Remembering Gallipoli in a Global Context: France and Senegal

Matthew Graves, Aix Marseille Univ, LERMA, Aix-en-Provence, France

Preprint of the article published by Monash University Publishing

It has become a commonplace to assert that the Dardanelles campaign has none of the resonance in French national history, public commemoration or popular culture of Gallipoli for Australia, New Zealand or even Great Britain. The French preference for the name of the Straits rather than the battlefield tends to blend the episode into the wider history of its strategic conception. Indeed, the impression that France's role has in some way been neglected or occluded has been a constant of accounts of the campaign since the earliest records, most recently in relation to the involvement of French Empire troops. On closer inspection, however, the idea that the Dardanelles belongs to the forgotten history of the First World War proves to be more trope than truth. 1 In the century since the French Navy and Army joined the British attempt to force the Straits and seize Constantinople, the history of the campaign has never fallen durably out of the public domain, but then neither has it become a singular focus for remembrance of the First World War, while the attention paid to it has fluctuated in intensity and emphasis.

A popular misconception is that France played only a supporting role in the Allied offensive under General Hamilton's command. This is not borne out by the statistics: in spite of General Joffre's initial reluctance to divert manpower from the Western Front, France eventually committed a battleship squadron to the naval assault and a march division of 20,000 troops to the Gallipoli landings. 2 The Corps Expéditionnaire d'Orient (CEO) commanded by General Albert d'Amade had a substantial empire component and was organised into a metropolitan brigade, including an African regiment of Zouaves and Legionnaires (175 th RI and 1 st RMA) and a colonial brigade made up of a cavalry regiment from North Africa (1 st RCA) with two mixed regiments (4 th and 6 th RIC) combining Senegalese and European batallions in a proportion of two to one. By late 1915, following the diversion of French units to Salonika and the Balkan Front, the black troops from Senegal and the Caribbean formed the majority of 1 The trope persists in the literature of recent exhibitions about Albert Londres' war reporting (Vichy, 11 June -21 August 2011) and the Eastern Front (Marseille, 14 November 2014 -17 May 2015) which describe the Dardanelles and Balkan campaigns 1915-19 as the "forgotten front" and their veterans as the "forgotten soldiers" of the Great War. 2 If the French commitment were to be qualified, it is to the extent that the battleships were ageing and none of the troops were drawn from the Western Front. the rearguard. 3 Overall, through the reinforcement and rotation of the original expeditionary division, some 80,000 French troops served at Gallipoli in the course of the campaign. Of these, more than 14,000 were killed or reported missing in action and 40,000 wounded -a casualty rate of 68%. 4 Why then has the history of the Dardanelles remained stuck in the minor key of public memory? The key reason is contextual. From the French perspective, Turkey was the antechamber to a longer and (unlike Gallipoli) ultimately victorious campaign in the Balkans, Greece and Macedonia from 1915-18, which was itself overshadowed in scale, immediacy and impact by the war of attrition on the Western Front. Contrary to the Australian or New Zealand experience, the principal theatre of French operations pre-dated the Dardanelles expedition and was located largely on home soil. As a front line nation under partial occupation, France has remembered the geography of the Great War as much as its history.

From this underlying explanation flow the other reasons identified by Jauffret and Dutton for the loss of the memory of the Dardanelles in France in the immediate postwar years: the belated breakthrough in Southeastern Europe was not be allowed to outshine the victories on the Western Front; the continuation of the campaign beyond the Danube after 1918 to fight the Bolsheviks was politically sensitive; the casualties on the Gallipoli peninsula palled into insignificance compared to those incurred on the battlefields of France and Belgium; and the mistrust and resentment between allies jockeying for postwar influence in the Eastern Mediterranean cast the Entente in a poor light.

