bounded densities, see Appendix C.

• The proof of this result uses compactness arguments, therefore we do not have any explicit control on the error between the limit solution and the system in terms of ε.

Contents of the paper.

In Sect. 2, we collect various uniform estimates ε. Section 3 is devoted to passing to the limit ε → 0. Some open problems are drawn in Sect. 4. The Appendix contains different mathematical tools and lemmas used throughout the proofs.

Literature review and relevancy of the system. Phase transitions in fluids

In [START_REF] Takata | A simple kinetic model for the phase transition of the van der Waals fluid[END_REF], Noguchi and Takata consider a kinetic model to capture the dynamics of phase transition for the Van der Waals fluid. The model reads as follows

∂ t f + ξ • ∇ x f + F • ∇ ξ f = A( )( M -f ), A( ) > 0, F = F 1 + F 2 , F 1,2 = -∇ 1,2 , 1 = -ω L or -κ , 2 = -C 1 log(1 -) + 1 - -C 2 , C 1 , C 2 ∈ R,
for some kernel ω L and κ > 0. 1 is a combination of short range repulsion and long range attraction. 2 is a short range interaction potential. The authors state that full details of intermolecular collisions are not considered and that the collision term on the right-hand side plays just a thermal bath role which is less desirable for a physical/mechanical justification of the system. This problem has been adressed formally in [START_REF] Giovangigli | Kinetic derivation of Cahn-Hilliard fluid models[END_REF][START_REF] Takata | Kinetic model for the phase transition of the van der waals fluid[END_REF]. From the mathematical point of view, it raises new and interesting different difficulties and it can be definitely seen as a next step in our work. Nevertheless, the authors show that with this thermal bath term, the system exhibits the essential features of phase transition dynamics, both theoretically and numerically. By placing themselves in the framework of the strong interaction, they find a rescaling of the first equation of the system and obtain Equation (1) of VCH. Then, setting A ≡ 1 and letting ε → 0 they obtain formally that in the limit (that we refer to as the hydrodynamic limit), the macroscopic density satisfy

∂ t - -div( ∇( 1 + 2 )) = 0.
Noting that = div( ∇ log( )) they obtain the Cahn-Hilliard equation with degenerate mobility

∂ t -div( ∇( 1 + 2 )) = 0, 2 = 2 + log( ).
This model presents several mathematical difficulties. First of all, we are not aware of any existence result concerning the Vlasov equation when the potential is a function of the density . In the Vlasov-Poisson system, one has = -1 and there is a gain of two derivatives. For the existence of classical solutions for Vlasov-Poisson we refer to [START_REF] Horst | On the asymptotic growth of the solutions of the Vlasov-Poisson system[END_REF][START_REF] Lions | Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system[END_REF][START_REF] Pfaffelmoser | Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data[END_REF][START_REF] Schaeffer | Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions[END_REF]. A second difficulty comes from the rigorous passage to the limit. Indeed, the bound provided by the energy do not provide enough compactness. For instance, one cannot apply the averaging lemma 12 on this system because the functions are not bounded in L 1 uniformly in ε. For these reasons, we add the convolutions ω S in ( 1)-( 4) and provide a rigorous mathematical framework to establish the hydrodynamic limit of this model when 2 = 0. It would be possible to prove a similar result when

2 = ω S f ( ) where | f ( )| ≤ C| log | for C small enough.
Our work also provides a generic model to obtain different nonlocal and degenerate equations of Cahn-Hilliard/thin-film type as the hydrodynamic limit of kinetic models. For other kinetic models modeling phase transitions, we refer to [START_REF] Giovangigli | Kinetic derivation of Cahn-Hilliard fluid models[END_REF][START_REF] Grmela | On the approach to equilibrium in kinetic theory[END_REF][START_REF] Kobayashi | Numerical analysis of vapor-liquid two-phase system based on the Enskog-Vlasov equation[END_REF]. Kinetic theory The main purpose of kinetic theory is to provide a description of the evolution of a gas or plasma, and more generally a many-particle system made up of N similar individual elements, in the limit when N tends to infinity which corresponds to the so-called thermodynamical limit. In the kinetic theory, the density of particles is described with the probability measure

f ≡ f (t, x, ξ), t ≥ 0, x ∈ R d , ξ ∈ R d ,
such that, for every infinitesimal volume dx dξ around the point (x, ξ) in the phase space, the quantity f (t, x, ξ) dx dξ is the number of particles which have position x and velocity ξ at fixed time t. For this reason, f is a nonnegative function and integrable in both space and velocity variables, but it is not directly observable. Nevertheless, at each point of the domain it provides all measurable macroscopic quantities which can be expressed in terms of microscopic averages:

(t, x) = R d f (t, x, ξ) dξ (macroscopic density), J (t, x) = R d ξ f (t, x, ξ) dξ (flux).
It is clear that such a statistical description makes sense only with a very large number of particles, and as a consequence, all kinetic equations are expected to approximate the true dynamics of gases just in the thermodynamical limit. Rescaling the time and space with a parameter ε, i.e. t → ε 2 t, x → εx and sending ε → 0 is called the hydrodynamic limit. It allows us to find a rigorous derivation of macroscopic models from a microscopic description of matter. For hydrodynamics on the Vlasov-Poisson-Fokker-Planck system, we refer to [START_REF] El Ghani | Diffusion limit of the Vlasov-Poisson-Fokker-Planck system[END_REF][START_REF] Goudon | Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: analysis of the twodimensional case[END_REF]. Our aim is to obtain an equation on the macroscopic density and to relate it to a known model that has applications in fluid dynamics or biology, i.e. the Cahn-Hilliard equation. The Cahn-Hilliard equation Equation ( 11) is an example of a Cahn-Hilliard type equation that is widely used nowadays to represent phase transitions in fluids and living tissues [START_REF] Ebenbeck | On a Cahn-Hilliard-Brinkman model for tumor growth and its singular limits[END_REF][START_REF] Ebenbeck | Cahn-Hilliard-Brinkman systems for tumour growth[END_REF][START_REF] Elbar | Degenerate Cahn-Hilliard and incompressible limit of a Keller-Segel model[END_REF][START_REF] Frieboes | Three-dimensional multispecies nonlinear tumor growth-II: tumor invasion and angiogenesis[END_REF][START_REF] Frigeri | On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials[END_REF][START_REF] Garcke | A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis[END_REF][START_REF] Garcke | A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport[END_REF][START_REF] Lowengrub | Nonlinear modelling of cancer: bridging the gap between cells and tumours[END_REF][START_REF] Perthame | Relaxation of the Cahn-Hilliard equation with singular single-well potential and degenerate mobility[END_REF][START_REF] Wise | Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method[END_REF]. Originally introduced in the context of materials sciences [START_REF] Cahn | On spinodal decomposition[END_REF][START_REF] Cahn | Free energy of a nonuniform system. I. Interfacial free energy[END_REF], it is currently applied in numerous fields, including complex fluids, polymer science, and mathematical biology. For the overview of mathematical theory, we refer to [START_REF] Miranville | The Cahn-Hilliard equation[END_REF].

Cahn-Hilliard equation takes the form of

∂ t = div (b( )∇ ( ( ) -δ )) → ∂ t = div (b( )∇μ) , = -δ + I ( ), ( 12
)
where represents the relative density of one component = 1 /( 1 + 2 ), b( ) is the mobility, I is the interaction potential while is the quantity of chemical potential. We obtain a nonlocal version of the Cahn-Hilliard equation. The nonlocality comes from the convolution of the Laplace operator with a smooth kernel ω S concentrated around the origin. There is a different possibility to approximate this operator nonlocally, we refer for instance to [START_REF] Davoli | Nonlocal-to-local convergence of Cahn-Hilliard equations: neumann boundary conditions and viscosity terms[END_REF][START_REF] Melchionna | From nonlocal to local Cahn-Hilliard equation[END_REF], where the authors prove the convergence of a nonlocal Cahn-Hilliard equation with constant mobility to a local Cahn-Hilliard equation and to [START_REF] Elbar | Degenerate Cahn-Hilliard equation: from nonlocal to local[END_REF] for a similar result with degenerate mobility. In our case, because of the degenerate mobility, it is not clear that we can pass from the nonlocal Equation ( 11) to a local one by sending ω S to a Dirac mass.

Entropy, Energy, and Uniform Estimates

The analysis relies on various uniform bounds in ε which use an initial data that satisfies

R 2d (1 + |x| + |ξ | 2 + | log f 0 |) f 0 (x, ξ) dx dξ < +∞, (13)
sup α≤1 1 α 2 R 2d ω L α (y)[ 0 ω S (x) -0 ω S (x -y)] 2 dx dy < +∞. ( 14
)
Then, we begin with proving the bounds Theorem 3 (Uniform estimates). With the assumptions ( 13) and ( 14), the following uniform estimates hold for ε ∈ (0, 1):

(A) { f ε } in L ∞ t L 1 x,ξ and { ε } in L ∞ t L 1 x , (B) { f ε | log( f ε )|} and { f ε |ξ | 2 } in L ∞ t L 1 x,ξ , (C) { ε | log( ε )|} in L ∞ t L 1 x , (D) ( ε M-f ε ) (log( ε M)-log( f ε )) ε 2 in L 1 t,x,ξ , (E) ε M-f ε ε in L 1 t,x,ξ , (F) {J ε } and {J ε log 1/2 log 1/2 max(J ε , e)} in L 1 t,x , (G) { f ε |x|}, { ε |x|} in L ∞ t L 1 x,ξ and L ∞ t L 1 x respectively.
Moreover, { ε } and {J ε } are weakly compact in L 1 t,x . The proof of these estimates uses a fundamental property of energy dissipation. To show that, we define the energy (kinetic+potential) and the Helmholtz free energy respectively as

E(t) := R 2d |ξ | 2 f ε dx dξ + 1 2α 2 R 2d ω L α (y)[ ε ω S (t, x) -ε ω S (t, x -y)] 2 dx dy, ( 15
)
F (t) := R 2d [2D log( f ε ) + |ξ | 2 ] f ε dx dξ + 1 2α 2 R 2d ω L α (y)[ ε ω S (t, x) -ε ω S (t, x -y)] 2 dx dy. ( 16
)
The Helmholtz free energy satisfies the Theorem 4 (Free energy dissipation). The free energy F(t) is dissipated as

d dt F(t) = - 2D ε 2 R 2d f ε -ε M(ξ ) log f ε -log ( ε M(ξ )) dx dξ = -2D R 2d D ε dx dξ, ( 17
)
where the dissipation term is defined as

D ε (t, x, ξ) := 1 ε 2 f ε -ε M(ξ ) log f ε -log ( ε M(ξ )) ≥ 0. ( 18
)
This theorem can be seen as a combination of relations for both the total energy and the entropy of the system.

