Mathilde Bouteiller
email: mathilde.bouteiller@etu.univ-rouen.fr

M Gallique

A Kosta

J Hardouin

Y Konto-Ghiorghi

Corinne Barbey

X Latour

A Chane

M G J Feuilloley

A Mérieau

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

INTRODUCTION

T6SSs are macromolecular machineries that are utilized by bacteria to inject diverse effectors into prokaryotic and/or eukaryotic cells by using a contractile sheath (TssB/C). These systems are constituted by a membrane complex, a baseplate and a tip that is expelled in association with effectors (figure n°1). Nowadays, the most studied effectors exhibit antibacterial activity acting on peptidoglycan, lipid membrane or DNA.

The Flagella are extracellular appendages, formed by a basal body and a long filament (polymer of flagellin) connected by a hook (FlgE) (figure n°2) that are assembled in a four-tiered transcriptional regulatory in Pseudomonas [START_REF] Dasgupta | A four-tiered transcriptional regulatory circuit controls flagellar biogenesis in Pseudomonas aeruginosa[END_REF]. Briefly, genes that are clustered in class I to class III permit the expression of the basal body and the hook, whereas class IV genes are involved in the filament and flagellar motor production. In order to obtain a functional flagella, the switch between the expression of class III to class IV genes occurs when the hook achieves a correct length. In that case, the anti-sigma factor (FlgM) is secreted through incomplete flagellar apparatus, releasing the sigma factor FliA, that permits the expression of class IV genes. Here we study different deletion mutants (phenotypes, proteomic and transcriptomic analysis) to identify T6SS and flagellar cross-talk.

The lack of T6SS formed with Hcp1, in ∆hcp1 or ∆tssC deletion mutants, disturbs flagellar formation.

The lack of T6SS formed with Hcp1 inhibits the sigma factor FliA activity

The overexpression of fliA gene restores the secretion of flagellin in ∆hcp1 supernatant, allowing the production of flagella and by the way the "swimming" motility.

Concerning motility, only FliA activity is disturbed by the lack of T6SS formed with Hcp1. The lack of T6SS formed with Hcp1 blocks the secretion of the anti-sigma factor FlgM. FliA is not free in the cytoplasm of ∆hcp1 mutant, and there is no expression of flagellar class IV genes.

Here we show that the deletions of hcp1 or tssC genes induce the loss of flagella and decrease the expression of flagellar class IV genes, that is regulated by the sigma factor FliA. The overexpression of this sigma factor, in ∆hcp1 or ∆tssC strains, restores the wild type phenotype such as flagellin secretion, flagella proper assembly and by the way "swimming" motility. By western blot and LC-MS-MS analysis, we show that the anti-sigma factor FlgM is not secreted by ∆hcp1 deletion mutant and that could impact free FliA in the cytoplasm. This result may explain the less expression of flagellar class IV genes.

On the first hand, we assume that anti-sigma factor FlgM could be secreted through the T6SS formed with Hcp1 in Pseudomonas fluorescens MFE01. This hypothesis will be checked by analyzing the effect of FlgM overexpression on prey bacteria motility using strains that are immobilized when they are co-cultivated with MFE01 [START_REF] Decoin | A Pseudomonas fluorescens type 6 secretion system is related to mucoidy, motility and bacterial competition[END_REF]. On the other hand, we forecast that the complex "sigma-factor FliA/anti-sigma-factor FlgM" is stabilized by T6SS deletion mutation. In order to test this hypothesis we will study phenotypes of the double mutant ∆hcp1-∆flgM.

  flagellar class IV genes is disturbed in MFE01∆hcp1. MFE01, ∆hcp2 and ∆hcp3 possess flagella, contrary to ∆tssC and ∆hcp1

Figure n°1 :

 n°1 Figure n°1 : Schematic representation of T6SS (adapted from Gallique M. et al., 2017b).

Figure n°2 :

 n°2 Figure n°2 : Schematic representation of flagellum (unpublished).

Figure n°3 :

 n°3 Figure n°3 : Electronic microscopy of MFE01, and different T6SS deletion mutants. Red arrows show flagellar filament (Gallique et al., in progress).

Figure n°4 :

 n°4 Figure n°4 : Expression of flagellar genes in MFE01 ∆hcp1 +EV/+hcp1/ +fliA in comparison to wild type strain containing empty vector (EV; A.). Data shown represent the mean ± SEM. Significances of differences between mean values were assessed using Mann-Whitney test (n=6) with significance set as * P < 0.05; **P<0,005; ns : no significance. Identification of genes regulated by the sigma-factor FliA (B.) (Gallique et al., in progress).

Figure n°5 :1

 n°5 Figure n°5 : Phenotypes analysis of ∆hcp1 or ∆tssC that overexpress fliA. (A.) SDS-PAGE on supernatant proteins (B.) Electronic microscopy. Red arrows show flagellar filament (C.) Motility on LB 0,3% agar plate (Gallique et al., in progress).

  Pseudomonas fluorescens MFE01 strain possesses a functional T6SS (Decoin et al., 2014). All genes encoding T6SS membrane complex, baseplate and the sheath are clustered. The presence of three different orphan genes encoding the tube (hcp1, hcp2, hcp3) allows to provide various declinations of T6SS. The T6SSs containing Hcp2 or Hcp3 exhibit an antibacterial activity (Decoin et al., 2014; Gallique et al., 2017), contrary to the T6SS formed by Hcp1 (Decoin et al., 2015) that acts on bacterial motility. We already showed that the deletion of hcp1 gene suppresses MFE01∆hcp1 motility and by the way no flagellin is secreted in this deletion mutant.

Figure n°6 :

 n°6 Figure n°6 : Western blot on supernatant protein of MFE01 or ∆hcp1 after a translational fusion of flag sequence with flgM gene. The anti-Flag primary antibody, and a secondary antibody coupled with alkaline phosphatase was used to highlight Flag-FlgM protein. (unpublished).


Mathilde Bouteiller
email: mathilde.bouteiller@etu.univ-rouen.fr

M Gallique

A Kosta

J Hardouin

Y Konto-Ghiorghi

C Barbey

X Latour

A Chane

M G J Feuilloley

A Merieau

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

T6SSs are macromolecular machineries that are utilized by bacteria to inject diverse effectors into prokaryotic and/or eukaryotic cells by using a contractile sheath (TssB/C). These systems are constituted by a membrane complex, a baseplate and a tip that is expelled in association with effectors (figure n°1). Nowadays, the most studied effectors exhibit antibacterial activity acting on peptidoglycan, lipid membrane or DNA.

Flagella are extracellular appendages, formed by a basal body and a long filament (polymer of flagellin) connected by a hook (FlgE) (figure n°2) that are assembled in a four-tiered transcriptional regulatory in Pseudomonas [START_REF] Dasgupta | A four-tiered transcriptional regulatory circuit controls flagellar biogenesis in Pseudomonas aeruginosa[END_REF]. Briefly, genes that are clustered in class I to class III permit the expression of the basal body and the hook, whereas class IV genes are involved in the filament and flagellar motor production. In order to obtain a functional flagella, the switch between the expression of class III to class IV genes occurs when the hook achieves a correct length. In that case, the anti-sigma factor (FlgM) is secreted through incomplete flagellar apparatus, releasing the sigma factor FliA, that permits the expression of class IV genes. Here we study different deletion mutants (phenotypes, proteomic and transcriptomic analysis) to identify T6SS and flagellar cross-talk.

The lack of T6SS formed with Hcp1, in ∆hcp1 or ∆tssC deletion mutants, disturbs flagellar formation.

The lack of T6SS formed with Hcp1 inhibits the sigma factor FliA activity.

The overexpression of fliA gene restores the production of flagella and by the way the "swimming" motility.

Only FliA activity is disturbed by the lack of T6SS formed with Hcp1.

Here we show that the deletions of hcp1 or tssC genes induce the loss of flagella and decrease the expression of flagellar class IV genes, that are regulated by the sigma factor FliA. The overexpression of this sigma factor, in ∆hcp1 or ∆tssC strains, restores the wild type phenotype such as flagellin secretion (data not shown), flagella proper assembly and by the way "swimming" motility. By western blot and LC-MS-MS analysis, we show that the anti-sigma factor FlgM is not secreted in ∆hcp1 deletion mutant and could bind free FliA in the cytoplasm. This result may explain the less expression of flagellar class IV genes.

Our first hypothesis (1) is that this anti-sigma factor FlgM could be secreted through the T6SS formed with Hcp1 in Pseudomonas fluorescens MFE01. The second hypothesis (2) is that a toxin secreted through T6SS formed by Hcp1 could stabilized the FliA/FlgM complex. Our last hypothesis (3) is that this toxin could interact directly with FliA, and inhibits by the way its activity. In order to test these hypothesis, interactomic studies with FlgM and Hcp1 protein by co-immuno-precipitation are in progress.

FLGM INTERACTIONS

The wild type strain is able to secrete FlgM anti-sigma factor, in contrary to hcp1 deletion mutant. FlgM seems to be strongly associated with other protein(s) during protein extraction.

(3) The toxin interacts directly with FliA, and inhibits its activity

(1) FlgM is secreted through Hcp1-T6SS

(2) The toxin secreted through Hcp1-T6SS stabilizes FliA/ FlgM complex

The Pseudomonas fluorescens MFE01 strain possesses a functional T6SS [START_REF] Decoin | A Type VI Secretion System Is Involved in Pseudomonas fluorescens Bacteriam Competition[END_REF]. All genes encoding T6SS membrane complex, baseplate and the sheath are clustered. The presence of three different orphan genes encoding the tube (hcp1, hcp2, hcp3) allows to provide various declinations of T6SS. The T6SSs containing Hcp2 or Hcp3 exhibit an antibacterial activity [START_REF] Decoin | A Type VI Secretion System Is Involved in Pseudomonas fluorescens Bacteriam Competition[END_REF]Gallique et al., 2017a), contrary to the T6SS formed by Hcp1 that acts on bacterial motility [START_REF] Decoin | A Pseudomonas fluorescens type 6 secretion system is related to mucoidy, motility and bacterial competition[END_REF]. We already showed that the deletion of hcp1 gene suppresses MFE01∆hcp1 motility and by the way no flagellin is secreted in this deletion mutant.

Figure n°1 :

 n°1 Figure n°1 : Schematic representation of T6SS (adapted from Gallique M. et al., 2017b).

Figure n°2 :

 n°2 Figure n°2 : Schematic representation of flagellum (unpublished).

Figure n°3 :

 n°3 Figure n°3 : Electronic microscopy of MFE01, and different T6SS deletion mutants. Red arrows show flagellar filament (Bouteiller M. et al., in progress).

Figure n°4 :

 n°4 Figure n°4 : Expression of flagellar genes in MFE01 ∆hcp1 +EV/+hcp1 in comparison to wild type strain containing empty vector (EV). Data shown represent the mean ± SEM. Significances of differences between mean values were assessed using Mann-Whitney test (n=6) with significance set as * P < 0.05; **P<0,005 (Bouteiller M. et al., in progress).