The historiography of the Dardanelles can be divided into four distinct phases encompassing shifting perceptions of the campaign. The veterans' memoirs which began to appear within a year of the events; the first-and second-hand accounts of the inter-war years; post-Second World War histories; and the soldiers' diairies rediscovered and published in the 1990s and 3 Saletes finds that the Senegalese made up more than half of the CEO from March to December 1915 -5,500 Tirailleurs of 10,700 frontline troops -though the latter total appears to be an underestimate. predecessors. In L'Expédition des Dardanelles au jour le jour, the diplomat François Charles-Roux acknowledges the Senegalese and Martiniquais troops (albeit with metropolitan paternalism as 'our blacks') and reports their suffering from the cold at the onset of winter, 9 while Captain F.J. Deygas underlines the subsidiary role played by France, laying responsibility for the failures of Gallipoli squarely at Britain's door 10 -a recurrent theme of the period. Both Deygas' book and the fourth edition of Gouin's (1931) The message is clear. The war was not won on the Western Front but in the East, where the French Army took the lead, but there is a sub-text in the early 1930s: the wars of the future will be won by movement. If this argument invites renewed interest in the Dardanelles, it subsumes it within the wider campaign in the Balkans to which it remains a static and unsuccessful prelude. Moreover, it is difficult to assess the public readership of books like these. The first editions of French library books on the Dardanelles are frequently found to have uncut pages or languish unconsulted in basement storage rooms. Gouin's had a limited print run (1500 plus 120 in two collectors' editions), was lavishly illustrated, published regionally (in Marseille) and clearly marketed at libraries and veterans as a souvenir edition. French battleship Bouvet in the aborted attempt to force the Straits symbolised the French Navy's resentment of the British Admiralty prior to Mers-el-Kébir. [START_REF] Guiffan | Histoire de l'Anglophobie en France: de Jeanne d'Arc à la Vache Folle[END_REF] The gap decades in French publications catalogued by the Bibliothèque Nationale de France (1946-64 and 1971-98) give a rough measure of the eclipse. [START_REF] Although | s Gallipoli was translated into French in the late 1950s[END_REF] With communicative memory beginning to fade, Jean Vizern's Dardanelles 1915 (1964) is the imaginative reconstruction -in Socratic dialogue -of a departed comrade-in-arms' frontline experience [START_REF] Vizern | [END_REF] , while Jean Giraudoux's rough field notes published posthumously in 1969 revisit the playwright's earlier wartime memoir (1917). [START_REF] Giraudoux | Souvenirs des Dardanelles[END_REF] In 1965, Radio France marked the 50th anniverary of the Gallipoli campaign with a two hour broadcast featuring Vizern. [START_REF]Les Dardanelles[END_REF] By the early 1980s, the Dardanelles occupies just a single paragraph in the 30 pages dedicated to the First World War in the standard history textbook for state high schools. [START_REF] Bouillon | Histoire : Le Monde Contemporain 1914-1945[END_REF] Even in later specialist histories of 'France's Great War', the campaign is briefly summarised as part of diversionary strategy in the Eastern Mediterranean. [START_REF] Duroselle | La Grande Guerre des Français[END_REF] The resurgence of interest in the Dardanelles in France in the late 1990s followed the 'awakening' of Anzac memorialism, but possessed its own commemorative dynamic. [START_REF] Rechniewski | Quand l'Australie invente et réinvente une tradition : L'exemple du débarquement de Gallipoli (avril 1915)[END_REF] To mark the 80th anniversary of the the Armistice, Le Monde featured a profile of the last of the 'Tirailleurs sénégalais' (Senegalese skirmishers), Abdoulaye Ndiaye, evoking the 'sensitive memory' of the so-called 'Black Force' from former French West Africa whose invalidity and veterans' pensions had been frozen on independence. [START_REF] Abdoulaye | dernier tirailleur sénégalais[END_REF] The issue was dramatised in the television film The Debt (2000). If Ndiaye was awarded the Légion d'Honneur alongside other surviving Great War veterans in 1998, it was only after Jacques Chirac's state visit to Senegal in 2005 that FWA pensions were thawed and revalued. As late as 2003, Marc Michel could write that 'the role of these [black] troops in the Dardanelles operation has been largely obliterated '. 21 For the 90th anniversary of the Dardanelles landings, Michèle Alliot-Marie, the defence minister, prefaced a collection of first-hand accounts assembled by the Dardanelles and Eastern Front National Memorial Association, the veterans' society founded in 1920 which merged with its counterparts for Macedonia-Serbia and the Levant in the early 1980s. 22 It was the latest in a series of manuscripts unearthed by researchers to be published at the turn of the new millenium. The move to rehabilitate theatres of operations 'too long overlooked' 23 has been part of a trend towards the 'internationalisation of memory' since the second Chirac presidency, involving bilateral agreements on shared memory with former allies and adversaries, with a meridional dimension that emphasises the geo-strategic importance of the Mediterranean to France. 24 The upsurge in interest in the Dardanelles coincided with Chirac's declarations in favour of Turkish membership of the European Union, a position that was shows a wreath-laying ceremony at the cemetery, its coverage presents the event from the Turkish perspective with no explicit mention of France's involvement. [START_REF]27 For the symbolism of the Monument aux Morts des Armées d'Orient, see Emmanuel Laugier, 'Les Monuments du Souvenir[END_REF] In France's official Great War centenary programme, 'la Mission du Centenaire', the Gallipoli-Dardanelles campaign is briefly cited in relation to French participation in events planned to mark the Anzac centenary in Australia and New Zealand and then only in the same breath as the Western Front. [START_REF] Prost | Centenaire de la Première Guerre Mondiale[END_REF] President François Hollande made no mention of Gallipoli or the Dardanelles in his speech on 7 November 2013 to launch the centenary commemorations, unlike his British counterpart David Cameron who announced that the Anzac centenary would be co-opted into the British programme in 2015. [START_REF] Hollande | Allocution pour le Lancement des Commémorations du Centenaire de la Première Guerre Mondiale[END_REF] The French Embassy in Turkey announces