Proposition 5 (Total energy dissipation). The total energy E(t) is dissipated as

d dt E(t) = 1 ε 2 R 2d |ξ | 2 ε M(ξ ) -f ε dx dξ. ( 19
)
Proof. By multiplying (1) by |ξ | 2 and taking the integrals with respect to x and ξ we obtain

ε 2 R 2d |ξ | 2 ∂ t f ε dx dξ + ε R 2d |ξ | 2 ξ • ∇ x f ε dx dξ + ε R 2d |ξ | 2 F ε ∇ ξ f ε dx dξ = R 2d |ξ | 2 [ ε M(ξ ) -f ε ] dx dξ. ( 20
)
For integrable solutions, the second term on the left-hand side vanishes. Furthermore, with integration by parts, the above equation reduces to

ε 2 d dt R 2d |ξ | 2 f ε dx dξ -2ε R 2d ξ F ε f ε dx dξ = R 2d |ξ | 2 [ ε M(ξ ) -f ε ] dx dξ. ( 21
)
By recalling [START_REF] El Ghani | Diffusion limit of the Vlasov-Poisson-Fokker-Planck system[END_REF], the second term can be rewritten as

-2ε R 2d ξ F ε f ε dx dξ = -2ε 2 R d α,ε div J ε dx = 2ε 2 R d α,ε ∂ t ε dx. ( 22
)
We now want to prove that 2

R d α,ε ∂ t ε dx = 1 2α 2 d dt R 2d ω L α (y)[ ε ω S (t, x) -ε ω S (t, x -y)] 2 dx dy. ( 23
)
First, by recalling (2), 2

R d α,ε (t, x)∂ t ε (t, x) dx = 2 R d L α,ε (t, x)∂ t ε (t, x) dx +2 R d S α,ε (t, x)∂ t ε (t, x) dx. ( 24
)
As regards the first term on the right-hand side 2

R d L α,ε (t, x)∂ t ε (t, x) dx = - 2 α 2 R d [ω L α ω S ω S ε ](t, x)∂ t ε (t, x) dx = - 2 α 2 R d [ω L α ε ω S ](t, x)∂ t [ ε ω S ](t, x) dx = - 1 α 2 R 2d ω L α (y)[ ε ω S ](t, x -y)∂ t [ ε ω S ](t, x) dx dy - 1 α 2 R 2d ω L α (y)[ ε ω S ](t, x)∂ t [ ε ω S ](t, x -y) dx dy = - 1 α 2 d dt R 2d ω L α (y)[[ ε ω S ](t, x) • [ ε ω S ](t, x -y)] dx dy
The second term on the right-hand side can be handled similarly and gives 2

R d S α,ε (t, x)∂ t ε (t, x) dx = 2 α 2 R d [ω S ε ](t, x)∂ t [ ε ω S ](t, x) dx = 1 α 2 d dt R d [ ε ω S ] 2 (t, x) dx = 1 2α 2 d dt R 2d ω L α (y) [ ε ω S ] 2 (t, x) +[ ε ω S ] 2 (t, x -y) dx dy.
By summing up the two previous identities we get [START_REF] Lowengrub | Nonlinear modelling of cancer: bridging the gap between cells and tumours[END_REF], which, inserted in [START_REF] Kobayashi | Numerical analysis of vapor-liquid two-phase system based on the Enskog-Vlasov equation[END_REF], concludes that

ε 2 d dt R 2d |ξ | 2 f ε dx dξ + ε 2 2α 2 d dt R 2d ω L α (y)[ ε ω S (t, x) -ε ω S (t, x -y)] 2 dx dy = R 2d |ξ | 2 [ ε M(ξ ) -f ε ] dx dξ.

Proposition 6 (Entropy relation).

The following estimate holds:

d dt R 2d f ε log f ε dx dξ = 1 ε 2 R 2d [ ε M(ξ ) -f ε ] log f ε dx dξ. ( 25
)
Proof. By multiplying (1) by

(1 + log f ε ) we obtain ε 2 d dt ( f ε log f ε ) + εξ • ∇ x f ε (1 + log f ε ) + εF ε ∇ ξ f ε (1 + log f ε ) = [ ε M(ξ ) -f ε ](1 + log f ε )
By taking the integrals with respect to x and ξ , the second and third terms in the above equation vanish and we obtain

ε 2 d dt R 2d f ε log f ε dx dξ = R 2d [ ε M(ξ ) -f ε ] log f ε dx dξ as announced.
With these two estimates, we can finally prove Theorem 4.

Proof of Theorem 4. From Propositions 5 and 6, we get the following result:

d dt F (t) = 1 ε 2 R 2d |ξ | 2 [ ε M(ξ ) -f ε ] dx dξ + 2D R 2d [ ε M(ξ ) -f ε ] log f ε dx dξ = 1 ε 2 R 2d ε M(ξ )|ξ | 2 dx dξ - R 2d |ξ | 2 f ε dx dξ +2D R 2d [ ε M(ξ ) -f ε ] log f ε dx dξ . ( 26
)
Using ( 6), we know that log

( ε M(ξ )) = log ε + C -|ξ | 2 2D
for some constant C. Inserting this expression of |ξ | 2 in the first two terms on the righthand side of (26), we obtain

2D ε 2 R 2d ε M(ξ ) -f ε log ε + C -log( ε M(ξ )) dx dξ = 2D ε 2 R 2d ε M(ξ ) -f ε -log( ε M(ξ )) dx dξ.
Added to the third term on the righthand side of (26), we obtain the announced result.

In order to prove Theorem 3, a major difficulty is to estimate the flux J ε defined by [START_REF] El Ghani | Diffusion limit of the Vlasov-Poisson-Fokker-Planck system[END_REF]. We start by establishing a useful inequality, recalling the notation [START_REF] Goudon | Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: analysis of the twodimensional case[END_REF].

Lemma 7 (Pointwise estimates on J ε ). For every 0 < r ≤ 1 and (s,

x) ∈ (0, T ) × R d , we have |J ε (s, x)| ≤ r ε D ε (s, x, •) L 1 ξ + C 1 r d exp 2C M r 2 ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ . Proof. For r > 0, we decompose J ε (s, x) = J (1) ε (s, x) + J (2) ε (s, x), with J (1) ε = 1 ε log( fε ε M ) ≥ |ξ | r ξ( f ε -ε M(ξ )) dξ, J (2) ε = 1 ε log( fε ε M ) ≤ |ξ | r ξ( f ε -ε M(ξ )) dξ.
From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

For J

(1) ε , we write

|J (1) ε (s, x)| ≤ r ε log( fε ε M ) ≥ |ξ | r log f ε ε M ε M f ε ε M -1 dξ ≤ r ε D ε (s, x, •) L 1 ξ .
For J (2) ε , we use the Cauchy-Schwarz inequality and, with B(ξ

) := |ξ | r (exp( |ξ | r )-1)
,

|J (2) ε (s, x)| ≤ R d |ξ | 2 ε M B(ξ ) dξ 1/2 1 ε 2 | log( fε ε M )|≤ |ξ | r ε M f ε ε M -1 2 B(ξ ) dξ 1/2 .
Because M(ξ ) is a Gaussian and ε depends only on (t, x), we obtain

|J (2) ε (s, x)| ≤ ε (s, x) 1/2 R d |ξ | 2 M(ξ ) B(ξ ) dξ 1/2 (I 1 + I 2 ) 1/2 .
Here we have split the second integral according to the sign of log( f ε ε M ). When it is negative, we may write, since B(ξ ) ≤ 1,

I 1 := 1 ε 2 fε ε M ≤1 ε M f ε ε M -1 2 B(ξ ) dξ ≤ 1 ε 2 R d ε M f ε ε M -1 log f ε ε M dξ = D ε (s, x, •) L 1 ξ .
The second term is defined as

I 2 := 1 ε 2 0≤log( fε ε M )≤ |ξ | r ε M f ε ε M -1 2 B(ξ ) dξ.
Since log is a concave function, for A > 1 and y ∈ [1, A], we have y -1 ≤ log(y) A-1 log(A) . We choose A = A(ξ

) := exp( |ξ | r ) and y = f ε ε M so that y ∈ [1, A] means exactly 0 ≤ log( f ε ε M ) ≤ |ξ | r .
Then, I 2 can be estimated as follows

I 2 ≤ 1 ε 2 R d ε M f ε ε M -1 log f ε ε M r (exp( |ξ | r ) -1) |ξ | B(ξ ) dξ = D ε (s, x, •) L 1 ξ .
Therefore, for some constant C M , defined through M(ξ ), we have

|J (2) ε (s, x)| ≤ C ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ R d r |ξ | exp -|ξ | 2 C M exp |ξ | r -1 dξ 1/2 .
It remains to treat the integral factor that we denote by I 3 and for r smaller than 1,

I 3 = R d r |ξ | exp -|ξ | 2 C M exp |ξ | r -1 dξ ≤ C r d exp 2C M r 2
where C does not depend on r . This can be seen by splitting the integral in the zones {|ξ | ≤ 2C M r } and {|ξ | ≥ 2C M r }. Finally, we obtain

|J ε | ≤ r ε D ε (s, x, •) L 1 ξ + C 1 r d exp 2C M r 2 ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ .
From this lemma, we deduce the following L 1 bounds on J ε Proposition 8

(Estimate on J ε in L 1 x ). With the decomposition of Lemma 7, J ε (s, x) = J (1) ε (s, x) + J (2) ε (s, x), we have • |J (1) ε (s, x)| ≤ ε D ε (s, x, •) L 1 ξ , • |J (2) ε (s, x)| ≤ C ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ , • J (2) ε (s, •) log 1/2 + |J (2) ε (s, •)| L 1 x ≤ C ε (s, •) log + ε (s, •) L 1 x + D ε (s, •, •) L 1 x,ξ , • J ε log 1/2 log 1/2 max(J ε , e) L 1 t,x ≤ C( D ε (s, •, •) L 1 x,ξ , ε (s, •) log + ε (s, •) L 1 t,x
).

The first two estimates are similar to [9, Proposition 7.1] for the Vlasov-Poisson-Fokker-Planck system. Here, we have additionally included the last two controls and we give a different proof.