Figure n°5 :1

 n°5 Figure n°5 : Phenotypes analysis of ∆hcp1 or ∆tssC that overexpress fliA. Motility on LB 0,3% agar plate (Bouteiller M. et al., in progress).

Figure n°6 :

 n°6 Figure n°6 : Western blot on different protein fractions of MFE01 or ∆hcp1 after a translational fusion of flag sequence with flgM gene. An anti-Flag primary antibody, and a secondary antibody coupled with alkaline phosphatase were used to highlight Flag-FlgM protein. (Bouteiller M. et al., in progress).

Figure n°7 :

 n°7 Figure n°7 : Western blot on different concentrated intracellular protein complexes of MFE01 or ∆hcp1 after a translational fusion of flag sequence with flgM gene. An anti-Flag primary antibody, and a secondary antibody coupled with alkaline phosphatase were used to highlight Flag-FlgM protein. (Bouteiller M. et al., in progress).


Mathilde Bouteiller
email: mathilde.bouteiller7@univ-rouen.frannabelle.merieau@univ-rouen.fr

M Gallique

M Feuilloley

A Merieau

A Merieau Feuilloley

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Deletions of tssC, hcp1, vgrG1 or tec1 genes induce accumulation of a toxin that supresses "swimming" ability.

Principle Results Hypothesis

Translational fusions are realized between T18 or T25 domain, and interest proteins Both of these constructions are introduced in E. coli BTH101

Interaction between Hcp1, VgrG1 and Tec1 (Type 6 effector chaperon).

One toxin is secreted by T6SS containing Hcp1 protein, and is involved in motility inhibition. VgrG1 Cterm domain interacts with Tec1, VgrG1 Nterm domain and it-self. VgrG1 Nterm domain interacts with Hcp1.

Phenotypical analysis

MFE01 and different deletion mutants were spotted around Pseudomonas fluorescens MFN1032 on a LB 0,3% agar plate. ∆tssc is a sheath mutant devoid of a functional T6SS.

MFN1032 is sequestered by MFE01, but deletion mutants are unable to inhibit MFN1032 motility.

MFE01 and different deletion mutants were spotted on a LB 0,3% agar plate. Secreted proteins of MFE01, and deletion mutants were separated by SDS-PAGE 18%, and identified by MALDI-TOF.

Flagellin

The toxin associated with Hcp1 impacts motility by supressing flagellin secretion. An other effector, which is accumulated in ∆tec1 or ∆vgrG1 impacts flagellar activity, probably by acting on flagellum assembly.

P. fluorescens

MFE01 inhibits MFN1032 « swimming », by T6SS formed by Hcp1. VgrG1 and Tec1 are essential for this T6SS activity.

Pseudomonas fluorescens MFE01 is an environmental strain, considered as a biocontrol agent that possess a type 6 secretion system (T6SS). T6SSs are contractile nanomachines involved in the secretion of effectors directly in other cells (bacteria or eukaryotic cells). T6SS is composed of a membrane complex, a baseplate and a tip that is expelled thanks to a contractile sheath (Figure 1). Effectors are also secreted with the proteins that compose the tip. Nowadays, the most studied effectors exhibit antibacterial activity, acting on peptidoglycan, lipid membrane, or DNA (Durand et al., 2014).

In Pseudomonas fluorescens MFE01, genes encoding T6SS membrane complex, baseplate and sheath are clustered. There are at least three different orphan genes encoding the tube (hcp1, hcp2 and hcp3). It has already been demonstrated that T6SS of MFE01 protects potatoes tubers against the phytopathogen Pectobacterium atrosepticum (Decoin et al., 2014). Two different hcp genes (hcp2 and hcp3) are involved in antibacterial activity, in contrary to the other one (hcp1) that acts on bacterial motility (Decoin et al., 2015). Bacterial motility is allowed thanks to an extracellular appendage, the flagella. This complex structure consists of basal body and long filament (polymer of flagellin) connected by a "hook" (Figure 2).

We study the genetic environment and the interaction between three proteins, resulting of the expression of hcp1, vgrG1 and tec1 genes, in order to elucidate the structure of T6SS containing Hcp1. Deletion mutants were already constructed and phenotypical analysis are ongoing to understand the mechanisms of the motility decrease. References :

  is observed in ∆hcp1, flagellin is secreted by vgrG1 and tec1 deletion mutants.

Figur e 1 :

 1 Figur e 1 : Schematic representation of T6SS (Gallique et al., 2017).

  Decoin, V. et al. A Type VI Secretion System Is Involved in Pseudomonas fluorescens Bacterial Competition. PLOS ONE 9, e89411 (2014). Decoin, V. et al. A Pseudomonas fluorescens type 6 secretion system is related to mucoidy, motility and bacterial competition. BMC Microbiol 15, (2015). Durand, E., Cambillau, C., Cascales, E. & Journet, L. VgrG, Tae, Tle, and beyond: the versatile arsenal of Type VI secretion effectors. Trends Microbiol. 22, 498-507 (2014). Gallique, M., Bouteiller, M. & Merieau, A. The Type VI Secretion System: A Dynamic System for Bacterial Communication? Front Microbiol 8, 1454 (2017).


S Bailly

S Giordano

L Landrieu

N Chehata

CROP-ROTATION STRUCTURED CLASSIFICATION USING MULTI-SOURCE SENTINEL IMAGES AND LPIS FOR CROP TYPE MAPPING

Keywords: Classification, Temporal regularization, Conditional Random Fields, Agriculture, Monitoring, Sentinel

Automatic analysis of Sentinel image time series is recommended for monitoring agricultural land use in Europe. To improve classification capacities, we propose a temporal structured classification combining Sentinel images and former vintages of the Land-Parcel Identification System. Inter-annual crop rotations are learned and combined with the satellite images using a Conditional Random Field. The proposed methodology is tested on a 233 km 2 study area located in France and with a 25 categories national nomenclature. The classification results are globally improved.

INTRODUCTION

Sentinel image time series exhibit unprecedented characteristics that are perfectly tailored with agriculture monitoring. Spatial resolution of both optical and radar images are very consistent with a parcel-level approach. More remarkably, the high temporal frequency of acquisition (5-6 days) and the fact that this program is at least maintained until 2030 pave the way to chronicle short and long-term evolutions. In Europe, several use-cases for agricultural monitoring with Sentinel images have been proposed [START_REF]Towards future copernicus service components in support to agriculture?[END_REF] such as for example crop monitoring, controlling Common Agricultural Policy (CAP) payments with remote sensing, updating and quality control of the Land Parcel Identification System (LPIS) or precision farming at the farm-level. In the LPIS, the information on the agricultural parcels (cover types and areas) are usually derived from annual declaration by the farmers followed by manual controls by national paying agencies.

In this context, recent studies have assessed the classification accuracy that can be obtained with Sentinel images on agricultural land covers [START_REF] Immitzer | First experience with sentinel-2 data for crop and tree species classifications in central europe[END_REF][START_REF] Inglada | Assessment of an operational system for crop type map production using high temporal and spatial resolution satellite optical imagery[END_REF][START_REF] Kussul | Parcel-based crop classification in ukraine using landsat-8 data and sentinel-1a data[END_REF][START_REF] Gisat | Czech agriculture national demonstrator[END_REF]. However, in all of them the nomenclature is restricted to few and most frequent annual crops. Classes known to be challenging to determine with remote sensing such as permanent grassland or permanent crops are not taken into account. Furthermore when a parcel-based approach is used, small parcels are usually ignored (< 250 ha in [START_REF] Kussul | Parcel-based crop classification in ukraine using landsat-8 data and sentinel-1a data[END_REF]).

To overcome this issue, we believe that modelling the spatio-temporal structures concerning agricultural parcels in combination with the Sentinel image time series, can lead to significant gains in accuracy with more exhaustive nomenclatures. These structures are of different types and can concern for example (i) the choice of a crop type on a given parcel (annual or multi-annual crop rotations), (ii) local similarities in events (ploughing, seeding or harvesting dates) or (iii) management practices for the parcels belonging to a same farmer. In this paper, we investigate the modelling and the integration in the classification of inter-annual crop rotation information.

Two different approaches can be used to model rotations. The first one consists in using a priori agronomist expert knowledge. Several models have been proposed and assessed [START_REF] Dury | Models to support cropping plan and crop rotation decisions. a review[END_REF]. However, the models are strongly dependent on the study area and have no adaptation capacity to environmental or agricultural management changes. The second approach is learning rotations as way of overcoming the limitations driven by the expert knowledge a priori approaches. The data that can be used for learning are the former versions of LPIS. This method has recently been studied by [START_REF] Osman | Assessment of a markov logic model of crop rotations for early crop mapping[END_REF]. Nevertheless, only very few studies have focused on the integration of crop rotation information into classification pipelines. A prediction model based on the Markov logic is proposed by [START_REF] Osman | Assessment of a markov logic model of crop rotations for early crop mapping[END_REF]. However, there is no combination with remote sensing observations.

The objective of this paper is then to propose a method to integrate the crop rotation temporal structure into a Sentinelbased classification process and to assess the capacity of the method to improve classification results. The methodology is explained in Section 2, the experimental set-up (study area and datasets) along with result analysis are proposed in Section 3. Conclusions are drawn in Section 4.

METHODOLOGY

The classification method proposed is made up with a data term (Section 2.1) combined with a Conditional Random Fields (CRF) for temporal structured prediction (Section 2.2).

Parcel-based multi-source and multi-temporal classification

The data term is obtained with common state-of-the art methods. Sentinel images were automatically downloaded from the French national mirror Peps [START_REF] Cnes | Plateforme d'exploitation des produits sentinel[END_REF] (Sentinel-1) in the Ground Range Detected (GRD) format and from the national downstream service Theia [START_REF] Theia | Pôle thématique surfaces continentales theia[END_REF] (Sentinel-2) in ground geometry and calibrated in Top of Canopy (TOC) reflectance. Dual polarisation GRD Sentinel-1 images were first calibrated to σ 0 radar backscattering coefficient. Orthorectification is performed using the SRTM digital terrain model and the georeferencing information supplied with the GRD files. The speckle is slightly removed using a simple 5 × 5 Lee filter [START_REF] Lee | Digital image enhancement and noise filtering by use of local statistics[END_REF]. An extra radar feature (

σ0 V H σ0 V V
) is computed. This ratio is known as being more independent to the slope and moisture content effects of each agricultural parcel [START_REF] Veloso | Understanding the temporal behavior of crops using sentinel-1 and sentinel-2-like data for agricultural applications[END_REF]. For optical images, the missing data (clouds) are filled in using provided cloud masks with a multi-temporal spline interpolation. Average and standard deviation of these 3 radar channels and the 10 optical reflectance bands are then computed for each date of the satellite time series for each agricultural parcel geometry of the LPIS. At the parcel-based level, a Random Forest classifier is used. Decision tree votes are gathered to model the data term probabilities p(y n |x n ) of Equation 2, y n being the agricultural classes considered and x n the parcel-based multi-source and multitemporal time serie features.