  Jean-Loup Saletes, 'Les Tirailleurs Sénégalais dans la Grande Guerre et la Codification d'un Racisme Ordinaire', Guerres Mondiales et Conflits Contemporains 244 no. 4 (2011): 135. 4 The most precise figure, derived by Jauffret from the French necropolis at Seddul-Bahr, is 14,340 dead. Jean-Claude Jauffret, 'L'expédition des Dardanelles vue du côté français', in Les Armes et la Toge, ed. Jean-Claude Jauffret (Montpellier: Centre d'Histoire Militaire et d'Etudes de Défense Nationale de Montpellier, 1997), 351. Dutton gives a similar estimate. David Dutton, 'Docile Supernumerary : A French Perspective on Gallipoli' in Gallipoli, ed. Jenny MacLeod (London: F. Cass, 2004), 87. Contemporary reports put the French dead and missing at 10,000. 2000s which demonstrate a resurgence in interest for the expeditionary corps and its successor, l'Armée d'Orient. The first veterans' memoirs began to appear before the war's end. One of the earliest, Dardanelles -Serbie -Salonique. Impressions et souvenirs de guerre (1916), written by army surgeon Joseph Vassal (and prefaced by General d'Amade) highlights the shortcomings of first-hand testimony in wartime: the account was heavily censored. 5 It nevertheless ran to six editions and an English translation was published by Heinemann in 1916 under the title Uncensored Letters from the Dardanelles (written by a French medical officer to his English wife). Vassal's book and Combats d'Orient (1917) by Captain Ricciotto Canudo, the French man of letters of Italian origin who coined the expression 'the Seventh Art', set a trend in combining their accounts of the Dardanelles with its sequel on the Balkan Front where the latter served in the Foreign Legion. 6 Such early memoirs are, as Gustave Gouin puts it in the first edition of L'Armée d'Orient, des Dardanelles au Danube 'notes and correspondence written in the thick of the action … for historians to come'. 7 They evoke the heroism and the hardships of trench warfare in the exposed French sector at the tip of the Gallipoli peninsula between Cape Helles and Sedd-el-Bahr. They cite the places that might have become household names akin to the lieux de mémoire of the Anzac sector -Kum Kalé, the village on the eastern shore of the Straits seized in the diversionary attack of 25 April only to be evacuated, or Kérévés Déré, the 'shrapnel gully' in which repeated French offensives foundered 8 -had they acquired equivalent narrative purchase or momentum. Unlike the manuscript diaries of the late-twentieth century, these carefully crafted memoirs are often the work of literary stylists with a self-conscious classical frame of reference: the campaign is a modern Iliad in the shadow of the antique Troy; Kum Kalé lies at the mouth of the Scaramander where Agamemnon's 1000 ship fleet alighted; The River Clyde, run aground to shield the landing at Sedd-el-Bahr, is a Trojan Horse of iron. With increased distance from events, the uncensored writers of the immediate post-war and inter-war years introduced a critical perspective largely absent from their wartime 5 Joseph Vassal and Albert-Gérard Léo d'Amade, Dardanelles, Serbie, Salonique, Impressions et Souvenirs de Guerre (avril 1915 -février 1916) (Paris: Plon-Nourrit, 1916). 6 Ricciotto Canudo, Combats d'Orient (Paris: Hachette et cie, 1917). 7 Gustave Gouin and Fernand Detaille, L'Armée d'Orient, des Dardanelles au Danube (Marseille: Detaille, 1 st ed. 1917, 4 th ed. 1931), 1. 8 For an account of how forbidding a barrier Kérévés Déré was to the French, see Capitaine Elie, Historique du 58e Régiment d'Infanterie Coloniale (Toulon: impr Mouton & Combe, 1921), 7.