Proof. The first two estimates are a direct consequence of Lemma 7. The third estimate follows from the inequality, for u ≥ 1, v ≥ 0 and uv ≥ 1,

(uv) 1/2 log 1/2 (uv) ≤ u log u + √ 2v.
The last result is given for the sake of completeness and its technical proof is postponed to Appendix D. This concludes the proof of Proposition 8.

With these estimates, we can now prove the main result of this section.

Proof of Theorem 3. Estimate (A) follows by mass conservation. The next bounds are deduced from the energy equality ( 16)-( 17) which we write as

R 2d 2D log( f ε (t)) + |ξ | 2 f ε (t) dx dξ + 2D t 0 D ε (s, •, •) L 1 x,ξ
ds ≤ F(0), [START_REF] Perthame | Relaxation of the Cahn-Hilliard equation with singular single-well potential and degenerate mobility[END_REF] where we ignore the nonnegative interaction term as it does not help in this computation. It is standard, see Appendix A, to conclude from this inequality that

R 2d 2D| log( f ε (t))| + 1 2 |ξ | 2 f ε (t) dx dξ + D D ε L 1 t,x,ξ ≤ F(0) + C ε L ∞ t L 1 x , x f 0 L 1 x,ξ . ( 28
)
The estimates (B) and (D) follow immediately. Then, estimate (E) follows from estimate (D) and the Csiszár-Kullback Inequality, see Lemma 13. Estimate (C) is also very standard and we reproduce the proof from [18, Lemma 2.1]. We consider the convex function ψ( ) = log( ) and apply the Jensen inequality. We obtain

ε log( ε ) = ψ( ε ) = ψ R d f ε M M dξ ≤ R d ψ f ε M M dξ = = R d f ε M log f ε -log M(ξ ) M dξ = R d f ε log f ε + |ξ | 2 2D dξ + C R d f ε dξ.
From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

The conclusion follows by taking the absolute values of both sides and integrating with respect to x. Finally, estimate (F) is a direct consequence of Proposition 8, whereas (G) follows from (40). Concerning the weak compactness of { ε }, it follows from estimates (C) and (G). Then, the weak local compactness of {J ε } is a direct consequence of Proposition 8 and the Dunford-Pettis theorem. Indeed, with the notations of Lemma 7, J 1 ε converges strongly to 0 in L 1 t,x . For J 2 ε we first have the weak local compactness in L 1 t,x thanks to the third estimate of Proposition 8, bound (C) and the Dunford Pettis theorem.

To prove the global weak compactness we only need to prove it for J

(2)

ε . We recall that, from Lemma 7, we have

|J (2) ε (s, x)| ≤ C ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ .
Therefore we can estimate with the Cauchy-Schwarz inequality

J (2) ε |x| 1/2 L 1 t,x ≤ C ε |x| 1/2 L 1 t,x D ε 1/2 L 1 t,x,ξ
which yields global weak compactness in L 1 t,x with the Dunford-Pettis theorem. This ends the proof.

The Limit ε → 0

We now perform the analysis allowing us to prove Theorem 1. We take α = ε where the parameter α defines the long range potential (4). Note, however, that different scaling between α and ε could possibly be considered. Recalling the mass balance equation ( 9) and the ξ -moment equation [START_REF] Elbar | Degenerate Cahn-Hilliard and incompressible limit of a Keller-Segel model[END_REF], our aim is to take the limit ε → 0 in these equations, and establish the relations

∂ t (t, x) + div J (t, x) = 0, ( 29
)
J (t, x) = -D∇ (t, x) -∇ (t, x), = -δ [ω S ω S ], (30)
which are equivalent to [START_REF] Elbar | Degenerate Cahn-Hilliard equation: from nonlocal to local[END_REF]. A significant contribution comes from Theorem 3. The entropy bound for ε , see (C), and the L 1 bound on J ε , see Proposition 8, we immediately conclude that

• After extractions, ε and J ε (t, x) admit weak limits in L 1 t,x , and J , see also Theorem 3,

• The equation ( 29) holds in the distributional sense.

The latter estimate on J ε also tells us that ε 2 ∂ t J ε (t, x) converges to 0 in the distributional sense. Therefore, establishing the equation (30) from equation [START_REF] Elbar | Degenerate Cahn-Hilliard and incompressible limit of a Keller-Segel model[END_REF], is reduced to proving the two local weak limits in

L 1 t,x R d ξ ⊗ ξ f ε (t, x, ξ) dξ → D (t, x) I, ε ∇ ε → ∇ (t, x).
These follow directly from the following three lemmas Lemma 9. We have

(0,T )×R d R d ξ ⊗ ξ( f ε -ε M) dξ dx dt --→ ε→0 0. Lemma 10. The sequence { ε } is precompact in L p t L 1 x for every 1 ≤ p < ∞. Lemma 11. The potential ε (t, x) satisfies, uniformly in ε ∈ (0, 1), ε ∞ ≤ C, ∇ ε ∞ ≤ C. ( 31
)
Moreover, we have for every

1 ≤ p < ∞ the strong convergence in L p t L ∞ x , ε (t, x) -→ (t, x), ∇ ε (t, x) -→ ∇ (t, x), (t, x) := -δ [ω S ω S (t, x)]. ( 32
)
The end of the proof of Theorem 1 is thus to establish these results.

Proof of Lemma 11. Recalling the expressions of both long-range and short-range potentials and that α = ε, we see that

ε (t, x) = - 1 ε 2 R d ω L ε (z) ω S ω S ε (t, x -z) -ω S ω S ε (t, x) dz.
Let now set y = z ε , so that from (5) we deduce that

ε (t, x) = - 1 ε 2 R d ω L (y) ω S ω S ε (t, x -εy) -ω S ω S ε (t, x) dy.
Because the convolution terms are smooth (say W 3,∞ ), we may use the Taylor expansion and obtain

ε (t, x) = 1 ε R d ∇ x [ω S ω S ε (t, x)] • y ω L (y) dy - R d D 2 x [ω S ω S ε (t, x)]y • y ω L (y) dy + O(ε)
where the term O(ε) converges to 0 in L ∞ since it is controlled by

Cε R d |y| 3 ω L (y) D 3 x ω S ω S ε (t, •) ∞ dy,
and we recall the uniform bound (A). Moreover, recalling (5), we see that the first term in the right-hand side vanishes and the Hessian matrix reduces to the Laplacian, so that

ε (t, x) = -δ x ω S ω S ε (t, x) + O(ε) ( 33
)
from which we directly conclude from (A)

|| ε || ∞ ≤ C uniformly in ε ∈ (0, 1).
As far as ∇ ε is concerned, the properties of convolution with respect to derivatives gives

∇ ε (t, x) = - 1 ε 2 R d ω L ε (z) ∇ω S ω S ε (t, x -z) -∇ω S ω S ε (t, x) dz,
so that the L ∞ t,x bounded on ∇ ε follows from the previous argument assuming now that ω S ∈ W 4,∞ .

It remains to show that

ε → strongly in L p t L ∞
x , the convergence of ∇ ε uses the same arguments. The convergence follows from (33) since we have

ε (t, x) -(t, x) = -δ ω S ω S ( ε -)(t, x) + O(ε),
so that, thanks to the above control of the term O(ε) and properties of the convolution,

ε - L p t L ∞ x ≤ C ε -L p t L 1 x + Cε. ( 34
)
Using Lemma 10, we obtain the result.

Proof of Lemma 10. This result is a consequence of the compactness averaging lemma in kinetic theory [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF][START_REF] Perthame | A limiting case for velocity averaging[END_REF]. Here, we use the following variant from [

ε∂ t h ε + ξ • ∇ x h ε = h ε 0 + ∇ ξ • h ε 1 .
Then, for all

ψ ∈ C ∞ 0 (R d ), R d (h ε (t, x + y, ξ) -h ε (t, x, ξ)) ψ(ξ) dξ L 1 t,x → 0,
when y → 0 uniformly in ε.

To prove Lemma 10, we cannot apply this averaging lemma directly on

{ f ε } because { f ε } is not bounded in L 2 t,
x,ξ and we follow the argument in [START_REF] El Ghani | Diffusion limit of the Vlasov-Poisson-Fokker-Planck system[END_REF] which follows idea of renormalized solutions [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: global existence and weak stability[END_REF]. We fix ν > 0 and we consider the functions 2 . Now we multiply (1) by β ν ( f ) and obtain

β ν ( f ) = f 1+ν f with derivative β ν ( f ) = 1 (1+ν f )
ε∂ t β ν ( f ε ) + ξ • ∇ x β ν ( f ε ) = ( ε M -f )β ν ( f ) ε -∇ ξ • (F ε β ν ( f ε )).
We verify assumptions of Lemma 12. From (A) we see that

h ε = β ν ( f ε ) is bounded in L 1 t,x,ξ ∩ L ∞ t,x,ξ and hence in L 2 t,x,ξ by interpolation. The L 1 t,x,ξ bound on h ε 0 = ( ε M-f )β ν ( f ε ) ε is deduced from (E) and the L ∞ t,x,ξ bound on β ν ( f ε ). Finally, since F ε is bounded in L ∞ t,x and β ν ( f ε ) is bounded in L 1 t,x,ξ we see that h ε 1 = -F ε β ν ( f ε ) is bounded in L 1
t,x,ξ . The assumptions of Lemma 12 are satisfied and we obtain

R d (β ν ( f ε )(t, x + y, ξ) -β ν ( f ε )(t, x, ξ)) ψ(ξ) dξ L 1 t,x
→ 0, when y → 0, uniformly in ε. As this is true for all ν > 0, Lemma 15 implies

R d ( f ε (t, x + y, ξ) -f ε (t, x, ξ)) ψ(ξ) dξ L 1 t,x → 0, ( 35
)
when y → 0, uniformly in ε.

The final step is to remove the weight ψ in the convergence (35) using uniform bound on { f ε |ξ | 2 }. To this end, consider a sequence of functions

{ψ n (ξ )} n in D(R d ) such that ψ n (ξ ) = 1 for |ξ | ≤ n and ψ n (ξ ) = 0 for |ξ | ≥ n + 1. Then, R d ( f ε (t, x, ξ)(1 -ψ n (ξ )) dξ L 1 t,x ≤ |ξ |≥n f ε (t, x, ξ) |ξ | 2 n 2 dξ L 1 t,x ≤ f ε |ξ | 2 L 1 t,x,ξ n 2
and similarly for the term with f ε (t, x + y, ξ). Hence, we may choose first n large enough and then for such n apply (35) to deduce

ε (x + y) -ε (x) L 1 t,x = R d ( f ε (t, x + y, ξ) -f ε (t, x, ξ)) dξ L 1 t,x → 0, ( 36
)
when |y| → 0, uniformly in ε > 0. This yields compactness in space.