Temporal-structured classification of parcels

In order to model the temporal structure of the crop rotations, we used the graphical model given in Figure 1. This first order Markov chain can be generalized to greater orders. However, because of the specifications of LPIS data, only the first order has been applied and assessed in this paper. Indeed, the LPIS are known at the parcel-block level (i.e. a group of contiguous parcels with the same operator but possibly different land covers) before 2015. The agricultural parcel class of a given n year is modeled by a random variable Y n . We consider this variable only influenced by the previous year variable Y n-1 . The satellite observations of each n year X n are taken into account with a Conditional Random Field (CRF). With such a modelling and under the hypothesis of the independance between transitions Y n and observations X n , the factorisation of the graphical model given in Figure 1 can be written with Equation 2.

X 1 ∅ X 2 ∅ X n-1 ∅ X n S1-2 Y 1 LPIS parcel-block Y 2 LPIS parcel-block Y n-1 LPIS parcel Y n LPIS parcel ...
p θ (y|x) ∝ p(y 1 ) × p(y 1 |x 1 ) × n i=2 (p(y i |y i-1 ) × p(y i |x i ))
(1) Nevertheless, the use of X = {X 1 , X 2 , ..., X n-1 } in dashed line in Figure 1 is not necessary. The former LPIS versions give a robust knowledge of the realization of the random variables Y = {Y 1 , Y 2 , ..., Y n-1 }. Indeed, errors concerning crop types in the initial declaration by farmers are in minority and furthermore this information is thereafter controlled by National Paying Agencies. Therefore, we propose a simpler form given in Equation 2. The data term is estimated with the Random Forest method (Section 2.1) and a temporal structure term p(y n |y n-1 ) is necessary.

p θ (y n |x n , y 1 , . . . , y n-1 ) ∝ p(y n |y n-1 ) temporal structure × p(y n |x n ) data term (2)
To estimate the regularization term p(y n |y n-1 ), we learn a transition matrix M built up with a series of parcel-block LPIS. An hypothesis of spatio-temporal homogeneity of the parcels is considered. Only geometrically stable and pure in terms of land cover parcel-blocks are used. Equation 3shows how to estimate the p(y n |y n-1 ) with past agricultural land cover transitions.

M k,l = n k,l n k
with n k,l the number of transitions k → l (for all parcels of all years considered) and n k the number of occurences of the agricultural class k (for all the parcels of the n-1 first years)

(3)

RESULTS

Experimental Set-up

The method has been tested on a 233 km 2 study area located 60 km at the East of Paris. A national 25 classes land cover classification was used to estimate the transition matrix of Equation 3 on a French department administrative area (5 915 km 2 ). 36 891 geometrically stable and pure in terms of land covers parcel blocks contribute to the estimation of the M matrix.

As for the data term, only the land cover classes present on the study area were learned. This agricultural classes and the number parcels considered are Corn (350 parcels), Barley (158), Other Cereals (889), Rape seed (85), Protein (peas) (76), Fiber plants (76), Forage crops (46), Permanent grassland (725), Fruit trees (30), Vegetables (131). These classes are categories and can possibly gather a large number of agricultural land covers (16 different classes for Other Cereals) on the department area. The average area of the parcels on the study area is 4.5 ha with high standard deviation (σ = 6.5 ha). The 2016 parcel-based LPIS has been used for training and validation of the data term.

Table 1 gives the number of features used for the estimation of the data term. Sentinel-2 acquisition technical problems on the particular orbit covering the study area had limited the amount of optical images available. Table 1. Features used for the estimation of the data term.

In the following a configuration named unstructured corresponds to the results of the parcel-based classification. The Random Forest was trained with 100 decision trees using all aggregated attributes from Table 1 and a maximum depth of the trees of 15. Then a second configuration called structured in the rest of the paper corresponds the product of the two probabilities defined in Equation 2for classification. The 2015 parcel-based LPIS was needed to determine the necessary previous land cover observed on a parcel. 5 years (2010-2014) of the parcel-block LPIS were used for the estimation of the transition matrix. For the unstructured and temporal structured classification, 50% of the parcels were used for training and 50% for validation, and the results proposed are the average of 10 iterations.

The classification was assessed with global metrics (overall accuracy (OA), weighted (F-score-w) and unweighted Fscore by the number of samples of each classes) and per-class metrics (F-score, recall and precision).

Result analysis

The transition matrix estimated contains very relevant information. For instance, strong inter-annual crop rotation are observed. The probability to grow Other Cerals (common wheat) after Rape seed is 97.09% or Protein (peas) to Other Cereals (94.85%). The permanent grassland remains stable (94.45%), such as permanent crops like Fruit trees (83.65%).

Table 2 shows the global results after 10 iterations. The results are proposed only with the radar features or the optical features (Table 1) or with their aggregation. Taking into account the temporal structure improves all the global metrics for all the feature configurations. In OA, we observe a gain of +2.7% for the radar configuration, +4.6% for the optical and +2.6% for the combined optical and radar one).

However the improvement of classification accuracy is not homogeneous for the different agricultural classes. Table 3 evaluates the effect of taking into account the temporal structure on each class, with the optical and radar features.

First, we can notice that there is a strong improvement on the stable over time agricultural classes such as Perma- Then, for annual crops that are used with regular rotation patterns, the class metrics are improved. (Protein (peas) (+0.4%), Rape seed (+1%), Other Cereals (+0.7%). However, when the transitions are less respected by farmers, the temporal structure can cause a loss in the classification accuracy. This is the case for Corn (-6.3%) or Barley (-8.2%). For the Barley class, the decrease (Recall) is partially due to the misclassification of the Fiber Plants class. For Corn and Barley the decrease could be explained by the fact that these crops are used in 3-years rotation, a structure that the 1st-oder Markov chain can not capture completely.

Finally, there is a major drawback with the Fiber plants class (-97.4% in F score). For that particular case, the data term is reliable (97.4% in F-score) and after temporal regularization almost all the Fiber plants are classified as Barley. The result could be explained by the fact that there is a very small amount of geometrically stable Fiber plants on the area. The strong general transition rule learned for the Fiber plants could not be useful for those specific parcels.

CONCLUSION

In conclusion, we proposed a framework to take into account the crop rotation temporal structure in the classification. The classification is globally improved. The method is based on learning both data term and temporal structure. As a result, it can be easily applied to other areas. Indeed, the LPIS are produced at national scale each year. Results have however to be produced on numerous and diverse areas. Several im- provement of the method can be proposed. Greater order of the Markov chain could be tested in the future with the upcoming parcel-based versions of the LPIS. Other structures could also be modeled such as spatial relationship between parcels.The exhaustiveness and the few mistakes contained in the LPIS give valuable label information. The approach proposed is this paper will be tested in prototype of operational systems. In this article we only assessed the effet of temporal structure after 1-year of satellite observations. The objective would be to apply the method to partial time series to propose a pre-filled LPIS declaration for the farmers

Fig. 1 .

 1 Fig. 1. Conditional Random Field.

Sentinel- 1

 1

Table 2 .

 2 Global classification metrics after 10 iterations. Almost all the Fruit trees validation samples were classified as Permanent grasslands in the unstructured configuration. The temporal structure is also useful to separate classes that have similar data terms. This is for example the cases for Forage crops (+70.5% in F score) and Permanent grasslands. These two classes are very similar in the optical and radar features. As a result, Forage crops are almost for the most part classified as permanent grasslands. The temporal structure enables to make the distinction.

		Unstructured
	Configuration	Overall	Unweighted Weighted
		Accuracy	F-score	F-score
		OA	F-score	F-score-w
	Radar	0.892	0.734	0.878
	Optical	0.824	0.624	0.809
	Radar+Optical	0.890	0.744	0.885
		Structured
	Radar	0.919	0.776	0.911
	Optical	0.870	0.675	0.853
	Radar+Optical	0.916	0.762	0.906
	nent grasslands (+8.7% in F score) and Fruit trees (+93% in
	F score).

Table 3 .

 3 Effect on temporal structure on Class Metrics

	Class	F-score Recall Precision
			Unstructured
	Corn	0.941	0.929	0.953
	Barley	0.898	0.937	0.862
	Other cereals	0.947	0.956	0.9378
	Rape seed	0.959	0.975	0.944
	Protein	0.949	0.932	0.968
	Fiber plants	0.974	1	0.950
	Forage crops	0	0.1	0
	Permanent grasslands	0.868	0.814	0.930
	Fruit trees	0.010	0.090	0
	Vegetables	0.895	0.914	0.877
	Class		Structured
	Corn	0.878	0.831	0.935
	Barley	0.816	0.785	0.849
	Other cereals	0.954	0.941	0.968
	Rape seed	0.969	0.985	0.954
	Protein	0.953	0.967	0.939
	Fiber plants	0 0	0.1	0
	Forage crops	0.705	0.778	0.648
	Permanent grasslands	0.955	0.943	0.967
	Fruit trees	0.940	1	0.892
	Vegetables	0.451	0.970	0.297


Bonnemain Thibault

Lptms

Denis Ullmo

Thierry Gobron

Schrödinger apprach to Mean Field Games with negative coordination

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled objects in interaction. Here we consider such systems when the interaction between controlled objects are negatively coordinated and analyze the behavior of their solutions using the correspondence which have been evidenciated with the non linear Schrödinger equation. When the system is conned, we rely on the existence of an ergodic state which notion has been shown previously to characterize most of the dynamics for long optimization times. In the case of an unbounded domain, such an ergodic state does not exist, and we show the existence of a scaling solution that can play a similar role in the analysis.

I. INTRODUCTION

Mean Field Games are a powerful framework introduced a little more than ten years ago by Lasry and Lions [START_REF] Lasry | Jeux à champ moyen. I le cas stationnaire[END_REF] to deal with complex problems of game theory when the number of players becomes large. Their applications are numerous, ranging from nance [START_REF] Lachapelle | Les cahiers de la chaire/n 16[END_REF] to sociology [2] and engineering science [START_REF] Arman | Load shaping via grid wide coordination of heating-cooling electric loads: A mean eld games based approach[END_REF], when tackling optimization issues involving many coupled subsystems.

Important mathematical eorts and progresses in this eld have been made recently, for one part on the coherence of the theory [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications[END_REF][START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications[END_REF], with important results on the existence and uniqueness of a solution to these problems [START_REF] Cardaliaguet | Notes on mean eld games[END_REF], and in the study of the convergence of a many player game to its mean eld counterpart [START_REF] Bensoussan | The master equation in mean eld theory[END_REF], and on the other part on the development of eective numerical schemes [3] granting the opportunity for more application oriented studies, and, especially in the more recent years, in the extension of the theory to more complex framework [].

However, even for simple Mean Field Games, the constitutive equations of such models are dicult to analyze. Few exact solutions exists and the numerical schemes, while quantitatively accurate, do not provide a complete understanding of the underlying Mean Field Games mechanisms. This lack of understanding of the behaviour of a Mean Field Games (ie of the solutions of the corresponding system of dierential equations) is most presumably slowing down the appropriation of these techniques by researchers concerned primarily by the application to sociology, economy, or engineering sciences.