  are prefaced by Marshal Franchet d'Esperey, the commander of the Allied army in the Macedonian campaign, who develops a revisionist narrative: the exploits of the Armée de l'Orient have been unjustly neglected; by opening a new front and successfully pursuing a strategy of manoeuvre they precipitated the collapse of the Central Powers and hastened the Armistice.

  Deygas' l'Armée d'Orient dans la guerre mondiale 1915-19 was printed by leading Parisian publisher, Payot, and is addressed to a 'forgotten' constituency of veterans, the 'Poilus d'Orient'. Charles-Roux's is one of the few post-war first hand accounts to focus exclusively on the Dardanelles and is cited as authoritative by contemporaries as the insider view of a French HQ staff officer. 11 It is between the Second World War and the decades of the 'second memory boom' that historiography of the Dardanelles undergoes a hiatus. The history of the present and the greater 'débâcle' of 1940 overshadow the more distant disaster of 1915-16. In Vichy France, the Dardanelles were cast in an Anglophobic light: Jean Guiffan notes that the sinking of the 9 François Charles-Roux, L'expédition des Dardanelles au jour le jour (Paris: A. Colin, 1920), 292-3. 10 F-J. Deygas and Louis Félix Marie François Franchet d'Esperey, L'Armée d'Orient dans la Guerre Mondiale (Paris: Payot, 1932), 37. 11 The scion of a dynasty of Marseille industrialists, François Charles-Roux was a career diplomat attached to the French staff at Gallipoli as a translator. Claudine Roubaud, 'François Charles-Roux, un Marseillais dans l'Armée d'Orient', Marseille et la Grande Guerre, Marseille 246 (2014): 58-60.

  reversed in August 2005 over Turkey's refusal to recognise the Greek Cypriot government. Nonetheless, the Dardanelles remains a poor relation to the Second World War lieux de mémoire, especially the anniversary of Operation Anvil-Dragoon 15 August 1944, a tradition initiated by Charles de Gaulle in 1964, followed by Prime Minister Chirac in 1974. The landing in Provence is celebrated in France as 'the other D-Day', not just for its sweeping success in liberating the South of France but because, unlike the Normandy landings, Free French forces (drawn largely from North Africa) played a leading role: the 1st French Army under General de Lattre de Tassigny made up two thirds of the 11 Allied divisions deployed. More recently, its commemoration has been assiduously observed by Presidents Sarkozy and Hollande, surfing on the popular interest stimulated by the film Indigènes (2006), 25 as an opportunity for memorial diplomacy with leaders from France's former North and West 21 Marc Michel, Les Africains et la Grande Guerre: l'appel à l'Afrique (1914-18) (Paris: Karthala, 2003), 85. My translation. 22 The association participates in the relighting of the eternal flame at the Arc de Triomphe every 25 April. 'Association Nationale pour le Souvenir des Dardanelles et Fronts d'Orient'. Accessed 4 November 2014. Available from : http://www.fronts-dardanelles-orient-levant.tk/. 23 Association Nationale pour le Souvenir des Dardanelles et Fronts d'Orient, préf. de Michèle Alliot-Marie, Dardanelles, Orient, Levant (Paris: l'Harmattan, 2005), 5. 24 Olivier Wieviorka, La Mémoire Désunie: le Souvenir Politique des Années Sombres, de la Libération à Nos Jours (Paris: Editions du Seuil, 2010), 266. 25 Rachid Bouchared's Franco-Maroccan, Belgian and Algerian coproduction about the destiny of four North African soldiers who landed with the 1st French Army was entitled 'Days of Glory' in its English language version. African colonies. In terms of popular culture, the Dardanelles has inspired nothing comparable to Capitaine Conan (1996), Bertrand Tavernier's feature film about a unit of skirmishers on the Eastern Front. The only artwork featured in the centenary mission's statement, an oil painting by Henry Valensi 'Expression des Dardanelles' (1917), has none of the instant familiarity of Sidney Nolan's canvases. Formal commemoration of the Dardanelles lacks a focus comparable to Toulon's Mont Faron memorial. The French cemetery in Sedd-el-Bahr was inaugurated by the former CEO commander General Gouraud in 1930, but while an Agence France Presse report of the commemoration of the 99th anniversary of 'The Battle of the the Dardanelles at Gallipoli'