From Lemma 16 we know that { ε } is also compact in time, and as a result

T -h 0 R d | ε (t + h, x + k) -ε (t, x)| dx dt ≤ T -h 0 R d | ε (t + h, x + k) -ε (t + h, x)| dt dx + T -h 0 R d | ε (t + h, x) -ε (t, x)| dt dx ≤ θ(h, k),
where θ(h, k) → 0 whenever |h|, |k| → 0 uniformly in ε. This provides the equicontinuity of { ε } in L 1 t,x which provides us with local compactness in x. From (G) in Theorem 3 we know that

sup 0<ε<1 (0,T )×R d |x ε (t, x)| dt dx < ∞,
and we obtain the strong convergence of the density in L 1 t,x by Fréchet-Kolmogorov theorem, see also [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]. Using Estimate (A) we obtain by interpolation and [31, Theorem 1] the strong convergence in L p t L 1

x for every 1 ≤ p < ∞ and this concludes the proof of Lemma 10.

Proof of Lemma 9. We adapt the proof of Lemma 7. We write

R ε := R d ξ ⊗ ξ( f ε -ε M) dξ ≤ R d |ξ | 2 | f ε -ε M| dξ ≤ log( fε ε M ) ≥ |ξ | 2 r |ξ | 2 ε M f ε ε M -1 dξ + log( fε ε M ) ≤ |ξ | 2 r |ξ | 2 ε M f ε ε M -1 dξ = I 1 + I 2 ,
where r is chosen later. For the first term, we just write

I 1 ≤ r R d log f ε ε M ε M f ε ε M -1 dξ ≤ r ε 2 D ε L 1 ξ .
The term I 2 is decomposed in two parts: where f ε ≥ ε M and f ε < ε M. The resulting integrals are called I A 2 and I B 2 . We only discuss I A 2 as I B 2 can be treated similarly as it was discussed in Lemma 7. We use the Cauchy-Schwarz inequality to obtain

I A 2 ≤ 0≤log( fε ε M )≤ |ξ | 2 r |ξ | 4 ε M B(ξ ) dξ 1/2 • • 0≤log( fε ε M )≤ |ξ | 2 r ε M f ε ε M -1 2 B(ξ ) dξ 1/2 =: I A,1 2 • I A,2 2 ,
where, as before, B(ξ

) = log(A) A-1 = |ξ | 2 r (exp( |ξ | 2 r )-1)
, with

A = A(ξ ) := exp( |ξ | 2 r ).
As in the proof of Lemma 7, we have the inequality log(y) ≥ (y -1) log(A) A-1 which yields with

y = f ε ε M I A,2 2 ≤ 0≤log( fε ε M )≤ |ξ | 2 r ε M f ε ε M -1 log f ε ε M dξ 1/2 ≤ ε D ε 1/2 L 1 ξ . Now we choose r such that M(ξ ) exp( |ξ | 2 r ) = C exp(-a|ξ | 2 )
for some a > 0. Then, we have

R d |ξ | 4 M(ξ ) B(ξ ) dξ ≤ r R d |ξ | 2 M(ξ ) exp |ξ | 2 r dξ ≤ Cr R d |ξ | 2 exp(-a|ξ | 2 ) dξ =: C 2 .
It follows that

I A,1 2 ≤ C 1/2 ε . Finally we get R ε ≤ r ε 2 D ε L 1 ξ + Cε 1/2 ε D ε 1/2 L 1 ξ
and, using the Cauchy-Schwarz inequality, the proof of Lemma 9 is concluded. This also concludes the proof of Theorem 1.

Conclusion

We proved that macroscopic densities { ε } formed from solutions of the Vlasov-Cahn-Hilliard equation (1) converge to the solutions of non-local degenerate Cahn-Hilliard [START_REF] Elbar | Degenerate Cahn-Hilliard equation: from nonlocal to local[END_REF]. It is an open question whether one can obtain a local version of this equation by sending short-range interaction kernel ω S to the Dirac mass δ 0 . One expects in the limit the local degenerate Cahn-Hilliard equation:

∂ t -D -div( ∇ ) = 0 (37)
where = -δ . One can try to perform this limit either on equation [START_REF] Elbar | Degenerate Cahn-Hilliard equation: from nonlocal to local[END_REF] or directly on (1), by sending ω L α * δ 0 , ω S * δ 0 together, see Fig. in the nondegenerate Cahn-Hilliard. On the other hand, when one tries to pass to the limit from ( 11) to (37), the entropy is available but it yields estimates only on

( ω S ) in L 2 t L 2 x , ∇ √ in L 2 t L 2 x .
The minimal required information allowing to pass to the limit seems to be strong compactness of {∇ } in L 2 t L 2

x . Moreover, it is also open to prove whether we can add the "usual" double-well Cahn-Hilliard interaction potential in the system. In fact, as far as this modification is concerned, it is not even clear if there exists a solution to the Vlasov-Cahn-Hilliard equation when the potential is a function of the density .

A. Useful Inequality and Lower Bound on the Energy

We recall two lemmas which have been used in the proof of Theorem 3. The first one is a variant of the Csiszar-Kullback inequality.

Lemma 13. Let f, g ≥ 0 with f 1 = g 1 . Then, f -g 2 1 ≤ f 1 R d ( f -g) (log f -log g)
The second lemma is used to control f log -( f ) from f log f , which immediately establishes the Inequality (38).

Lemma 14. Let log -( f ) := max{-log( f ), 0}. Then R 2d 2D log -( f ε (t)) f ε (t) dx dξ ≤ C ε L 1 t,x , x f 0 L 1 x,ξ + R 2d |ξ | 2 4 f ε (t) dξ dx +D t 0 ||D ε (s, •, •)|| L 1 x,ξ ds. ( 38
)
Proof of Lemma 13. Let f 1 = g 1 = 1. Usual the Csiszar-Kullback inequality gives us

f -g 2 1 ≤ 2 R d f log f g .
By symmetry of the (LHS) we have

2 f -g 2 1 ≤ 2 R d f log f g + 2 R d g log g f = 2 R d ( f -g)(log f -log(g)).
The general case follows by rescaling.

Proof of Lemma 14. We proceed as in [9, Proposition 5.1]. We divide the domain in two parts:

1 := f ε > exp - |x| 4 - |ξ | 2 8D , 2 := f ε ≤ exp - |x| 4 - |ξ | 2 8D , On 1 , log -( f ε ) is bounded so that we have f ε log -( f ε ) ≤ |x| 4 + |ξ | 2 8D f ε , while on 2 , f ε ≤ 1 so that √ f ε log -( f ε ) is bounded by some constant C. Hence, f ε log -( f ε ) ≤ C f ε ≤ C exp - |x| 8 - |ξ | 2 16D . It follows that R 2d log -( f ε (t)) f ε (t) dx dξ ≤ R 2d C exp - |x| 8 - |ξ | 2 16D + |x| 4 + |ξ | 2 8D f ε (t) dξ dx. (39)
Now, we only need to bound the term R 2d |x| 4 f ε (t) dξ dx. For this, we first observe that

d dt R d |x| f ε (t) dξ = 1 ε R d f ε (t) x |x| ξ dξ = x |x| J ε ≤ 2 D ε (s, •, •) L 1 ξ + C ε ,
where we have used Proposition 8 and Young's inequality (with ε ≤ 1). Therefore, for all t ≥ 0

R 2d |x| f ε (t) dξ dx ≤ R 2d |x| f 0 dξ dx + C ε L 1 t,x + 2 t 0 D ε (s, •, •) L 1 x,ξ ds. ( 40
)
Finally, equation (39) simplifies to give the desired result (38).

B. Criteria for Compactness

Lemma 15 (Compactness of

β ν ( f n ) implies compactness of f n ). Let { f n (t, x, ξ)} be a sequence such that { f n } and { f n log f n } are bounded in L 1 t,x,ξ . Let ψ(ξ ) ∈ C ∞ c (R d ).
Suppose that for all ν > 0 and all ε > 0, there exists δ(ν, ε) such that,whenever |y| ≤ δ(ν, ε),

R d (β ν ( f n (t, x + y, ξ)) -β ν ( f n (t, x, ξ ))) ψ(ξ ) dξ L 1 t,x ≤ ε.
Then, for all ε > 0 there exists δ(ε) > 0 such that

R d ( f n (t, x + y, ξ) -f n (t, x, ξ)) ψ(ξ) dξ L 1 t,x ≤ ε.
Proof. First, we observe that

|β ν (s) -s| ≤ s 1 + s ν -s = νs 2 1 + ν s ≤ min(ν s 2 , s).
Therefore, for M and ν to be chosen later

R d ( f n (t, x + y, ξ) -β ν ( f n (t, x + y, ξ ))) ψ(ξ ) dξ L 1 t,x ≤ ≤ ψ ∞ ν f n (t,x+y,ξ )≤M f 2 n (t, x + y, ξ) dξ dx dt + ψ ∞ f n (t,x+y,ξ )≥M f n (t, x + y, ξ) dξ dx dt ≤ ψ ∞ ν M f n 1 + ψ ∞ f n log f n 1 log M . Similarly, R d ( f n (t, x, ξ) -β ν ( f n (t, x, ξ ))) ψ(ξ ) dξ L 1 t,x ≤ ψ ∞ ν M f n 1 + ψ ∞ f n log f n 1 log M .
Let ε > 0. First, we choose ν and M such that

ψ ∞ ν M f n 1 + ψ ∞ f n log f n 1 log M ≤ ε 3 .
Then, we take δ(ν, ε/3) such that

R d (β ν ( f n (t, x + y, ξ)) -β ν ( f n (t, x, ξ ))) ψ(ξ ) dξ L 1 t,x ≤ ε/3
when |y| ≤ δ(ν, ε/3). The conclusion follows by the triangle inequality.