It is therefore useful to study a small set of paradigm Mean Field Game problems, which, in the spirit of the Ising problem of statistical mechanic, are simple enough to be analyzed, and understood in the sense a physicist would give to that word but complex and rich enough to shed some light on the behaviour of a set of mechanisms that will characterize a much larger class of Mean Field Games. Quadratic Mean Field Games, for which the connection to non-linear Schrödinger (NLS) equation can be use to make a link with a eld very familiar to physicists, are a good candidate for that role, and have been studied by some of us in the regime of strong positive coordination [START_REF] Swiecicki | Schrödinger approach to mean eld games[END_REF][START_REF] Ullmo | Quadratic mean eld games[END_REF].

In this paper, we would like to extend these previous studies by considering in details the negative coordination regime. This will in particular allow us to tackle in depth one of the conceptual diculties posed by Mean Field Games, namely the one associated with the forward-backward structure of the equations, which imposes in some sense a kind of non-locality in time of the problem. Indeed, since a forward equation, specied by its past, is coupled to a backward equation, specied by its future, the behaviour at any given time appears a priori aected by what is going on during the entire duration of the Mean Field Game process. We shall moreover focus on the long optimization time limit, and choose a conguration (typically a very narrow initial distribution of agents) such that the system we consider goes through dierent regimes in which the weight of disorder, interactions between the agent, and personal preferences on the location have dierent relative importance, and where this issue of time non-locality will be of particular importance.

The structure of this paper will be the following : In section II, we review briey the Mean Field Game formalism and its connection with the non-linear Schrödinger equation and discuss the associated hydrodynamic representation. We shall also address in that section the question of conserved quantities. In section III we then consider the corresponding ergodic state which role in the long optimization time limit is fundamental both because it describes a signicant part of the agents dynamics, but also, as we shall argue, because its existence provides a major simplication even for the transient dynamics as it eectively decouples the nal and initial boundary conditions of the problem. In section IV we study in details two important limiting regimes. Finally in section V we consider the full dynamics of the problem, and in particular address the important question of the matching of the dierent regime. Section VI contains a summary of our results and some concluding remarks.

II. NEGATIVE COORDINATION QUADRATIC MEAN FIELD GAMES A. The Mean Field Game equations

We start with a denition of what we call a "negative coordination quadratic Mean Field Games". This game involves a set of N players, or agents, which are assumed identical in every respect except for the value of their state variable X i ∈ R d representing what is supposed to be their relevant characteristics in the problem at hand (physical position, amount of a given resource, social status, etc..).

In the simplest case, these state variables follow Langevin dynamics

dX i t = a i t dt + σdW i t , (1)
where the deterministic drift velocity a i t is a control parameter xed by the agent and represents its strategy, σ is a constant and each of the d component of W i is a white noise of variance 1. Each agent adapts his strategy in order to minimize a cost functional that reects his preferences

c[a i ](t, x i t ) = T t L(X i τ , a i τ ) -V [m t ](X i τ ) dτ noise + c T (X i T ) noise . (2)
In this expression, • noise means an average over all realisation of the noise for a trajectories starting at x i t at time t, L(x, a) is a running cost depending on both state and control, c T (x) is the nal cost depending on the state of the agent at the end of the optimization period T , and V [m t ](X) is both a function of the agent's state X and a functional of the density of agents m t in the state space,

m t (x) = 1 N i δ(x -X i (t)) .
(3)

The mean eld assumption is expressed by the fact that V [m](x) only depends on the density of agents and not on each of their individual position. The quadratic aspect refers to the fact that we restrain our study to running cost that depend only on the square of the control parameter, namely L(x, a) = µa 2 /2. Finally, we describe by negative short range coordination the fact that the potential is a linear function of the density V [m](x) = gm(t, x) + U 0 (x), where g represents the strength of the interactions and U 0 (x) is a function representing the intrinsic interest for the players of having a state variable with value x (proximity to various facilities or resources, trending market, etc.. ). We stress that with our sign convention, V [m](x) has to be understood as a gain (not a cost), and thus in particular negative g implies repulsive interactions, and to be conning, U 0 (x) has to be large and negative at large distance.

As a result of the dependence of the players' cost function in the density of agents, Mean Field Games are essentially characterized by the coupling of diusion and optimization. As we are interested in the large number of players limit, one can identify the density m(t, x) with its average over noise, and it is therefore natural to focus on the Fokker-Planck equation associated with the Langevin equation (1).

And, because each agent tries to minimize its cost function (2), we may dene the value function u(x, t) = min a c[a](t, x), that is shown, using linear programming [],

to evolve according to Hamilton-Jacobi-Bellman equation []. As a consequence, the study of Mean Field Games can be reduced to that of a system of coupled PDEs

[25? ]        ∂ t u(t, x) = 1 2µ [∇u(t, x)] 2 - σ 2 2 ∆u(t, x) + gm(t, x) + U 0 (x) [HJB] ∂ t m(t, x) = 1 µ ∇ [m(t, x)∇u(t, x)] + σ 2 2 ∆m(t, x) [FP]
.

This system of equation furthermore presents rather atypical boundary conditions as the optimization is made with a specic goal (the terminal cost) in mind, hence u(T, x) = c T (x), and the diusion describes the evolution of initial distribution m(0, x) = m 0 (x). This Forward-Backward structure, which is also imposed by the signs of the Laplacian terms of both equations, is one of the main challenges of such problems and we aim, in this paper, to provide a better understanding of its implications through the discussion of limiting regimes and various approximation schemes.

In this respect, a very important concept is the one of ergodic state introduced by Cardialaguet et al. [START_REF] Cardaliaguet | Long time average of mean eld games with a nonlocal coupling[END_REF]. In the long optimization time limit T → ∞ that we are considering here, and under some assumption that are veried for our problem, it is possible to show that for most of the duration of the game the system will be well approximated by m(x, t) m er (x) u(x, t) u er (x) + λt

(for 0 t T ) , (5)
with m er (x) and u er (x) fullling the time independent equations

       λ = 1 2µ [∇u er (x)] 2 - σ 2 2 ∆u er (x) + gm er (x) + U 0 (x) 0 = 1 µ ∇ [m er (x)∇u er (x)] + σ 2 2 ∆m er (x) , (6)
and λ a constant that can be determined through the normalisation of m.

As we shall see, this notion is instrumental to the way we look at a Mean Field Game problem. The ergodic state for the quadratic games we consider will be studied in section III.

B. Alternative representations

Even if the forward-backward nature of Eqs. ( 4) constitute the main challenge of mean eld games, the coupling of a Fokker-Planck equation with an Hamilton-Jacobi-Bellman equation is not something physicists are particularly used to dealing with and poses its own challenges. In the special case of quadratic mean eld games, however, there exists a way to recast the problem into something physicists are more familiar with [START_REF] Thibault Bonnemain | Universal behavior in non stationary Mean Field Games[END_REF][START_REF] Guéant | Mean eld games equations with quadratic hamiltonian: a specic approach[END_REF][START_REF] Ullmo | Quadratic mean eld games[END_REF]. We discuss now these alternative forms of the Mean Field Game equations.

Schrödinger representation

Proceeding as in [START_REF] Ullmo | Quadratic mean eld games[END_REF] we can introduce two new variables (u(t, x), m(t, x)) → (Φ(t, x), Γ(t, x)) dened through

   u(t, x) = -µσ 2 log Φ(t, x) m(t, x) = Γ(t, x)Φ(t, x) , (7)
where the rst equation is a classic Cole-Hopf transform [START_REF] Hopf | The partial dierential equation u t + uu x = u xx[END_REF] and the second corresponds to an "hermitization" of Eq. ( 4). In terms of the new variables (Φ, Γ) the Mean Field Game equations reads

       -µσ 2 ∂ t Φ = µσ 4 2 ∆Φ + (U 0 + gΓφ)Φ +µσ 2 ∂ t Γ = µσ 4 2 ∆Γ + (U 0 + gΓφ)Γ . ( 8
)
As for the original form of the Mean Field Games equations this system has a forward backward structure implied both by the sign dierence between the time derivative and the Laplacian term and by the mixed initial and nal boundary conditions

Φ(t = T, x) = exp [-c T (x)/µσ 2 ], Γ(t = 0, x) = m 0 (x)/Φ(0, x).
Through these transformations the system (4) exhibits a mapping onto the nonlinear Schrödinger equation

i ∂ t Ψ = - 2 2µ ∆Ψ -(U 0 + gρ)Ψ , (9)
under the formal correspondence µσ 2 → , Φ(x, t) → Ψ(x, it), Γ(x, t) → [Ψ(x, it)] and ρ ≡ ||Ψ|| 2 → m ≡ ΦΓ. Equations ( 8) dier from non-linear Schrödinger in a few ways. Obviously they retain the forward-backward structure characteristic of mean eld games, and, because of how they are constructed, the functional space of which their solutions Φ and Γ are elements is also dierent than the one we, as physicists, are used to. Because of how they are constructed, Φ and Γ are actually dened as non-periodic, positive functions, while Ψ would be complex valued. Those dierences are signicant but they are not important enough to undermine the value of this mapping. Non-linear Schrödinger equation has been studied for decades in the various elds of non-linear optics [START_REF] Kaup | Perturbation theory for solitons in optical bers[END_REF], Bose-Einstein condensation [START_REF] Pitaevskii | Bose-Einstein Condensation[END_REF] or uid dynamics [START_REF] Kharif | Rogue Waves in the Ocean[END_REF]. Several methods have been developed along the years to deal with this equation and most can be adapted to mean eld games.

Hydrodynamic representation

Starting from the non-linear Schrödinger representation of equations (4) it is possible, through a Madelung-like substitution as described in [START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF], to exploit the "hermitized" nature of the previous transformations and make yet another change of variables

   Φ(t, x) = m(t, x)e K(t,x) Γ(t, x) = m(t, x)e -K(t,x) , ( 10
)
reformulating the problem into a more transparent one. If we dene a velocity v as

v = σ 2 ∇K = σ 2 Γ∇Φ -Φ∇Γ 2m = - ∇u µ -σ 2 ∇m 2m , (11)
it is easy from equations (8) to obtain a continuity equation along with its associated Euler equation

     ∂ t m + ∇.(mv) = 0 ∂ t v + ∇ σ 4 2 √ m ∆ √ m + v 2 2 + gm + U 0 µ = 0 , (12)
typical of hydrodynamics. This system closely resembles the original mean eld game equations (4) but can prove to be more convenient when performing some approximations (small noise limit) or applying some specic methods of resolution.