  There are no major monuments to the Dardanelles in France. The sole memorial to the broader campaign in the Eastern Mediterranean is the 'Monument to the Heroes of the Army of the East and Foreign Fields' in Marseille inaugurated by President Doumergue in 1927. 27 That site of memory was instrumentalised diplomatically in the early 1930s in efforts to build the Little Entente around France's former Balkan allies: Alexander I of Yugoslavia was assassinated on the Canebière on 9 October 1934 within minutes of paying homage there to the 'Poilus d'Orient'.


Christophe Viel
email: christophe.viel@onera.fr

Hélène Piet-Lahanier
email: helene.piet-lahanier@onera.fr

Michel Kieffer
email: michel.kieffer@lss.supelec.fr

Sylvain Bertrand
email: bertrand.sylvain@onera.fr

H * Lahanier

New state estimator for decentralized event-triggered

Keywords: Multi-agent system, event-triggered, consensus, undirected communication graph

INTRODUCTION

Consensus is an important problem in cooperative control: several agents have to be synchronized to the same state Reza Olfati-Saber and [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF]; [START_REF] Ren | Consensus tracking under directed interaction topologies: Algorithms and experiments[END_REF]; [START_REF] Cao | Distributed coordinated tracking with reduced interaction via a variable structure approach[END_REF]; Garcia et al. (2014c,b). In distributed cooperative control, consensus usually requires significant exchange of information between neighbouring agents so that each agent can properly evaluate its control law. This communication may be either permanent, as in Reza Olfati-Saber and [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF]; [START_REF] Ren | Consensus tracking under directed interaction topologies: Algorithms and experiments[END_REF], or takes place at discrete time instants, which is much more practical. In the latter case, communications may occur periodically, as in [START_REF] Garcia | Cooperative control with general linear dynamics and limited communication: Periodic updates[END_REF], may be intermittent Wen et al. (2012a,b); Guanghui [START_REF] Wen | Consensus of second-order multi-agent systems with delayed nonlinear dynamics and intermittent communications[END_REF], or may be eventtriggered as in [START_REF] Dimarogonas | Eventtriggered control for multi-agent systems[END_REF]; [START_REF] Jiangping | Distributed event-triggered tracking control of secondorder leader-follower multi-agent systems[END_REF]; [START_REF] Dimarogonas | Distributed event-triggered control for multiagent systems[END_REF]; [START_REF] Fan | Distributed event-triggered control of multi-agent systems with combinational measurements[END_REF]; Garcia et al. (2014c); [START_REF] Zhang | Distributed event-triggered control for consensus of multiagent systems[END_REF].