Lemma 16. The sequence { ε } from Lemma 10 is compact in time, i.e.

lim |h|→0 T -h 0 R d | ε (t + h, x) -ε (t, x)| dx dt = 0 uniformly in ε.
The proof of this lemma uses a sequence (ϕ δ ) δ>0 ∈ C ∞ c (R d ) of standard mollifiers with mass 1 such that ϕ δ (x) = 1 δ d ϕ( x δ ) with ϕ of mass 1 and compactly supported. Moreover

∇ k ϕ δ L 1 (R d ) ≤ C δ k ,
and for any function g ∈ L p (R d ),

g ϕ δ L p (R d ) ≤ ϕ δ L 1 (R d ) g L p (R d ) .
Proof. We know that

∂ t ε + ∇ • J ε = 0
where J ε is bounded uniformly in L 1 t,x , see Proposition 8. Using the mollifiers with δ depending on h to be specified later on, we first notice that

T -h 0 R d | ε (t + h, x) -ε (t, x)| dx dt ≤ T -h 0 R d | ε (t, x) -ε (t, •) ϕ δ (x)| dx dt + T -h 0 R d | ε (t + h, x) -ε (t + h, •) ϕ δ (x)| dx dt + T -h 0 R d | ε (t + h, •) ϕ δ (x) -ε (t, •) ϕ δ (x)| dx dt.
For the first and second terms, the computations are the same, hence, we only present it for the first term. Using the properties of the mollifiers and the compactness of ε in space, we want to prove that

T -h 0 R d | ε (t, x) -ε (t, •) ϕ δ (x)| dx dt ≤ θ(δ).
where θ(δ) → 0 when δ → 0 uniformly in ε. We write

T -h 0 R d | ε (t, x) -ε (t, •) ϕ δ (x)| dx dt = T -h 0 R d R d ϕ(y)( ε (t, x) -ε (t, x -δy)) dy dx dt.
Then we use Fubini's theorem and the fact that ϕ is compactly supported in some compact set K we obtain

T -h 0 R d R d ϕ(y)( ε (t, x) -ε (t, x -δy)) dy dx dt ≤ K τ δy ε -ε L 1 ((0,T )×R d ) dy.
where τ x is the translation operator in x variable. Now we use the compactness in space obtained in (36), so that

K τ δy ε -ε L 1 ((0,T )×R d ) dy ≤ |K | sup y∈K τ δy ε -ε L 1 ((0,T )×R d ) ≤ θ(δ).
Therefore the first and the second term are bounded by θ(δ) where θ(δ) → 0 when δ → 0 uniformly in ε. It remains to study the third term. The third term reads

T -h 0 R d | ε (t + h, •) ϕ δ (x) -ε (t, •) ϕ δ (x)| dx dt = T -h 0 R d t+h t ∂ t ε (s, •) ϕ δ (x)ds dx dt = T -h 0 R d - d i=1 t+h t J i ∂ i ϕ δ (s, x)ds dx dt ≤ d i=1 T -h 0 R d t+h t |J i ∂ i ϕ δ (s, x)| ds dx dt,
where we used J ε = (J i ) i=1,...,d . We perform the change of variables v = s-t h , use Fubini's theorem and obtain

T -h 0 R d t+h t |J i ∂ i ϕ δ (s, x)| ds dx dt = h 1 0 R d T -h 0 |J i ∂ i ϕ δ (vh + t, x)| dt dx dv.
Then we use the change of variables τ = vh + t and obtain

h 1 0 R d T -h 0 |J i ∂ i ϕ δ (vh + t, x)| dt dx dv = h 1 0 R d T +h(v-1) vh |J i ∂ i ϕ δ (τ, x)| dτ dx dv ≤ h δ J ε L 1 t,x
.

Using the L 1 t,x bound on J ε and taking δ = h 1/2 we conclude.

C. Uniqueness in L ∞

Let d ≥ 3. We are interested in the uniqueness of these solutions in the class of functions such that

∈ L ∞ t,x ∩ L ∞ t L 1 x ∩ C w t L 1 x ( 41
)
where C w t L 1 x denotes the space of weakly continuous in time functions with values in L 1

x . In this class, the definition of distributional solutions of Theorem 1 can be formulated as follows: for every test function ϕ

∈ C ∞ c ([0, T ) × R d ) we have, with - R d 0 ϕ(0, x) dx - T 0 R d ∂ t ϕ dx dt = D T 0 R d ϕ dx dt - T 0 R d ∇ ( ) • ∇ϕ dx dt,
where ( ) = -δ (ω S ω S ) and ∈ L ∞ t L 1 x . By interpolation belongs to every L p t,x , 1 ≤ p ≤ ∞ and so is ∇ (ρ). Therefore this formulation implies

T 0 ∂ t , ϕ = D T 0 R d ϕ dx dt- T 0 R d ∇ ( ) • ∇ϕ dx dt, ( 42
) for every ϕ ∈ L 1 t W 1,1 x ∩ L 1 t Ḣ 2
x where •, • denotes the dual pairing between Ḣ -2 and Ḣ 2 . Let 1 , 2 be two solutions as above with same initial data which satisfy 1 , 2 ∈ L ∞ t,x . The goal is to prove that 1 = 2 . We substract Equation (42) for 2 and 1 . Writing = 2 -1 , we obtain

T 0 ∂ t , ϕ = D T 0 R d ϕ dx dt -T 0 R d ∇ ( 2 ) • ∇ϕ dx dt dt -T 0 R d 1 ∇ ( ) • ∇ϕ dx dt. ( 43
)
We want to test (43) with ϕ(t) = -N * where N is the Newtonian potential so that -ϕ = . This is an admissible test function. Indeed,

∂ x i ,x j ϕ ∈ L ∞ t L 2
x by the Calderon-Zygmund theory cf. Lemma 1]. Therefore, testing (43) with ϕ we obtain

∇ϕ L ∞ t L 2 x ≤ C ∇N L d d-1 ,∞ L ∞ t L 2d d+2 x . Finally, ϕ ∈ L ∞ t,x cf. [1,
1 2 R d |∇ϕ(T )| 2 dx + D T 0 R d 2 = - T 0 R d ∇ ( 2 ) • ∇ϕ - T 0 R d 1 ∇ ( ) • ∇ϕ.
We denote by I 1 and I 2 the two terms of the right-hand side. Using -ϕ = and the formula ϕ∇ϕ = ∇ • (∇ϕ ⊗ ∇ϕ) -1 2 ∇|∇ϕ| 2 we obtain

I 1 = T 0 R d ϕ∇ϕ • ∇ ( 2 ) ≤ C T 0 R d |D 2 ( 2 )||∇ϕ| 2 ≤ C T 0 R d |∇ϕ| 2 .
as |D 2 ( 2 )| can be bounded as in Lemma 11 only in terms of 2 L ∞ t,x . For I 2 we recall that 1 is bounded in L ∞ t,x . Using the Cauchy-Schwarz inequality it remains to see that ∇ ( ) L 2 ≤ C ∇ϕ L 2 which can be achieved by definition of ( ) and ϕ and the fact that convolutions commute with derivatives. Therefore

I 2 ≤ C ∇ϕ 2 L 2 t,x .
Combining the previous results we obtain

∇ϕ(T, •) 2 L 2 ≤ C T 0 ∇ϕ 2 L 2 ,
so that ∇ϕ 2 L 2 = 0 and the proof is concluded.

D. Estimate on J ε log 1/2 log 1/2 max( J ε , e)

From Lemma 7 we recall that for 0 < r ≤ 1

|J ε (s, x)| ≤ r ε D ε (s, x, •) L 1 ξ + C 1 r d exp 2C M r 2 ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ .
We can make further simplifications: applying a simple rescaling of r , ignoring ε, estimating 1 r d ≤ exp( 1 r d ) and changing r = 1 α we can assume For such α > 1 we have

|J ε (s, x)| ≤ C α D ε (s, x, •) L 1 ξ + C exp α 2 ε (s, x) 1/2 D ε (s, x, •) 1/2 L 1 ξ . (44
v 1 α + exp α 2 u v = v 1 α + 1 2α 3 ≤ 2v α .
Then,

2v α ≤ ⎧ ⎨ ⎩ 2 √ 2 v log 1/2 + log 1/2 + v if v ≥ u log 1/2 + v, 2 u log 1/2 + v if v < u log 1/2 + v. ( 46
)
Proof. The first statement is a consequence of simple calculus and we only have to prove that the minimum is attained for α > 1. This follows from

v u = 2α 3 exp(α 2 ) ≤ exp(2α 2 ) ⇒ 1 2 log v u ≤ α 2 . ( 47
)
As v > e 2 u, we deduce α > 1.

We proceed to the estimates on v 2α . Suppose that v ≥ u log We are left with the case v < u log 1/2 + v. In this case we estimate directly using α > 1:

2v α ≤ 2v ≤ 2 u log 1/2 + v.
We proceed to estimating J ε log 1/2 log 1/2 max(J ε , e) in L 1 t,x . Let us observe that we can always restrict the set of integration to the points (t, x) where D ε L 1 ξ is arbitrarily large. Indeed, given M ≥ e, we estimate

+ T 0 R d J ε log 1/2 log 1/2 max(J ε , e) 1 D ε L 1 ξ >e 2 ε 1 J ε >M .
The first integral is bounded by J ε L 1 t,x log 1/2 log 1/2 M. For the second integral, we note that (44) implies that J ε ≤ C ε so this integral is finite because we can use Young's inequality and log x ≤ x to get ε log 1/2 log 1/2 max( ε , e) ≤ ε + 1 2 ε log max( ε , e).

In the third integral, by estimate (44) with α = 2, we have D ε L 1 ξ ≥ M C for some constant C. It follows that D ε L 1 ξ can be assumed to be arbitrarily large by taking sufficiently large M. This allows us to apply Lemma 17.

Splitting the domain of integration for two subsets as in Lemma 17, it is sufficient to prove that the following functions

P 1 ε := D ε L 1
ε log + D ε L 1 ξ ≤ ε log ε + ε + D ε L 1 ξ .

1 .Fig. 1 .

 11 Fig. 1. Relation between three types of the degenerate Cahn-Hilliard equations

[ 4 ,

 4 Theorem 3.5, Chapter 3]. Moreover, as ∇N ∈ L d d-1 ,∞ (i.e. weak L p spaces) we can use Young's convolutional inequality to deduce

)Lemma 17 .

 17 To choose the best α in the inequality above, we let u= ε , v = D ε (s, x, •) L 1 ξ so that we can estimate |J ε (s, x)| ≤ C v min Let v ≥ e, u ≥ 0, v > e 2 u. The minimum in (45) is attained for α > 1 which is the unique solution of 2 α 3 exp(α 2 ) = v u .

J 1 ξ

 1 ε log 1/2 log 1/2 max(J ε , e)1 J ε ≤M + T 0 R d J ε log 1/2 log 1/2 max(J ε , e) 1 D ε L ≤e 2 ε

  24, Lemma 4.2].