C. Action, and conserved quantities

The system of equations ( 8) can be derived from an action S dened as

S[Γ, Φ] ≡ T 0 dt R dx µσ 2 2 (Γ∂ t Φ -Φ∂ t Γ) - µσ 4 2 ∇Γ.∇Φ + U 0 + g 2 ΓΦ ΓΦ , (13) so that
Eq. ( 8)

⇔      δS δΦ = 0 δS δΓ = 0 . ( 14
)
The existence of an action underlying the dynamics has two consequences. First, and as we shall see in section IV, this action can serve as the basis of a variational approach. Second, because Eqs. ( 4) are time translation invariant, this implies through Noether theorem that there exist a corresponding conserved quantity that, by analogy with physical systems, we shall call energy.

Depending on the considered regime of approximation, either the Schrödinger or hydrodynamic representation may prove to be more convenient. As such, we provide the reader with two alternative expressions for the energy of the game

E = R dx - µσ 4 2 ∇Γ.∇Φ + U 0 + g 2 ΓΦ ΓΦ = R dx - µσ 2 2 m v σ 2 + ∇v + σ 2 (∇m) 2 4m + U 0 + g 2 m m . ( 15
)
Continuing on with the analogy with physical systems, the rst, σ dependent, term of each integrand can be interpreted as a kinetic energy, while the U 0 term corresponds to potential energy and the g term to interaction energy.

In the following sections, we are going to consider dierent regimes of approximation, which will be characterized by a dierent balance between the various components of the energy. The conservation of the total energy, and the fact that transitioning from one regime to another implies a transfer between one kind of energy to another, will help us providing a global picture, across the various regimes, of the Mean Field Game dynamics.

III. STATIC MEAN FIELD GAME : THE ERGODIC STATE

The notion of ergodic state is crucial in Mean Field Games theory, and its importance is twofold. To start with, it corresponds to a simpler, static, problem, which for the vast majority of the game provides a good approximation of the exact behaviour of Eqs. [START_REF] Bensoussan | The master equation in mean eld theory[END_REF]. But it also allows for the short time and long time dynamics (leading to or leaving the ergodic state) to essentially decouple. Rather than having to nd a solution of Eqs. ( 4) for arbitrary boundary conditions m 0 (x) and c T (x), the beginning of the game can be described by solving those equations with the same arbitrary initial condition m 0 (x) but a generic terminal condition : the ergodic state. Conversely, the end of the game can be described using the ergodic state as initial condition and c T (x) as terminal. As such, this notion of ergodic state reduces the problem (4) to two relatively simpler ones and it therefore make sense to address it rst. The aim of this section will be to discuss the ergodic solution,

the approximation schemes we use to describe this regime, and its stability.

A. Alternative representations in the ergodic state

In the strong interaction regime we focus on, the ergodic state can be approached equivalently within the NLS and the hydrodynamic representations. The two approaches lead to a very simple analysis, we present both below.

During the ergodic state, strategies become essentially stationary, as established by Eqs. [START_REF] Thibault Bonnemain | Universal behavior in non stationary Mean Field Games[END_REF]. As such, it is appropriate to introduce Ψ er (x), solution of the stationary NLS equation

-λΨ er (x) = µσ 4 2 ∆Ψ er (x) + U 0 (x)Ψ er (x) + g|Ψ er (x)| 2 Ψ er (x) , (16)
to describe Φ er = exp [-u er /µσ 2 ] and Γ er = m er /Φ er , ergodic solutions in the Schrödinger representation. Indeed, one can easily check from the denition of the ergodic state Eqs. [START_REF] Brazhnyi | Hydrodynamic ow of expanding bose-einstein condensates[END_REF], that both Φ er and Γ er follow the same equation [START_REF] Kharif | Rogue Waves in the Ocean[END_REF]. For the ergodic state, the system of time-dependent coupled PDEs Eqs. ( 8) reduces to the one, time-independent, ODE Eq. ( 16), and the connection with NLS equation is Thomas-Fermi approximation (dashed). In this case g = -2, σ = 0.4, µ = 1 and

U 0 (x) = -x 2 .
made even clearer. One can then check that

     Φ(t, x) = exp + λ µσ 2 t Φ er (x) Γ(t, x) = exp -λ µσ 2 t Γ er (x) , (17)
are solutions of the time dependent equation [START_REF] Cardaliaguet | Long time average of mean eld games with a nonlocal coupling[END_REF] and that Φ(t, x)Γ(t, x) = Φ er (x)Γ er (x) = m er (x) corresponds to the static ergodic density.

Similarly, we can introduce the ergodic equations of the hydrodynamic representation. Introducing v er as the ergodic velocity, Eqs. ( 12) readily become

     v er = 0 λ + σ 4 2 √ m er ∆ √ m er + gm er + U 0 µ = 0 , (18)
once again simplifying greatly the problem by getting rid of the time dependence but also of the coupling between the two solutions.

B. Bulk of the distribution: Thomas-Fermi approximation

One of the many interests of the Schrödinger representation is that we can exploit the large literature surrounding this equation. In the large interaction regime, a particularly popular way of tackling the stationary NLS (or Gross-Pitaevskii) equation is through the use of Thomas-Fermi approximation as described in [START_REF] Dalfovo | Order parameter at the boundary of a trapped bose gas[END_REF].

Noting L a charactersitic length scale of the problem, we can check that the kinetic energy behaves as

E kin = - R dx µσ 4 2 ∇Γ.∇Φ ∼ µσ 4 L 2 , ( 19
)
and that the interaction energy as

E int = R dx g 2 (ΦΓ) 2 ∼ g L . (20)
The ratio between kinetic and interaction energies, which is a good measure of the relative importance of the diusion and interaction processes, is then given by

E kin E int ∼ ν L , (21)
where

ν ≡ µσ 4 |g| (22)
has the dimension of a distance. In the context of the Non Linear Schrödinger equation, ν is known as the "healing length", and represents the typical lengthscale on which the interaction energy balances quantum pressure (or diusion in the context of MFG), and is named in this way because it is the minimum distance over which the wave function can tend to its bulk value (i.e. "heal") when subjected to a local perturbation.

In the limiting case where the kinetic energy is negligible in the bulk of the distribution, i.e. when the typical extension of the distribution is large in front of the healing length ν (something that, we assume, will happen because -strong -repulsive interactions will cause agents to spread despite the conning potential U 0 ), Eq.( 16) loses its dierential status and becomes a simple algebraic equation

-λ ≈ U 0 (x) + g|Ψ er (x)| 2 , (23)
which is easily solved as

Ψ TF (x) =        λ + U 0 (x) |g| 1/2 if λ > -U 0 (x) 0 otherwise , (24)
where the constant λ is then computed using the normalisation condition

∞ -∞ m er (x)dx = 1 . ( 25
)
The exact same approximation can also be obtained by neglecting the o(σ 4 ) term in Eqs. [START_REF] Lachapelle | Les cahiers de la chaire/n 16[END_REF], which yields

     v er = 0 m er = - λ + U 0 g , (26)
an expression that is perfectly equivalent to Eq. ( 24).

Such an approximation may seem naive at rst but actually yields rather good results. Let us take the example of quadratic external potential U 0 (x) = -µω 2 0 x 2 /2. [Note that, as mentioned above, U 0 (x) has to be understood as a gain and, to be conning has to reach its maximum value for a nite x and go to -∞ for large x, thus the negative sign.] We nd λ = 3|g| µω 2 0 /4

√ 2 2/3
, and we can see on Fig. 1 that, in the bulk, the approximation agrees perfectly with the exact result.

The tails of the distribution, for which densities is low, and thus interactions eects are small, cannot be described in this way however and call for a specic treatment.

C. Tails of the distribution: semi-classical approximation

If Thomas-Fermi approximation yields good results in the bulk of the distribution, i.e. for λ > -U 0 (x), it fails to describe regions where the density of agents is small.

When this density is suciently small however, that is in the tails of the distribution where λ + U 0 (x) is suciently negative, the problem simplies once again because the non-linear interacting term is negligible. In this context Eq. ( 16) reads and we will address it here through a semi-classical approximation. More specically, we look for solutions of Eq. ( 27) in the form Ψ SC (x) = ψ(x) exp S(x) √ µσ 4 up to the second order in σ 2 . As an example, we will once again look at the case of quadratic external potential U 0 (x) = -µω 2 0 x 2 /2, and compare the approximation to numerical results. In order to keep the core of the text concise, details of the computation are provided in Appendix (A). The semi-classical approximation yields

-λΨ(x) ≈ µσ 4 2 ∆Ψ(x) + U 0 (x)Ψ(x) , (27)
Ψ SC (x) = C µω 2 0 x 2 -2λ 1/4 exp λ µω 0 σ 2 x µω 2 0 2λ x 2 µω 2 0 2λ -1 -argcosh x µω 2 0 2λ , ( 28
)
where C is a constant numerically determined to match with the bulk of the distribution. This gives results in very good agreement with the real solution for x > 2λ µω 2 , as illustrated Fig ( 2).

One can note that, for this approximation scheme, the (so-called) turning point

x = X, where Ψ TF vanishes, is singular. This can be easily avoided by way of a Thomas-Fermi approximation (dashed), semi-classical approximation (dot) and uniform approximation (dash-dot). Parameters for this gure are to g = -2, illustrates how this uniform approximation Eq. ( 29) constitutes an improvement over the previous one Eq. (28).

σ = 0.4, µ = 1, U 0 (x) = -x 2 , C left = 0.
Ψ SC =          C left 8πS left 3U 0 1/2 cos π 3 J 1/3 (S left ) + J 1/3 (S left ) if x < X 2C right 8S right π |U 0 | 1/2 cos π 3 K 1/3 (S right ) if x > X , (29)
Depending on the external potential U 0 (x), computing this approximation may become somewhat involved. If so, the tails of the distribution can still be described by an Airy function, as discussed in [START_REF] Dalfovo | Order parameter at the boundary of a trapped bose gas[END_REF], using the consistently simpler, albeit less accurate, approximation method of linearising the potential around x ≈ X and looking at the asymptotic behaviour.

D. Some properties of the ergodic state

To conclude this section on the ergodic state, we shall describe here some of its properties that will become relevant when trying to connect it to the beginning (or end) of the game.

Final cost and energy

Something that may not appear clearly from the denition Eqs. ( 6) of the ergodic state, but becomes obvious when looking at its hydrodynamic counterpart Eqs. [START_REF] Ullmo | Quadratic mean eld games[END_REF], is that for quadratic Mean Field Games in the strong repulsive interaction regime, the value function u, becomes essentially at during the ergodic state

v er = 0 ⇔ u er = K er + o(σ 2 ) , (30)
where the o(σ 2 ) terms are the corrections to the Thomas Fermi approximation and K er is a constant. The Mean Field Games equations Eqs. ( 4) being invariant by translation of u, we will choose this constant K er to be zero for the remainder of this paper. This characteristic u er = 0 will then be used as an "eective" terminal condition when discussing the beginning of the game.

Another interesting aspect of the ergodic state is that it provides us with an easy access to the (conserved) energy E = E er of the system

E er = R dx g 2 m 2 er + m er U 0 dx < 0 , (31)
neglecting o(σ 4 ) terms of the "kinetic" energy. By denition, because interactions are chosen to be repulsive and the external potential to be conning (which implies it can be chosen negative for all x), the energy has to be negative.