Event-triggered communication is the most promising approach to save communication energy, while allowing a consensus to be reached. In [START_REF] Dimarogonas | Distributed event-triggered control for multiagent systems[END_REF], each agent performs an estimation of the state of other agents. Communications occurs when the error between the model and the actual state reaches some threshold. Nevertheless, the communication frequency increases close to consensus. Each agent is modeled as a double integrator in [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]. The triggering condition depends on a state-independent and exponentially decreasing threshold. With this method, communication frequency still increases close to consensus, but slower. For the general linear model of the dynamic of agents considered in [START_REF] Zhu | Event-based consensus of multi-agent systems with general linear models[END_REF]; Garcia et al. (2014c,a), the error between the measurement of an agent state and its estimate of this state is used to trigger communication. Measurement errors make this approach sensitive to perturbations. Such issues have been partly addressed in [START_REF] Hu | Eventtriggered consensus of multi-agent systems with noises[END_REF]; [START_REF] Cheng | A mean square consensus protocol for linear multi-agent systems with communication noises and fixed topologies[END_REF], which study the influence of perturbations on the state of each agent and propose an event-triggered method with a time-varying coefficients to mitigate the influence of the noise. This paper considers the event-triggered communication strategy developed in Garcia et al. (2014c,a), and briefly recalled in Section 2, to reach a consensus for identical agents with a fixed topology. It introduces in Section 3 a new agent state estimator to further reduce the amount of communications. The new estimator better represents the agent behaviour by accounting for the control input evaluated by each agent. To implement this strategy, each agent, instead of estimating the state of its neighbours only, will estimate the states of all agents of the network. Each agent is then able to estimate the control inputs applied by all other agents. Section 4 shows that the proposed approach allows to reduce the need for communications when the perturbations remain bounded. Finally, conclusions are drawn in Section 5.

PRELIMINARIES

We start introducing classical notations taken from [START_REF] Cortes | Distributed Control of Robotic Networks[END_REF]. The Kronecker product is denoted as ⊗. Note λ min (M ), λ min>0 (M ), λ max (M ) the smallest eigenvalue, the smallest strictly positive eigenvalue and the largest eigenvalue of a matrix M . Consider a network of N agents with fixed undirect communication graph G and fixed adjacency matrix A. The set of neighbours of an Agent i is

N i = {j ∈ N | (i, j) ∈ E, i = j}. N i is the cardinal number of N i .
In what follows, the state perturbation affecting Agent i is assumed additive with

d i (t) = m (t) + s i (t) , (1)
where m (t) ∈ R n is a bounded time-varying perturbation with m (t) ≤ M max identical for all agents and s i (t) ∈ R n is a bounded agent-specific perturbation with s i (t) ≤ S max ∀t , i = 1, . . . , N . The vector of all state perturbations is then

d (t) = 1 N ⊗ m (t) + s 1 (t) T . . . s N (t) T T
.

(2)

DECENTRALIZED CONTROL WITH NEW AGENT STATE ESTIMATOR

Here, as in Garcia et al. (2014c), undirected communication graph and fixed topology are considered. Agent dynamics is modelled as

ẋi (t) = Ax i (t) + Bu i (t) + d i (t) (3) u i (t) = c 1 F j∈Ni y i i (t) -y i j (t)
where

x i ∈ R n is the state of Agent i and u i ∈ R m is its control input, i = 1, . . . , N . A ∈ R n×n and B ∈ R n×m .
In Garcia et al. (2014c),

c 1 = c + c 2 with c = 1/λ 2 (L)
and c 2 ≥ 0 is a design parameter. F = -B T P where P is the symmetric positive semi-definite matrix, solution of the Riccati equation PA + A T P -2PBB T P + 2αP < 0, (4) with α > 0. The esimation y i j (t) is here expressed as

y i j t i j,k = x j t i j,k , (5) ẏi
j (t) = Ay i j (t) , ∀t ∈ t i j,k , t i j,k+1 , (6)
In order to reduce the number of messages broadcast by each agent, a new estimated dynamic is built to represent the agent behaviours by accounting for the control input evaluated by each agent and his dynamic behaviour. To implement this strategy, each agent, instead of estimating the states of its neighbours only, will estimate the states of all agents of the network. Each agent is then able to estimate the control inputs applied by all other agents. If the estimators perform well, this may reduce the need for communications.

In Theorem 1, we assume that the estimation errors are perfectly known by all agents. In Section 3.2, a communication protocol and an estimator of the estimation errors are introduced to allows a practical implementation of the proposed technique.

The following section describes the resulting decentralized control algorithm.

Estimation model

In this part, each agent is assumed to perform its own estimation of the states of all other agents. Note

y i = y iT 1 , y iT 2 , . . . y iT N T with y i ∈ R N n
is the vector of state estimates of all agents by Agent i. Define the estimation errors e i j = y i j -x j with y i j (t) is an estimate of the state of Agent j by Agent i and denote e i = e i i . Define also e = y -x = e T 1 , e T 2 , . . . e T

N

T with e ∈ R N n .