	Lemma 12. Assume that {h ε } is bounded in L 2 t,x,ξ , {h ε 0 } and {h ε 1 } are bounded in L 1 t,x,ξ .
	Moreover, suppose that

  (we use here v u > e 2 and v > e to write log + instead of log). In view of (47), this gives lower bound on α which implies

						1/2 + v. Then, we have
	log v ≥ log u + log log	1/2 + v ⇒ log 1/2 +	v u	≥ log	1/2 + log 1/2 + v
	2v α	≤	log	2 1/2 √ 2v + log 1/2 + v	.

  v = log 1/2 log 1/2 max(v, e)). For P 1 ε (this is the limiting case!), we restrict to the values of D ε L 1

			log	1/2 + log	1/2 +	ξ	D ε L 1 ξ	log	1/2 + log	1/2 +	⎛ ⎝	log	D ε L 1 ξ + log 1/2 1/2 +	ξ D ε L 1	⎞ ⎠ ,
	P 2 ε := ε log	1/2 +	D ε L 1 ξ	log 1/2 + log	1/2 +	ε log	1/2 +	D ε L 1 ξ	.
	are bounded in L 1 t,x (here, we use that log	1/2 + log	1/2
														ξ	so large that
	log 1/2 + log	1/2 +	D ε L 1 ξ	> 1. Then,
		log	1/2 + log 1/2 +	⎛ ⎝	log	D ε L 1 ξ + log 1/2 1/2 +	ξ D ε L 1	⎞ ⎠ ≤ log 1/2 + log	1/2 +	D ε L 1 ξ
	so that P 1 ε ≤ D ε L 1
	P 2 ε ≤ ε log	1/2 +		D ε L 1 ξ		log	1/2 +	ε log	1/2 +	D ε L 1 ξ	≤
	≤ ε log	1/2 +		D ε L 1 ξ		log	1/2 +	ε + ε log	1/2 +	D ε L 1 ξ	log	1/2 + log 1/2 +	D ε L 1

+ ξ .

For P 2 ε , we apply log x ≤ x,

√ x + y ≤ √ x + √ y and 2 x y ≤ x 2 + y 2 to get ξ ≤ ε log + ε + ε log + D ε L 1 ξ + ε log + D ε L 1 ξ so it is sufficient to prove that ε log + D ε L 1 ξ is bounded in L 1 t,x

. This follows from Fenchel-Young's inequality

Acknowledgements. J.S. was supported by the National Agency of Academic Exchange project "Singular

BPN/BEK/2021/1/00044. B.P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623).


Bobbi Fleiss

Helen Stolp

Valérie Mezger

Pierre Gressens

Central Nervous System Development

Keywords: Critical periods, synapses, neuronal networks, microglia, neuroinflammation

Brain development results from the realization of successive genetic programs at different ontogenic stages. The main stages of brain development include the production of neurons, their migration, the programmed death of a significant proportion of neurons, the production of axons and dendrites, the elimination of excess neurites, synaptogenesis, elimination and selective stabilization of synapses where microglia play a central role, the production of astrocytes and oligodendrocytes, myelination, and angiogenesis. These different stages of brain development and maturation are controlled by intrinsic factors (genetically determined) and modulated by extrinsic environmental factors. This modulation by environmental factors can bring into play epigenetic mechanisms.

Brain development results from the accomplishment of successive genetic programs during the different ontogenic stages. It starts with the individualization of the neural plate at the beginning of the 3rd week post-conception and is mostly complete by adolescence. However, some neuronal production persists lifelong. This late neurogenesis has been well described at the level of the olfactory bulbs and the dentate gyrus of the hippocampus, but its importance to the associative neocortex remains to be shown.

The principal stages of brain development can be summarized as follows: induction of the neuroectoderm, formation of the neural tube followed by the telencephalon, neurogenesis (production of neuronal progenitors and then of mature neurons), neuronal migration, programmed neuronal death, generation of neurites (axons and dendrites), elimination of superfluous neurites, synaptogenesis, elimination and selective stabilization of synapses, angiogenesis, gliogenesis (production of astrocytes and oligodendrocytes), and myelination (Fig 1). These different stages of the development and maturation of the brain are controlled by intrinsic factors (determined genetically) and modulated by extrinsic environmental factors. This modulation by environmental factors could bring epigenetic mechanisms into play. The perturbation of the unfolding of any of these different stages of brain development leads to a deficit in brain growth and/or brain malformations. The functional consequences to the child depend on the developmental stage in question. This raises the notion of 'critical periods' which we will touch on throughout this chapter; a critical period is a maturational stage in the lifespan of an organism during which the nervous system is especially sensitive to certain environmental stimuli.

Neuronal production and migration

The expansion of the cortex occurs through two processes that take place in parallel: a lateral expansion process that allows the surface of the cortex to grow, and a radial expansion process that leads to an increase in its thickness. The total number of neurons in a mature human brain is estimated at between three and a hundred billion. The cortex is initially generated from a radially oriented monolayer of proliferative neuroepithelial cells lining the walls of the lateral ventriclesthe ventricular zone (VZ). Around the 7th gestational week in humans a second proliferative zone, the inner subventricular zone (SVZ), appears, derived from precursors in the VZ [START_REF] Fietz | Cortical progenitor expansion, self-renewal and neurogenesis-a polarized perspective[END_REF] . The cells of the inner SVZ do not adopt a radial conformation.

More recently, a third proliferative zone has been identified in the developing neocortex: the outer SVZ, which appears in humans around gestational week 11. In humans, this outer SVZ displays prolonged proliferation. This difference in the behavior of neuronal precursors has been hypothesized to explain the impressive evolutionary expansion and folding observed in the surface of the neocortex.

These proliferative zones, situated on the dorsal side of the lateral ventricles, give rise to the excitatory (glutamatergic) neurons of the different cortical layers (the cortex in mammals consists of six layers), as well as potentially (as this is still controversial and seems to be species / cell subtype-dependent) a portion of inhibitory neurons (GABAergic interneurons).

The other inhibitory neurons derive from another proliferative structure located on the lateral wall of the lateral ventricles, the ganglionic eminence (which is also the source of thalamic neurons) and one located in the preoptic area. Migrating neurons can adopt a radial trajectory by migrating in contact with specialized glial cells, the radial glia, which serve to guide them 2, 3 .

After exiting the mitotic cycle, neurons migrate from the proliferative zones towards the future cortex [START_REF] Fietz | Cortical progenitor expansion, self-renewal and neurogenesis-a polarized perspective[END_REF][START_REF] Bystron | Development of the human cerebral cortex: Boulder Committee revisited[END_REF][START_REF]Dual origin of the mammalian neocortex and evolution of the cortical plate[END_REF][START_REF] Rakic | Neuronal migration and contact guidance in the primate telencephalon[END_REF][START_REF] Rakic | Neuron-glia relationship during granule cell migration in developing cerebellar cortex. A Golgi and electronmicroscopic study in Macacus Rhesus[END_REF][START_REF] Nieuwenhuys | The Human Central Nervous System[END_REF] . The first wave of migratory neurons forms the primitive cortical plate or preplate (Fig. 2). The second wave of migratory neurons then splits this primitive plate into two, around gestational week 7, giving rise to a three-layered structure: layer I, which contains the Cajal-Retzius neurons, is located just below the meninges, layer VI, which contains neurons that have already completed their migration, and finally the subplatea transient structure located below the future neocortex. Consecutive waves of migratory neurons subsequently cross the subplate and the cortical layers already in place, but stop below layer I, thus successively forming layers V, VI, III and II along what is known as an "inside-out" gradient [START_REF] Angevine | Autoradiographic study of cell migration during histogenesis of cerebral cortex in the mouse[END_REF] . Until recently, it was thought that the migration of neurons to the neocortex was complete by around gestational week 24 [START_REF] Sidman | Neuronal migration, with special reference to developing human brain: a review[END_REF] . However, recent studies suggest that GABAergic interneurons continue to be added to the neocortex practically until term [START_REF] Xu | Late development of the GABAergic system in the human cerebral cortex and white matter[END_REF] .

In animal models of encephalopathy of prematurity and in human post-mortem tissues of preterm neonates, a rarefaction of some subclasses of cortical interneurons has been demonstrated [START_REF] Verney | Neuronal damage in the preterm baboon: impact of the mode of ventilatory support[END_REF][START_REF] Stolp | Interneuron Development Is Disrupted in Preterm Brains With Diffuse White Matter Injury: Observations in Mouse and Human[END_REF][START_REF] Panda | Estrogen Treatment Reverses Prematurity-Induced Disruption in Cortical Interneuron Population[END_REF] .

Several molecules involved in controlling neuronal migration and their navigation to the appropriate destinations have been identified. These molecules can be schematically divided into four categories [START_REF] Diaz | The molecular and genetic mechanisms of neocortex development[END_REF][START_REF] Gressens | Pathogenesis of migration disorders[END_REF] .

(i) Cytoskeletal molecules, which play an important role in the initiation and progression of neuronal movement (extension of the apical process and nucleokinesis). Molecules controlling initiation include Filamin-A (an actin-binding protein that is implicated in periventricular nodular heterotopias) and Arfgef2 (a molecule that plays a role in vesicular trafficking and is involved in periventricular nodular heterotopias associated with microcephaly). Among the molecules controlling progression are Doublecortin (a 6 microtubule-associated protein -MAPimplicated in double cortex syndrome), Lis1 (a MAP implicated in Type 1 lissencephaly and Miller Diecker syndrome) and Alpha-1 Tubulin (involved in the formation of tubulin heterodimers).

(ii) Signaling molecules that play a role in lamination, such as Reelin (a glycoprotein implicated in a human disorder combining lissencephaly with cerebellar hypoplasia).

(iii) Molecules modulating glycosylation that provide a stop signal to migrating neurons, such as POMPT1 (protein O-mannosyltransferase, associated with Walker-Warburg syndrome), POMGnT1 (protein O-mannose beta-1,2-N-acetylglucosaminyltransferase, implicated in muscle-eye-brain disease) and Fukutin (a glycosyltransferase implicated in Fukuyama congenital muscular dystrophy). These three human disorders display Type 2 lissencephaly.

(iv) In addition to these three principal groups of molecules, neuronal migration can be modulated by other factors such as certain neurotransmitters (glutamate and GABA), molecules derived from peroxisomal metabolism and certain environmental factors (inflammation, ethanol, and cocaine).

Programmed neuronal death

Depending on the brain region under consideration, 15 to 50% of the neurons initially produced die through a physiological process termed programmed cell death or apoptosis.