What those two properties allow is for us to restrict our analysis of the transient states to games of negative energy and at costed terminal conditions, making for a simpler discussion of the time-dependent problem.

Approaching the ergodic state : stability analysis

To nish this section, we discuss the stability of the ergodic state. Focusing on the bulk of the distribution we will use the hydrodynamic representation as it is the better framework to deal with the small σ limit. Recalling Eqs. [START_REF] Ullmo | Quadratic mean eld games[END_REF], the expression of the ergodic state under this representation

     v er = 0 m er = - λ + U 0 g , (32)
we then apply small perturbations δm and δv to this stationary state and compute their evolution. Near the ergodic state Eqs. [START_REF] Dalfovo | Theory of Bose-Einstein condensation in trapped gases[END_REF] become

     δ m = -∇(m er δv) δv = - g µ ∇δm , ( 33
) implying δ m = g µ ∇(m er ∇δm) . ( 34
)
Assuming that δm = δm 0 e ωt , Eq. (34) amounts to the eigenvalue problem Dδm 0 =

-(µ/g)ω 2 δm 0 with D ≡ -∇(m er (x)∇•) . (35)
It is relatively straightforward to show that D is a real symmetric operator, implying its eigenvalues are real, and furthermore that all these eigenvalues are positive (cf Appendix. B). Noting i these real eigenvalues and ϕ i (x) the corresponding eigenvector, the linear modes in the vicinity of (m er , v er ) are

Q ± (i) = (δm (i) , δv ± (i) ) ≡ (ϕ i (x), ± -g/µ i ∇ϕ i (x)) , (36)
and they follow an exponential time dependence Q ±

(i) (t) = e ±ω i t Q ± (i) (0), with ω i = -g i /µ (remember g < 0).
This exponential behaviour highlights the fact that, as discussed in [START_REF] Ullmo | Quadratic mean eld games[END_REF] in a simpler (variational) context, the ergodic state should be understood as a unstable / hyperbolic xed point, which is this approached exponentially fast at small times, and left exponentially quickly near T .

Returning to the particular case of the quadratic external potential U 0 = -

µω 2 0 x 2 2 ,
and assuming as above that δm ∝ e ±ωt , we get

         -2 ω ω 0 2 δm = ∂ y (1 -y 2 )∂ y δm y = x µω 2 0 2λ , (37)
As shown on Fig ( 4), the eect of this perturbation is simply to add tails to the distribution of agents, which is qualitatively in good agreement with simulations.

IV. TIME DEPENDENT PROBLEM : THE BEGINNING OF THE GAME

As shown by Eqs. ( 19)-( 20)-( 21)- [START_REF] Víctor | Dynamics of bose-einstein condensates: Variational solutions of the gross-pitaevskii equations[END_REF], dierent length scales are associated with dierent dynamical regimes : very short distances L ν are dominated by diffusion, and for L ν interactions take over. The large interaction limit that we consider here essentially means that the healing length ν is much smaller than any characteristic feature of the one-body gain U 0 (x), and we will work under that hypothesis. However, as the size of the distribution of agents further increases, interaction eects become weaker (although the eects of diusion decrease even more rapidly) and, even in the large |g| limit, the ergodic state is characterized by a balance between the interaction energy E int and the potential energy E pot . The fact that this balance has to be reached is eventually what xes the typical size of the ergodic state distribution.

To consider a system which traverses all dynamical regimes, we will therefore assume that the initial distribution is extremely narrow (i.e. that its width Σ 0 is signicantly smaller that ν). The beginning of the game will therefore mainly consist in an expansion of this initial distribution, expansion that will go on until the balance between E int and E pot is reached. During that expansion we may neglect the eects of the external potential. In this section we will therefore study the set of equations [START_REF] Cardaliaguet | Long time average of mean eld games with a nonlocal coupling[END_REF] in the particular case of U 0 (x) = 0

       -µσ 2 ∂ t Φ(t, x) = µσ 4 2 ∂ xx Φ(t, x) + gΦ 2 (t, x)Γ(t, x) + µσ 2 ∂ t Γ(t, x) = µσ 4 2 ∂ xx Γ(t, x) + gΦ(t, x)Γ 2 (t, x) . ( 39
)
While it can be shown that this system is integrable (in the sense that there exists a canonical transform from (Φ, Γ) to action-angle variables) [], we will not attempt here to explicitly use this property and will approach the various limiting regimes through the use of variational ansätze. Furthermore, as we know (cf Section III D) that the value function of the ergodic state, which can here be interpreted as a nal cost for the beginning of the game, is essentially constant, we shall work below under the assumption that the terminal cost is essentially at.

A. Large ν regime : Gaussian Ansatz

When the extension of the distribution of agents is small in front of ν, the eects of diusion become dominant, and Eqs. (39) become simple heat equations, for which the Green's function has a Gaussian shape. It is therefore natural to tackle this regime using Gaussian variational approach [START_REF] Víctor | Dynamics of bose-einstein condensates: Variational solutions of the gross-pitaevskii equations[END_REF], as already applied to Mean Field Games in [START_REF] Ullmo | Quadratic mean eld games[END_REF].

Preambular denitions

Variational approximation amounts to minimizing the action on a small subclass of functions (here taken so that the distribution of agents is Gaussian), eectively reducing a problem with an innite number of degrees of freedom to one with a nite, easily manageable, number. As in [START_REF] Ullmo | Quadratic mean eld games[END_REF] we consider the following Ansatz

       Φ(x, t) = exp (-Λ t /4 + P t • x) µσ 2 1 (2πΣ t ) 1/4 exp - (x -X t ) 2 (2Σ t ) 2 (1 - Λ t µσ 2 ) Γ(x, t) = exp (+Λ t /4 -P t • x) µσ 2 1 (2πΣ t ) 1/4 exp - (x -X t ) 2 (2Σ t ) 2 (1 + Λ t µσ 2 ) , (40)
which indeed yields a Gaussian distribution centered in X t with standard deviation

Σ t m(t, x) = Γ(t, x)Φ(t, x) = 1 2πΣ 2 t exp - (x -X t ) 2 2(Σ t ) 2 , (41)
and where P t and Λ t respectively are the momentum and the position-momentum correlator of the system. Inserting this variational ansatz in the action 13 we get

S =

T 0 L(t)dt where the Lagrangian L = Lτ + Ẽkin + Ẽint + Ẽpot only depends on X t , P t , Σ t , Λ t and their time derivatives. This yields

       Lτ = Ṗt X t - Λ t 2Σ t Σt Ẽkin = P t 2µ + Λ 2 t -µ 2 σ 4 8µΣ 2 t Ẽint = g 4 √ πΣ t Ẽpot = R U 0 (x)m(t, x)dx . ( 42
)
As long as the density of players m(t, x) remains narrow enough that U 0 (x) can be linearised on the distance Σ t , we see that Ẽpot ≈ U 0 (X t ) and that the variable (X t , P t ) and (Σ t , Λ t ) decouple. As discussed in [START_REF] Ullmo | Quadratic mean eld games[END_REF] (X t , P t ) then follows the dynamics of a point particle of mass µ subject to the external potential U 0 (x). The discussion below, in which we assume U 0 (x) = 0, could also therefore be generalized straightforwardly to this situation (by just adding the motion of the center of mass).

2. Evolution of the reduced system (X t , Σ t ; P t , Λ t ) for U 0 (x) = 0

Minimizing the action with respect to each parameter yields the evolution equations

       Ẋt = P t µ Ṗt = 0 Σt = Λ t 2µΣ t Λt = Λ 2 t -µ 2 σ 4 2µΣ 2 t + g 2 √ πΣ t . ( 43
)
Under the assumption that U 0 (x) = 0, P t is a constant and is essentially a measure of the asymmetry of Φ(t, x) and Γ(t, x) as well as the drift of the center of mass of the density. If Φ(t, x) and Γ(t, x) are symmetric with respect to x = x 0 , P t = 0 and the center of mass does not move. For the sake of simplicity, let us focus on this conguration and let X t = x 0 = 0. The equations concerning (Σ t ; Λ t ) are more complicated but can be decoupled using the fact that the total energy of the system

Ẽtot = Ẽkin + Ẽint + Ẽpot is conserved, hence Ẽtot = µ Σ2 2 - µσ 4 8Σ 2 t + g 4 √ πΣ t .
(44)

Zero-energy solution

In the limiting case where U 0 (x) is negligible for all times (not just the initial expansion we consider here), and assuming an innitely long game, the distribution of agents will spread indenitely, tending towards a perfectly diluted state m(t, x) ≈ 0. This would correspond to an (asymptotic) ergodic state Σ erg → ∞ and Ẽtot → 0.

In that case the the evolution equation reads

Σt = 1 Σ t µσ 4 √ π -2gΣ t 4µ √ π , (45)
which can be integrated as

1 - 2Σ t √ πη 1 + Σ t √ πη -1 - 2Σ 0 √ πη 1 + Σ 0 √ πη = - 3t 2πτ , (46)
Σ 0 being the initial width of the distribution.

Finite-energy solutions

In practice, we know that the energy of the ergodic state computed in section III is negative, and therefore we are mainly interested in negative energy solutions. In that case, Σ t cannot grow past a certain value

Σ * = g √ π+ √ g 2 / √ π-8µσ 4 Ẽ tot 8 Ẽtot otherwise
Σt would become complex. Eq. ( 44) can be integrated as

F (8E tot , -2g/ √ π, µσ 4 ; Σ t ) -F (8E tot , -2g/ √ π, µσ 4 ; Σ 0 ) = t 2 √ µ , (47)
where F (a, b, c; x) is dened as For completeness, we also provide solution of Eq. ( 44) in the case of positive energy :

F (a, b, c; x) = xdx √ ax 2 + bx + c = b 2|a| 3/2 arcsin 4a 2 b 2 -4ac x + b 2a + √ ax 2 + bx + c a . ( 48
G(8 √ πE tot , -2g, µσ 4 √ π; Σ t ) -G(8 √ πE tot , -2g, µσ 4 √ π; Σ 0 ) = t 2 µ √ π , (49)
where G(a, b, c; x) is dened as

G(a, b, c; x) =            √ ax 2 + bx + c a - b 2|a| 3/2 argsinh 4a 2 b 2 -4ac x + b 2a if b 2 4ac > 1 √ ax 2 + bx + c a - b 2|a| 3/2 argcosh 4a 2 4ac -b 2 x + b 2a if b 2 4ac < 1 . ( 50
)
It can be worth noting that in the t → 0 limit, Eqs. (45) (47) and (49) yield similar behaviour for Σ t . This concludes our discussion of the large ν regime, next we will address the small ν one.