The new estimation y i j (t) is here expressed as

ẏi j (t) = Ay i j (t) + Bũ i j (t) , ∀t ∈ t i j,k , t i j,k+1 (7)
ũi j (t) = c 1 F p∈Nj y i j (t) -y i p (t)
y i j t i j,k = x j t i j,k , with t i j,k the time at which the k-th message sent by Agent j has been received by Agent i. It is assumed that there is no communication delay between agents. The time instant at which the k-th message has been sent by Agent j is denoted t j,k and t j,k+1 denotes the time instant for the (k + 1)-th message. The time of reception by Agent i of the -th message is t i , whatever the sending agent.

The estimate of their own state for all agents is y =

y 1 1 T . . . y N N T T . Define y i = y i i .
The problem considered consists in designing a control scheme to reach an bounded consensus, while limiting the communications between agents. For that purpose, times t i,k are chosen locally by Agent i using an event-triggered approach considering a threshold δ i depending on the state estimation error e i . Theorem 1. Assume that (A, B) is controllable and that the communication graph is connected and undirected with a fixed topology described by Laplace matrix L.

Then the agents described by the dynamics (3) achieve a bounded consensus where the difference between any two states of Agents i and j is bounded by

lim t→∞ x i -x j 2 ≤ N η βλ min (P) (8)
with η > 0 is a design parameter, if the following condition on the perturbation bound is satisfied:

Smax≤ N η λmax(P)λ 2 (L) -2αP+2(1-c1λ2(L))PBB T P (9)
and if communication events are triggered when δi > σz

T i Θz i + η (10) with Θ i = (2c 2 -b i N i (c 2 -c)) M and δi = c1 |zi-Niei| T M j∈N i |∆ij |+Ni(N -1) 1 2b i e T i Mei + 1 2b i +bi Ni j∈N i (∆ T ij M∆ij ) +2(c2-c)Niz T i Mei + 2c(Ni) 2 (1+bi)+ c 2 -c b i Ni +cNi(N -1) bi+ 3 b i e T i Mei and M = PBB T P, 0 < b i < 2c2 (c-c2)Ni if c 2 > c, b i > 0 otherwise. z i = j∈Ni y i -y i j . ∆ ij = y j i -y i i = y j
i -y i is the difference between the estimates of state of Agent i by Agents i and j.

If η = 0 and if there is no perturbation, the system achieves an assymptotic consensus. If t = 0, all agents are assumed to broadcast a message. When t > 0, communications are then triggered according to Theorem 1. The proof of Theorem 1 is detailed in Appendix A.

∆ ij estimation and communication protocol

Theorem 1 involves the ∆ ij s which are in practice not accessible at all time instants. Agent i knows precisely ∆ ij only when it receives (possibly with some delay) a packet containing y j i from Agent j. To allow each agent to perform an estimate of the ∆ ij s, one has to consider a communication protocol in which each agent transmits its own estimate of the state of all other agents, but also his estimates of all other agents (c.f. Communication protocol).

Note first Theorem 1 only uses ∆ ij such as j ∈ N i : we will only create the estimates of ∆ ij for Agents j ∈ N i .

For j ∈ N i and if the k-th packet has been transmitted by Agent j at time t j,k , then

(1) Agent i has access to y j i (t j,k ) and is able to evaluate exactly ∆ ij (t j,k ) = y j i (t j,k ) -y i (t j,k ). ( 2) Agent i is also able to reinitialize its estimate y i j (t j,k ) of the state of Agent j at time t j,k , and thus Agent j can reset ∆ ji (t j,k ) = 0 .