Approximately 70% of the neurons that disappear seem to die between gestational weeks 28 and 41 in humans [START_REF] Bhutta | Interactions of inflammatory pain and morphine in infant rats: long-term behavioral effects. Research Support[END_REF] . Programmed cell death is a complex mechanism governed by a balance between cell deathand survival-inducing signals, genetic programs involved in cell death or survival, effectors of cell death and inhibitors of these effectors [START_REF] Polleux | Pre-and post-mitotic events contribute to the progressive acquisition of area-specific connectional fate in the neocortex[END_REF] . Cells that have initiated programmed cell death eventually go onto to be phagocytosed by neighboring glial cells (microglia) without inducing inflammatory phenomena or scar formation. This process is facilitated by changes in the composition of the apoptotic cell membrane ('eat me signals') which are sensed by the microglia which occurs very early in the apoptotic process. Within the cell death process, the activation of caspases (proteolytic enzymes) in the form of a cascade is a key stage that culminates in DNA fragmentation and the death of the neuron. In addition, it is reported that direct interaction between healthy neurons and microglia can initiate cell death programs or even engulfment of the live cell which is then 'digested' by the microglia in a process called phagoptosis [START_REF] Brown | Microglial phagocytosis of live neurons[END_REF] .

Electrical activity appears to be a critical factor for neuronal survival. During the peak period of brain growth in rodents, the administration of substances that block electrical activity induces a serious aggravation of developmental neuronal death in various brain regions. These substances include NMDA receptor inhibitors (MK-801 or ketamine), GABA-A receptor agonists such as antiepileptics (phenytoin, phenobarbital, diazepam, clonazepam, vigabatrin and valproic acid) and anesthetics (a combination of midazolam, nitric oxide and isoflurane) [START_REF] Turski | Neuropathological sequelae of developmental exposure to antiepileptic and anesthetic drugs[END_REF] .

Organization of the central nervous system

Subplate neurons

Subplate neurons constitute a transient structure during brain development [START_REF] Kostovic | Correlation between the sequential ingrowth of afferents and transient patterns of cortical lamination in preterm infants[END_REF] . They are generated around gestational week 7, and the subplate itself appears around gestational week 10. This structure, located under the neocortical plate, reaches its maximum thickness between gestational weeks 22 and 36.

Subplate neurons express various neurotransmitters, neuropetides and growth factors.

They receive synapses and form connections with cortical and subcortical structures. These neurons play an important role during brain development: (i) they produce axons that project to the internal capsule, which act as guides for neurons from layers V and VI; (ii) between gestational weeks 25 and 32, they produce axons for the corpus callosum; and (iii) they constitute a "waiting zone" for thalamocortical axons (with which they establish transient? synapses) before the latter invade the neocortical plate to reach layer IV. This waiting area is necessary for the adequate targeting of thalamocortical afferents. Subplate neurons may be destroyed in preterm newborns presenting with lesions of the periventricular white matter [START_REF] Sarnat | A new classification of malformations of the nervous system: an integration of morphological and molecular genetic criteria as patterns of genetic expression[END_REF] , findings which have been confirmed in animal models of periventricular white matter injury [START_REF] Mcquillen | Selective vulnerability of subplate neurons after early neonatal hypoxia-ischemia[END_REF][START_REF] Sfaello | Patterns of excitotoxin-induced brain lesions in the newborn rabbit: a neuropathological and MRI correlation. Comparative Study[END_REF] . This damage to the subplate could participate in the thalamocortical connection anomalies demonstrated by MRI in preterm infants [START_REF] Ball | The influence of preterm birth on the developing thalamocortical connectome[END_REF] , as well as in the associated cognitive and/or motor disorders.

Axonal and dendritic growth

The final morphology of the mature neuron and the formation of connections between neurons depend on complex cellular dendrite-axons interactions [START_REF] Lanoue | Branching mechanisms shaping dendrite architecture[END_REF][START_REF] Mccormick | Mechanistic advances in axon pathfinding[END_REF] . Maturing neurons develop axons and dendrites that extend as a result reorganization of the cytoskeleton, based on intrinsic and extrinsic cues. Many of the extrinsic cues include secreted chemoattractant or chemorepulsive factors, and adhesion molecules on target neurons that facilitate synapses formation. The interaction between these different ligands and their receptors leads to modifications in calcium levels at the growth cone/dendritic tip, which play a key role in motility and orientation. This ontogenic stage occurs largely, but not exclusively, during the second half of pregnancy, and extends into the postnatal period.

The formation of corticospinal projections occurs at early stages from the subplate, and then from neurons that populate the deep cortical layers (V and VI). Corticothalamic projections also appear, from both the subplate and the deep cortical layers. Corticocortical projections appear later.

A number of transcription factors (e.g. Ctip2, Satb2, Fezf2) regulate the commissural, corticothalamic and corticospinal projections of the cortex.

Dendritic arborisation progresses at slightly different rates for each neuronal subtype, though the stereotypical early branch formation is generally established in the third trimester. Even though genetic programs control the molecular mechanisms of axonal and dendritic growth and determine the initial pattern of these connections, experience and the environment then sculpt this global pattern to generate the final set of connections (through brain plasticity).

The processes underlying cortical expansion are still not fully understood [START_REF] Rana | The Subplate: A Potential Driver of Cortical Folding?[END_REF][START_REF] Quezada | Development Of The Cerebral Cortex And The Effect Of The Intrauterine Environment[END_REF] , but recent advances have strengthened a link to differences in neurite density in addition to neuronal production [START_REF] Quezada | Genetic and microstructural differences in the cortical plate of gyri and sulci during gyrification in fetal sheep[END_REF] .

Synaptogenesis

In the occipital neocortex of the monkey, five successive waves of synaptogenesis have been described [START_REF] Bourgeois | Synaptogenesis, heterochrony and epigenesis in the mammalian neocortex[END_REF] . Based on data obtained from the human occipital cortex [START_REF] Lagercrantz | Organization of the neuronal circuits in the central nervous system during development[END_REF] , the following chronology has been proposed for the human cortex: (i) a first phase starting at gestational week 6-8 and limited to the deep layers, such as the subplate; (ii) a second phase starting at gestational week 12-17, with relatively few synapses produced in the cortex; (iii) a third phase starting at around the middle of pregnancy and finishing at around the 8th month after birth (this phase is characterized by a rate of production of new synapses estimated at around 40,000 per second in the monkey); (iv) a fourth phase that extends up to puberty and is also characterized by a high rate of synapse formation; and (v) a last phase that extends up to adulthood but is somewhat masked by the significant loss of synapses with age.

Experimentally, the two first phases are not influenced by the deprivation of sensory stimuli.

The third phase is partly dependent on sensory input whereas the fourth phase is strongly controlled by sensory stimuli and experience.

The concept of synaptic stabilization (by the elimination of non-stabilized synapses) was first proposed by Changeux and Edelman [START_REF] Changeux | Selective stabilisation of developing synapses as a mechanism for the specification of neuronal networks[END_REF] . During brain development, there is a systematic overproduction of labile synapses in successive phases that lead to redundant connections in a fairly random manner. This stage is mainly controlled by genetic factors.

Each wave of overproduction is followed by a period of stabilization of synapses that are of functional value and the elimination of redundant or useless ones by glial cells. This period of stabilization and elimination is strongly influenced by environmental stimuli and experience.

In this model, the moderate increase in the number of genes involved in synaptogenesis during the course of evolution has resulted in a richer substrate upon which the environment and experience act to generate a more complex network.

Glial proliferation, differentiation, and myelination

Astrocytes

Neocortical astrocytes are of two origins [START_REF] Gressens | The germinative zone produces the most cortical astrocytes after neuronal migration in the developing mammalian brain[END_REF] . At the end of neuronal migration, radial glial cells (which are both neural stem cells and play a role in guiding migrating neurons) are transformed into astrocytes, which then reside in the deep cortical layers and the white matter.

In contrast, the astrocytes of the superficial cortical layers are primarily derived from glial precursors that multiply in the SVZ and then migrate into the cortex.

In the human neocortex, astrocytic proliferation begins around gestational week 24, with a peak around gestational week 26-28. The exact date at which the production of astrocytes ends is not known, but it could be supposed that the major part of astrocytic production is over by the end of a normal pregnancy. This peak in the production of astrocytes around gestational week 26-28 could be of particular importance for preterm newborns. Indeed, astrocytes play several important roles, including in axonal guidance, the stimulation of neuronal growth, synapse formation, the transfer of metabolites between blood vessels and neurons, the establishment of the pattern of certain brain structures, the production of extracellular matrix components, the production of trophic factors, neuronal survival, myelination, and maintenance of the blood-brain barrier. For example, experimentally blocking astrocyte production temporarily in the neocortex of rodents induces an increase in programmed cell death in neurons and long-term changes in neocortical synaptic equipment [START_REF] Zupan | Prenatal blockade of vasoactive intestinal peptide alters cell death and synaptic equipment in the murine neocortex[END_REF] . In addition, recent studies show that astrocytes can acquire different phenotypes which depend upon the physiological or pathological conditions they are exposed to [START_REF] Liddelow | Neurotoxic reactive astrocytes are induced by activated microglia[END_REF] . These different phenotypes can have beneficial or toxic effects on neighboring cells.

Oligodendrocytes and myelination

Oligodendrocytes can be divided into four types depending on their degree of maturation 37 :

(i) Oligodendrocyte precursors that arise from the SVZ are bipolar and mitotically active; their differentiation into preoligodendrocytes occurs during their migration into the future/putative white matter.

(ii) Preoligodendrocytes that are multipolar cells that retain their proliferative capacity. This second cell type is predominant in the periventricular white matter during the second half of pregnancy.

(iii) Immature oligodendrocytes that are multipolar cells that appear during the third trimester and wrap axons in preparation for their myelination. (iv) The last stage is their differentiation into myelinating and extremely multipolar mature oligodendrocytes.

The oligodendrocyte precursors and preoligodendrocytes, the predominant varieties in the brain of preterm newborns, are extremely vulnerable to injury caused by inflammatory mediators including oxidative stress, the excitotoxic cascade, and hypoxic-ischemic insults [START_REF] Volpe | Neurobiology of periventricular leukomalacia in the premature infant[END_REF] . This death of preoligodendrocytes has been implicated in cystic periventricular leukomalacia. This type of severe lesion has progressively decreased in incidence, and has replaced by more diffuse lesions [START_REF] Ferriero | Imaging selective vulnerability in the developing nervous system[END_REF] . In this diffuse white matter lesion in preterm infants, the primary phenomenon is a blockade of oligodendrocyte maturation [START_REF] Verney | Neuronal damage in the preterm baboon: impact of the mode of ventilatory support[END_REF][START_REF] Billiards | Myelin abnormalities without oligodendrocyte loss in periventricular leukomalacia[END_REF][START_REF] Buser | Arrested preoligodendrocyte maturation contributes to myelination failure in premature infants. Comparative Study Research Support[END_REF] , in which the HIF [START_REF] Yuen | Oligodendrocyte-encoded HIF function couples postnatal myelination and white matter angiogenesis[END_REF] and the Wnt [START_REF] Fancy | Axin2 as regulatory and therapeutic target in newborn brain injury and remyelination[END_REF] pathways play a role.