B. Small ν regime : Parabolic ansatz

As we have shown in a previous paper [START_REF] Thibault Bonnemain | Universal behavior in non stationary Mean Field Games[END_REF], in the weak noise, innite optimization time, limit of the potential-free negative coordination Mean Field Game, the density of players quickly deforms to take the shape of an inverted parabola that scales with time. These parabolic solutions can be interpreted as arising from a low order approximation of a multipolar expansion in a electrostatic representation of the problem [START_REF] Thibault Bonnemain | Universal behavior in non stationary Mean Field Games[END_REF]. Furthermore, simulations indicate that, under the assumption that the variations of the terminal cost are small compared to ũ, (non scaling) inverted parabolas are still stable solutions of Eqs(39) with nite optimization time.

Imposing the normalisation condition { ∞ -∞ m(t, x)dx = 1 ∀ t} we thus consider the ansatz m(t, x) =      3(z(t) 2 -x 2 ) 4z(t) 3 if z(t) > x 0 otherwise , (51)
and look for a formal solution outside the singularities in the derivative at x = ±z(t).

It is worth mentioning that such an approach already exists in the realm of cold atoms [12] [6]. However dierences arise from the fact that we are dealing with complex time and from the forward-backward structure of Mean Field Games.

In practice, in this subsection, we shall discuss as an independent problem an eective potential-free (ie U 0 (x) = 0) game in the small ν regime. We furthermore assume that the nal condition, at t = T , is that of a at terminal cost c T (x) = 0 and that the initial density of agents, at t = 0, is essentially a Dirac delta function,

i.e. an inverted parabola of the form (51) with z(t = 0) = z 0 = 0. Note that, as we will still assume that the healing length ν is the smallest length size of the problem, this implies that we actually consider here the limit ν, z 0 → 0 with z 0 ν. In the context of the original game, this eective game will correspond to the expansion phase beyond the healing scale ν. How it will be coupled to the ergodic state or to the small ν regime will be examined subsequently, but as the conserved energy of the ergodic state is negative, we will consider more specically this regime.

Preambular denitions

While the Schrödinger representation along with the Gaussian variational ansatz were well-suited to describe a large ν regime, the hydrodynamic representation is actually more convenient to deal with the small noise limit. In the context of cold atoms, the equivalent of the o(σ 4 ) term in Eqs. ( 12) is considered to be safely negli-gible as long as the extension of the condensate is large in front of the healing length ν. Focusing on this weak noise regime (Thomas Fermi approximation) here amounts to studying the system

     ∂ t m + ∇(mv) = 0 ∂ t v + ∇ v 2 2 + g µ m + U 0 µ = 0 . ( 52
)
Going through Madelung substitution shows that we can get away with only neglecting o(σ 4 ) terms while absorbing o(σ 2 ) contributions in the denition of v Eq. ( 11), which is not as transparent from Eqs. ( 4).

As we shall see below, we can nd exact solutions of Eqs. (52) assuming the parabolic form (51), and, therefore, we shall not need to resort to the action ( 13) to derive the corresponding dynamics.

Elementary integration of the hydrodynamic representation

In the U 0 (x) = 0 limit the expression of the velocity associated to a parabolic distribution Eq. ( 51) can easily be extracted from the continuity equation in (52).

Integrating over [-∞; x] and taking into account that m vanishes at innity, we get

v(t, x) = z (t) z(t) x . ( 53
)
To derive the time evolution of z(t), we insert the explicit forms of m(t, x) and v(t, x)

in the second equation of Eqs. (52), yielding

z (t) = 3g 2µz(t) 2 . ( 54
)
This closely resembles what can be found when dealing with expanding Bose Einstein condensates (BEC) [START_REF] Brazhnyi | Hydrodynamic ow of expanding bose-einstein condensates[END_REF], one main dierence lying in the fact that the multiplicative constant in front of 1/z 2 is negative in the context of Mean Field Games but positive in the context of Bose Einstein condensates.

Eq. ( 54) can be integrated as

z (t) 2 = - 3g µ 1 z(t) + z * . (55)
For commodity the integration constant has been written as 3|g| /µz * and can take the value -1, 0 or 1. We shall see below the values -1, 0 or 1 of correspond to negative, 0 or positive energies, and that in the = -1 (negative energy) case, z * (> 0) can be interpreted as z( T ) for the eective game. In the BEC context, only the positive case is relevant [START_REF] Brazhnyi | Hydrodynamic ow of expanding bose-einstein condensates[END_REF], and the fact that, here, zero or negative have to be considered as well, which allows for new sets of solutions, constitutes another important dierence.

Characterisation of z(t)

To solve this equation, let us introduce two functions ξ + (y) > 0 and ξ -(y) ∈ [0; 1],

associated with +1 and -1 values of , implicitly dened through the relations

ξ + (y)(1 + ξ + (y)) -argsinh ξ + (y) = y ∀y > 0 , (56)
and

arcsin ξ -(y) -ξ -(y)(1 -ξ -(y)) = y ∀y ∈ [0, π 2 ] . (57)
We also dene a third function ξ 0 (t) given explicitly as

ξ 0 (y) = 3y 2 2/3 ∀y > 0 , (58)
which corresponds to the = 0 solution discussed in [START_REF] Thibault Bonnemain | Universal behavior in non stationary Mean Field Games[END_REF]. It is worth noting that all three functions are monotonous increasing functions of time and have the following

properties                  ξ + (0) = ξ -(0) = ξ 0 (0) = 0 ξ + (y) > ξ 0 (y) ∀y ξ 0 (y) > ξ -(y) ∀y ∈ ]0, π 2 ] ξ + (y) ≈ ξ 0 (y) ≈ ξ -(y) as y → 0
.

We can now write the dierent solutions of Eq. (55) in terms of the above functions. Even if we only consider repulsive interactions, because of the square power in Eq. (55), its solutions can either be increasing or decreasing. There are three families of increasing solutions

z(t) =          z * ξ + (αz -3/2 * t) if = 1 ξ 0 (αt) if = 0 z * ξ -(αz -3/2 * t) if = -1 , (59)
where α = -3g/µ. The reciprocal three families of decreasing solutions are irrelevant to our discussion as they will not ultimately lead to the ergodic state introduced section III. We still provide a succinct analysis of those solutions in appendix C for the sake of completeness.

Let us address how the boundary conditions of our eective game constrain the solution within the family (59). The aforementioned initial condition that the density of agents starts as a Dirac delta function imposes that z(t = 0) = 0 is already implemented in Eq .( 59). Consider now the the terminal boundary condition, i.e.

the fact that at T the terminal cost is at. Recalling that v = -∇u/µ + o(σ 2 ), the expression of the velocity (53), implies that the terminal cost c T (x) = u( T , x) can be constant only if the time derivative of z(t) is zero. According to Eq. (55), this is only possible if = -1 and z(t) = z * . Hence, the study of the eective game we consider here can be reduced to that of "-" type solutions and we deduce that z * = z( T ). Now, one can check easily from Eq. ( 57) that ξ -(π/2) = 1 (which is compatible with the fact that ξ -(y) ∈ [0; 1] is an increasing monotonous function dened for y ∈ [0, π/2]). From Eq. (59) we infer

z( T ) = z * ⇒ αz -3/2 * ( T ) = π 2 . ( 60
)
This yields a relation between the nal time of the eective game T and the nal extension of the distribution of players

T = πz 3/2 * 2α . (61)
The duration of the eective game, i.e. the time it takes to go from a narrow, deltalike initial density of agents, to a at terminal cost, thus determines the parameter z * , and therefore xes which member of the family Eq. (59) has to be considered.

Inserting Eq. (59) in the ansatz (51) and (53), directly yields explicit expressions for m ans v, which,as illustrated in Figure [START_REF] Brazhnyi | Hydrodynamic ow of expanding bose-einstein condensates[END_REF] provide satisfactory approximations, even though the noise σ, and thus the healing length ν, is not strictly zero (see captions for details). (full) and analytically (dashed). In this case, we have chosen g = -2, σ = 0.45 and µ = 1, meaning ν ≈ 0.02. The actual (numerical) game takes place from t = 0, when it starts as an inverted parabola of extension 0.4, to t = T = 20 when the terminal cost is at. The eective game starts at time t ≈ -0.07 as a Dirac delta function and its eective duration is T ≈ 20.07. The only dierence between the numerical results and the parabolic ansatz comes from the fact that σ is non-zero in the simulation. This gure also illustrates how the Thomas-Fermi approximation becomes more and more eective as the typical extension of the density becomes larger in front of ν.

Energy of the system

The energy plays a crucial role in the dynamics of the spreading of the players and its conservation will be the key property we will use to match the dierent regimes of approximation. Because we ultimately want to link this regime to the ergodic state described in Section (III), we will focus on negative energy only. In the potential free regime, the energy contains two terms, one is the kinetic energy (associated with the diusion term), the other comes from the interactions. Dropping the o(σ 4 ) term in the denition Eq. ( 15) of the kinetic energy, we thus have

E = E kin + E int , with        E kin = µσ 2 2 z -z m v σ 2 + ∂ x v dx E int = z -z gm 2 2 dx . ( 62
)
As the energy is conserved, it can be evaluated at any time, and particularly at the end of the eective game. If = 0, z → ∞ as t → ∞ and it becomes clear that, in this case, E = 0. A similar reasoning would show that, if = +1, E ∼ 1/ √ z * > 0.

When = -1, however, we can evaluate the energy at t = T , when z = z * and v = 0, which trivially implies that, at that point and within the Thomas-Fermi approximation, the kinetic energy is zero. Inserting Eq. ( 51) with z(t) = z * into the second equation of (62) we get

     E - kin ( T ) = 0 + o(σ 4 ) E int ( T ) = 3g 10z * , (63)
which, using Eq. ( 61) implies

E = 3g 10z * = 3g 10 2α T π -2/3 . ( 64
)
For the eective game we consider here narrow initial density, at terminal cost v( T ) = 0, small ν regime, individual gain U 0 (x) = 0 there is a strong link between the duration of the game T and the energy E. In some sense T monitors the dynamics of the spreading of the players completely, and takes the same role as Ẽtot did in the large ν regime. As such, nite games with at terminal cost correspond to non-0 energy and there is a one-to-one relation between T and E.

This nishes our analysis of the small ν regime, and more generally of the expansion regime. The next section will now address ways to relate those transient times to the ergodic state.

V. THE ENTIRE GAME

In this section, we shall examine how the previously discussed regimes of approximation couple with one another. We will start in section V A by rst addressing, once again, an eective game, in the vein of the one we studied in section IV B, but assuming a nite value of healing length so that players are initially distributed on a distance much smaller than ν. This will allow us to focus on the transition from a large to a small ν regime during the initial stages of the game. Then, in section V B we will consider the matching of the expansion phase, described by the eective game, with the ergodic state.

A. Matching small and large ν regimes

As mentioned above, we consider here, just as in section IV B, an eective potential-free game of duration TV , with at terminal cost and an initial distri- bution of agents which width Σ 0 is much smaller that the healing length ν. We furthermore assume that the optimization time is large enough so that, at the end of the game, the density of player has spread on a distance much larger than ν.

Under those assumptions, we can distinguish two main phases the eective game will go through: an initial phase which can be described by the Gaussian ansatz introduced section IV A and, at the end of the game, a terminal phase for which the density of agents will follow the parabolic ansatz of section IV B. Between those two phases, the density will transition from a Gaussian-like distribution to an inverted parabola. The precise shape of the density during the crossover is complicated to describe, and will not be addressed here, but we shall see that we can still describe the dynamics of the spreading of the players across the two regimes.

To proceed, let us introduce a couple of quantities that will characterise the dynamics. The rst one is the total energy E of the system, a conserved quantity, which is common to both regimes. The second is the time t tr at which the system will transition from the Gaussian regime to the parabolic one.

Seen from the Gaussian side of the transition, the transition time t G tr is dened by the condition

Σ(t G tr ) = ν , (65)
) √ 5 = ν , (67)
where z/ √ 5 is the standard deviation of the parabolic distribution Eq. ( 51), and t para 0 = TV -TIV the time at which the parabolic evolution appears to have started (from an initial Dirac delta shape) seen from the small ν side of the transition.

From Eq. ( 59

) = arcsin √ 5ν z * - √ 5ν z * 1 - √ 5ν z * , (68)
which, given the fact that z * and E are linked through Eq. ( 64) is actually a relation between t para tr and E.

The self-consistent condition t para tr = t G tr then implies that Eqs. (66)-(68) x both the energy E and the transition time t tr , and thus solve the game we are considering in this subsection.

Knowing the energy, as illustrated in Figure [START_REF] Cardaliaguet | Notes on mean eld games[END_REF], one can reconstruct the evolution of the variance of the Gaussian distribution at small times using Eq.(47) and, then, of the width of the inverted parabola using Eq. (59). Figure [START_REF] Cardaliaguet | Long time average of mean eld games with a nonlocal coupling[END_REF] gives further indication that both the Gaussian and parabolic ansatz yield good result to evaluate not only the spreading of the players but also the shape of the distribution in this conguration. The two regimes overlap when Σ t is of order ν and either approximation regime gives a fairly accurate description of the phenomenon. However, as we near the end of the game both approximations become less and less accurate due to the proximity of the terminal condition, which, because σ is non-zero, is not perfectly at, v T (x) = 0 + o(σ 2 ).

B. Matching transient and ergodic states

We now turn back to the complete game of Eqs. ( 4), or more specically the rst half of that game linking the initial distribution of agents to the ergodic state. We specialize moreover to the case of a narrow initial condition, of width Σ 0 ν, for the distribution of agents. It should also be noted that we will assume that the maximum of the external gain U 0 coincides with the center of mass of the initial distribution, so that we do not have to take its motion into account. The system will, therefore, initially go through an expansion phase, during which we will neglect the individual gain / potential U 0 (x), and will successively traverse the large ν and the small ν regimes before reaching the ergodic state. Our goal here is to understand how to connect those.

In this conguration, the energy E is completely xed by the ergodic state

E = E er = g 2 R m 2 er dx + R m er U 0 dx . (69)
The initial large ν expansion phase is therefore completely xed by E and Σ 0 through Eq. (47), which in turn xes the transition time t tr between the large and the small ν regimes through Eq. (66).

Once in the large ν regime, the energy E again xes the duration TIV of the eective game of section IV B. The only parameter that remains to be xed is the eective beginning time t para 0 of that eective game which is given by Eq. (68) (with, according to Eq. ( 64), z * = 3g/10E).

Naturally, because one has to take the external gain into account when nearing Here g = -2, σ = 0.4, µ = 1, ω 2 0 = 0.2, E = -0.36 and T = 15.

mentioned in section III B E er int ∼ g z er .

(70)

Hence, if interaction energy represents a set proportion p of the total energy, E er int = pE er , z er should be of order z * /p. And, noting that TIV ∼ z 3/2 * , we can infer that τ er should not be too far-o from TIV /p 3/2 . In the particular case of a quadratic external gain U 0 (x) = -µω 2 0 x 2 /2, we can easily compute the ratio between E er int and

E er pot E er int E er pot = 2 ⇒ E er int = 2 3 E , ( 71
)
result which is completely independent of the values of g, µ or ω 0 . The ergodic density is then an inverted parabola of width z = 3z * /2 and τ er is of order TIV (3/2) 3/2 . This is illustrated Fig. [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications[END_REF].

What the eective game provides, in this context, is not a quantitatively precise description but a good qualitative estimation of what actually happens during the beginning of the game.

VI. CONCLUSION

Mean Field Games constitute a challenge because of their unusual forwardbackward structure. In this paper we presented a simple, heuristic, yet ecient method to describe negatively coordinated Mean Fields Games in one dimension, leaning heavily on the notion of ergodic state introduced by Cardaliaguet [START_REF] Cardaliaguet | Long time average of mean eld games with a nonlocal coupling[END_REF]. The existence of this ergodic state proves to be of paramount importance as it allows the initial and nal conditions to essentially decouple. The problem of nding a way to link initial and nal conditions, both arbitrary, simplies as it becomes a problem of nding a way to link either to a generic ergodic state. Making rst use of the mapping to the non-linear Schrödinger equation as introduced in [START_REF] Ullmo | Quadratic mean eld games[END_REF], and then of the hydrodynamic representation from [START_REF] Thibault Bonnemain | Universal behavior in non stationary Mean Field Games[END_REF], we were able to identify dierent regimes of approximation and put forward adequate ansätze to reconstruct the whole game.

Results from those ansätze have been compared to numerical solutions, for parameters in their domain of application, and are highly satisfactory as well as easily computed. Increasing "+", decreasing or mixed type solutions can all be observed numerically, however they refer to congurations where variations of the terminal cost are important in front of ũ = µσ 2 and fall outside the scope of this article. Still, we mention them, once again, for the sake of completeness.

[1]

[2] Y. Achdou, P-N. Giraud, J.-M.

Figure 1 :

 1 Figure 1: Computational solution of the Gross-Pitaevskii equation (full) and

Figure 2 :

 2 Figure 2: Computational solution of the Gross-Pitaevskii equation (full), Thomas-Fermi approximation (dashed) and semi-classical approximation (dot). The inset shows the same curves in Log-Linear plot focusing on the tail of the distribution. Parameters for this gure are g = -2, σ = 0.4, µ = 1, U 0 (x) = -x 2 and C = 8.10 -4 .

Figure 3 :

 3 Figure 3: Computational solution of the Gross-Pitaevskii equation (full),

  [START_REF] Hopf | The partial dierential equation u t + uu x = u xx[END_REF] and C right = 0.07.uniform approximation[START_REF] Langer | On the connection formulas and the solutions of the wave equation[END_REF]

  where C left and C right are constants to be numerically determined, J γ stands for the Bessel function of the rst kind of order γ and K γ for the modied Bessel function of the second kind. Explicit expressions for the actions S left and S right , in the case of the quadratic gain U 0 (x) = -µω 2 0 x 2 /2, are provided in Appendix A. Fig.(3)

Figure 4 :

 4 Figure 4: First order Legendre polynomial of the second kind. Its eect on the ergodic state would be to add tails to the distribution.

20 ΣFigure 5 :

 205 Figure 5: Computational solution of the Gross-Pitaevskii equation (blue dot) and variational Ansatz (red dashed). The inset shows the time evolution of the numerical variance (full) and Σ as dened in Eq. (47). In this case g = -2, σ = 3.5, µ = 1 and T = 20.

Figure 6 :

 6 Figure 6: Computational solution of the Gross-Pitaevskii equation (dot) and parabolic ansatz (dashed). The inset shows the time evolution of z numerically

Figure 7 :

 7 Figure 7: Time evolution of the variance (left) and the width of the parabola (right). The numerical solution for the density of players has been numerically tted with a Gaussian and an inverted parabola, full curves are obtained through the extraction of the tting parameters. Dashed curves are obtained using either the Gaussian or parabolic ansatz with energy E = -9.95 × 10 -3 computed through the self-consistent condition. Parameters for this gure are g = -2, σ = 1.2, µ = 1, ν = 1, Σ 0 = 0.2 and T = 300.

Figure 8 :Figure 9 :

 89 Figure 8: Density of players at dierent times, numerical results are plotted (solid line) along with the Gaussian (dotted line) and the parabolic ansatz (dashed line). At the beginning of the game, Figs (8a) and (8b), the Gaussian ansatz is the most accurate. Then in the middle of the game, Figs (8c) and (8d), the parabolic constitutes a better approximation. At the end of the game, Figs (8e) and (8f), the parabolic ansatz becomes less and less accurate as we near the terminal condition.Here g = -2, σ = µ = 1, ν = 1, Σ 0 = 0.2 and T = 300, while E = -9.95 × 10 -3 has been computed through the self-consistent condition.

Appendix A : z

 : Derivation of the semi-classical approximation for quadratic external potential Order σ 0 At zeroth order Eq. (27) reduces to(U 0 (x) + λ)ψ(x) + (∂ x S(x)) 2 2 ψ(x) = 0 ,(A1)which can be analytically integrated given the external potential is not too complicated. Taking once again the example of a quadratic potential, we get S(x) = x)] 2 , and m er (x) are all positive quantities, this implies that i too has to be positive.Appendix C: Decreasing solutions of the eective gameAs mentioned in section IV B we provide here expressions for the decreasing families of solutions of the eective game * ξ + (αz-3/2 * (t 0 -t)) if = 1 ξ 0 (α(t 0 -t)) if = 0 z * ξ -(αz -3/2 * (t 0 -t)) if = -1 . (C1)Contrary to increasing solutions, decreasing solutions can only be dened on [0, t 0 ],and with t 0 < πz 3/2 * 2α if = -1. Using those properties we can also construct a mixed type solution by patching together an increasing "-" type solution with a decreasingone of same z * T m )) for 0 ≤ t ≤ T m z * ξ -( π 2 -αz -3/2 * (t -T m )) for T m ≤ t ≤ T ,(C2) with T m the the time at which the solutions starts decreasing, with T -πz 3/2 * 2α ≤ 0 ≤ T m ≤ πz 3/2 * 2α .

  Seen from the parabolic side of the transition, the energy E xes the duration TIV of the eective game of section IV B though Eq. (64). On this side of the transition,

	the transition time t para tr	is dened by the condition
			z(t para tr	-t para 0
	which through Eq. (47) provides a relation between E and t G tr
	F (8E, -2g/ √	π, µσ 4 ; ν) -F (8E, -2g/ √	π, µσ 4 ; Σ 0 ) =	t G tr 2 √ µ	.	(66)

  Lasry, and P.-L. Lions. A long-term mathematical model for mining industries. Appl. Math. Optim., 74:579618, 2016.

[3] Yves Achdou, Fabio Camilli, and Italo Capuzzo-Dolcetta. Mean eld games: numerical methods for the planning problem. SIAM Journal on Control and Optimization, 50(1):77109, 2012.

Order σ 2

At rst order in σ 2 Eq. ( 27) becomes

easily computed using Eq. (A1)

up to a multiplicative constant K obtained numerically.

Uniform approximation

In order to avoid the use of complex numbers, one has to consider dierent versions of the semi-classical action S, depending on the sign of (U 0 +λ). For x <