Myelination occurs during a prolonged period, persisting into childhood. The chronology and the degree of myelination vary according to the brain structure studied. There is no detectable myelination in the prosencephalon before the 7th month of pregnancy. Myelination in the telencephalon is most intense during the third trimester and postnatally and is mostly complete by the age of 2 to 3 years. The olfactory and auditory pathways and the sensorimotor cortex are the first to be myelinated, while projection and association pathways (in particular the prefrontal cortex) are the last [START_REF] Girard | MR imaging of brain maturation[END_REF] .

It is also worth noting that preoligodendrocytes are currently indistinguishable from another glial populationthe polydendrocytes, or NG2 glia. Polydendrocytes give rise to oligodendrocytes, but these proliferative, lineage plastic, motile cells also play independent roles in reposes to injury and form uni-directional synapses with neurons [START_REF] Sakry | The role of the NG2 proteoglycan in OPC and CNS network function[END_REF][START_REF] Nishiyama | NG2 cells (polydendrocytes) in brain physiology and repair[END_REF] . However, very little is known about the function of these cells due to the current lack of specific tools with which to study them: an issue that should be a research priority for understanding brain development and health.

Microglia

Microglia constitute 5-15% of the total number of brain cells. Microglia are derived from a pool of embryonic myeloid precursors that arise from the yolk sac and infiltrate the developing brain, distinct from tissue resident macrophage [START_REF] Ginhoux | The mysterious origins of microglia[END_REF] . They have distinct maturational stages, are self-renewing and can live for up to 4 years in the human brain [START_REF] Matcovitch-Natan | Microglia development follows a stepwise program to regulate brain homeostasis[END_REF] . During the first trimester of pregnancy in humans, microglia have an amoeboid morphology that evolves progressively towards an intermediate and then a mature phenotype with a small cell body and long processes. Around mid-pregnancy, microglial populations are principally detected in white matter fascicles such as the internal and external capsules, the corpus callosum, and axonal fascicle crossings [START_REF] Monier | Entry and distribution of microglial cells in human embryonic and fetal cerebral cortex[END_REF] . Experimental data support the hypothesis that these microglia play an important role in the origin of injury to these developing white matter tracts in preterm infants [START_REF] Verney | Neuronal damage in the preterm baboon: impact of the mode of ventilatory support[END_REF][START_REF] Tahraoui | Central role of microglia in neonatal excitotoxic lesions of the murine periventricular white matter[END_REF][START_REF] Van Steenwinckel | Decreased microglial Wnt/betacatenin signalling drives microglial pro-inflammatory activation in the developing brain[END_REF] .

Microglia play a suite of key roles in brain development [START_REF] Tay | Microglia across the lifespan: from origin to function in brain development, plasticity and cognition[END_REF] , including regulating synaptic and neurite pruning and thus connectivity, regulating programmed cell death including the formation of cortical layers, and the initiation of oligodendrocyte maturation impacting myelin production. Like astrocytes, microglia can acquire various phenotypes in response to injury or insult. Depending on brain regions and pathological conditions the microglia can be directly deleterious or can attempt to repair the brain. These diverse roles for microglia mean that during brain development they can directly injure the brain via their inflammatory responses, and by not being capable of participating in brain building due to 'distraction' by inflammatory activation. This is particularly relevant for preterm born infants, as the systemic inflammation commonly seen in these premature babies spreads to the brain and leads to the immune activation of microglia. This "double jeopardy" for the developing brain is an important contributor to the subsequent onset of cognitive and behavioral deficits seen in many formerly very premature babies [START_REF] Bokobza | Neuroinflammation in preterm babies and autism spectrum disorders[END_REF][START_REF] Wilson-Ching | Attention difficulties in a contemporary geographic cohort of adolescents born extremely preterm/extremely low birth weight[END_REF] .

The environment and epigenetics

The different stages of brain development are finely controlled by diverse genetic programs.

Nevertheless, experience, the environment, or stimulation can modulate, adapt or refine the initial pattern to allow the brain to adjust to its environment. These adaptive processes are the expression of the great plasticity of the developing brain and allow the acquisition of new skills throughout childhood and adolescence. This is true even regarding developmental stages that were initially thought to be almost completely controlled by genetic programming, such as the proliferation of neuronal precursors. Experimental studies have shown that a maternal factor (vasoactive intestinal peptide or VIP) can change the proliferative capacity of these precursors by up to 20% [START_REF] Passemard | VIP blockade leads to microcephaly in mice via disruption of Mcph1-Chk1 signaling[END_REF] . However, the environment can also have deleterious effects on brain maturation. Preterm newborns are exposed to numerous stimuli that exist only sparingly or not at all with a fetus of the same age, such as excessive or repeated sensory stimulation, painful stimuli, stress, several neuroactive drugs, as well as the withdrawal of maternal and placental factors. Any environmental factor or drug that affects the brain is capable of altering a few or several stages of brain development.

The major mechanisms by which these environmental factors (positive or deleterious) act on brain maturation depend on the epigenetic events that control the organization and compaction of the genome, and thus the expression of genes. Chromosomes in eukaryotes result from a complex between DNA and a protein sheath called "chromatin", whose basic unit is the nucleosome, around which the DNA is wrapped. Each nucleosome is composed of an octamer of histones (H2A, H2B, H3 and H4). The compaction of DNA -and thus the accessibility of genetic information to the transcriptional machineryis regulated on the one hand by covalent modifications of the DNA (including DNA methylation), and on the other by post-translational modifications that decorate the N-terminal regions (tails) of histones (acetylation, phosphorylation and methylation). The combination of these two types of events governs the opening or closing of chromatin, and the transcription or repression of the gene in question.

The methylation of DNA at CpG islands located in transcriptional regulatory regions of genes represses their transcription by blocking the binding of transcription factors to the DNA [START_REF] Jones | Functions of DNA methylation: islands, start sites, gene bodies and beyond. Research Support[END_REF] .

DNA methyltransferases (DNMTs) catalyze the methylation of DNA by using S-adenosylmethionine (SAM) as a methyl group donor. In addition, the information deposited by the methylation of the DNA is "read" and "translated" by proteins that recognize methylated DNA: the methyl-CpG-binding proteins (MBPs). These MBPs recruit histone-deacetylases, HDACs (and thus constitute the link between DNA methylation and the post-translational modifications of histones), as well as other chromatin remodelers, and contribute to the establishment of transcriptionally inactive chromatin. The crucial role of histone acetylation in brain development is illustrated, for example, by the role of the histone-acetyl-transferase CBP, whose mutation in the heterozygous state is characteristic of Rubinstein-Taybi syndrome in humans, and leads to anomalies of cortical gyration, a reduction of white matter and regional hypoplasia in association with intellectual disability [START_REF] Graff | Epigenetic regulation of gene expression in physiological and pathological brain processes[END_REF] .

16

The level of DNA methylation is much higher in the brain than in other tissues and is essential for brain development in mouse models. [START_REF] Jones | Functions of DNA methylation: islands, start sites, gene bodies and beyond. Research Support[END_REF] . DNA methylation especially affects neuronal differentiation and survival and the neurogenesis/gliogenesis "switch". In humans, several syndromes with intellectual disability have a developmental origin linked to the DNA methylation pathway. This is the case with Rett syndrome, which is associated with mutations of the MBP MeCP2, and ICF syndrome (combined immunodeficiency, instability of the pericentromeric heterochromatin and facial dysmorphia), caused by mutations in Dnmt3b [START_REF] Graff | Epigenetic regulation of gene expression in physiological and pathological brain processes[END_REF][START_REF] Xu | Chromosome instability and immunodeficiency syndrome caused by mutations in a DNA methyltransferase gene[END_REF] . The recent discovery that DNA methylation occurs in a non-CPG context, mostly in brain neurons after birth, has greatly modify our understanding of the role of this epigenetic mark in pathophysiological brain development, in particular for RETT syndrome [START_REF] He | Non-CG Methylation in the Human Genome[END_REF] .There are numerous imprinted genes (i.e. those expressed uniquely from the paternal or maternal allele), which are affected by epigenetic regulation, are expressed in the brain. The deregulation of their expression is often associated with intellectual disability. This is the case, for example, with the neurodevelopmental syndromes, Prader-Willi and Angelman syndromes [START_REF] Chamberlain | Neurodevelopmental disorders involving genomic imprinting at human chromosome 15q11-q13[END_REF] .

Interestingly, fetal stress modifies the pattern of DNA methylation of key genes involved in neurodevelopment and the integrity of the adult brain (such as BDNF, Reelin or GAD67) [START_REF] Graff | Epigenetic regulation of gene expression in physiological and pathological brain processes[END_REF][START_REF] Schang | The impact of epigenomic nextgeneration sequencing approaches on our understanding of neuropsychiatric disorders[END_REF] . Thus, women suffering from depression or anxiety-related disorders during the third trimester of pregnancy give birth to newborns in whom the promoter region of the glucocorticoid receptor gene is hypermethylated. Additionally, neuronal plasticity as well as learning and memory rely on the establishment of epigenetic events, and neuronal activity modifies patterns of DNA methylation [START_REF] Guo | Neuronal activity modifies the DNA methylation landscape in the adult brain[END_REF] .

In summary, one could speak of "neuro-epigenetics" and all that this entails in terms of future therapeutic possibilities, since an in-depth knowledge of epigenetic events and their reversibility opens the door to the search for compounds that could remodel the epigenome of the brain and reprogram neural cells. In this context, diets poor in methionine (the precursor of SAM) and treatment with HDAC inhibitors (such as valproic acid) have been widely prescribed for psychiatric disorders such as schizophrenia or bipolar disorder [START_REF] Graff | Epigenetic regulation of gene expression in physiological and pathological brain processes[END_REF] .  Graff J, Kim D, Dobbin MM, Tsai LH. Epigenetic regulation of gene expression in physiological and pathological brain processes. Physiological reviews 2011;91: 603-49.

Figure 1 .

 1 Figure 1. Schematic representation of the major ontogenic events taking place in the human

Figure 2 .

 2 Figure 2. Schematic illustration of mammalian neocortical formation and neuronal migration.