As a consequence, the estimates ∆i ij performed by Agent i may be updated at t j,k as follows ∆i

ij (t j,k ) = y j i (t j,k ) -y i (t j,k ) if t = t j,k , (11) ∆i ij (t i,k ) = 0 n if t = t i,k , (12) ∆i ij (t) = Zij (t -t M ) ∆i ij (t M ) , ∀t ∈ ]t M , t M+1 [ (13) where Zij (t) = E ij exp (Zt) E + ij with E ij = [0 n , . . . , 0 n , I n , 0 n , . . . , 0 n ] composed of N 2 -1 matrix 0 n×n and with I n×n is the ((i -1) * N + j) -th matrix element of E ij . Define E + ij the pseudo- inverse matrix of E ij such as E + ij E ij = I N 2 n . t M = max {t j,k , t i,k } and t M+1 = min {t i,k , t i,k+1 }, Z = Ãc + Ñ, Ãc = I N 2 ⊗ A + c 1 (N ⊗ I N ⊗ (BF)) with N = diag ([ N 1 N 2 . . . N N ]), Ñ = à ⊗ (BF) -Ā ⊗ (BF) with à = I N ⊗ 1 T N . * 1 T N ⊗ A and Ā = (I N ⊗ A (1, : ))
T . . . (I N ⊗ A (N, : )) T , where A (i, : ) is the i-th line of the adjacency matrix A. Calculation of ∆i ij (t) is detailed in Appendix A. At t = 0, all agents are assumed to broadcast a message so ∆ij (0) = 0 n . ∆i ij (t) increases with t when it received a packet containing y j i from Agent j: in this case, ∆ ij is overevaluated. However, when Agent i broadcasts a message at t i,k , since Agent j broadcasts a message at t j,k , 13 results in ∆i ij (t) = 0, so ∆ ij is underevaluated. Using the method described in this paper involves that all agents must estimate all other agent states even if they do not belong to their neighbourhood. When the graph is not fully-connected, an Agent i can not receive information from an Agent j if j / ∈ N i , so it can not adjust its estimation y i j . However, Agent i can receive the missing information from one of its neighbours k, if Agent k is a neighbour of Agent j. The communication protocol below describes how informations are relayed when an Agent broadcasts a new message: Agent i will send its vector y i at t i,k (instead of only y i i ).

Communication protocol:

Assume that, according to Theorem 1 , Agent i broadcasts a message at time t i j,k :

• Note Lt i = [t 1,k1 ...t N,k N ]
the list of the last reception times corresponding to y i .

• At t i,k , y i (t i,k ) = x i (t i,k
) and Agent i broadcasts y i (t i,k ) and Lt i . Agent j ∈ N i compares the times in Lt i of reception of the estimated states with the ones in its own list Lt j . The components of y j are replaced by those of y i if they have been received more recently. Conversely, Agent j broadcasts to Agent i the estimates of the states of other agents that have been more recently received than those of Agent i.

The final step consists in resetting ∆i to zero. • As the graph is assumed to be connected, all state estimates are iteratively broadcasted to Agent i.

Transmitting the estimation y i j instead of the information received allows to account for its dynamical evolution between the time of reception and the time of sending the new message.

EXAMPLE

This section illustrates Theorem 1 considering an unstable system. Comparison with the estimator develops in Garcia et al. (2014c) is achieved when possible. Perturbations will be considered. Consider the N = 5 third-order agents with unstable linear dynamics given by:

A = 0.48 0.29 -0.3 0.13 0.23 0 0 -1.2 -1 B = 2 0 -1.5 1 0 1
, P is obtained by solving (4) P = 4.8436 5.4783 -1.1082 5.4783 7.0514 -1.4299 -1.1082 -1.4299 0.3778 , and we take α = 2.5 for the evaluation of S max in (9). L 5 is the Laplace matrix associated to the graph with N = 5:

L 5 =      1 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 1     
.

The following initial states are used:

x

The added perturbation is s

i = [0, s i2 , 0] T with s i2 is a Gaussian noise (0, σ 2 b )
where σ 2 b = S max , and truncated such as To evaluate the efficiency of the proposed technique, the reduction ratio of broadcasted messages, expressed in % is approximated by R com = 100 * Nm N m . We set η = 0.1, c = 1 λ2(L) , and c 2 = 0.1. Euler integration with a step dt = 0.01 s is used. As system has been discretised, the minimum delay between the transmission of two messages by the same agent is set to dt = 0.01 s. Figure 1 illustrates the compared performance in terms of consensus errors and number of event-trigger communications for observer ( 7) and ( 6). When initial conditions are perfectly known and there is no perturbation, using observer (7) allows to limit the number of communications to the only initial one. When initial conditions are not perfectly known, the required number of communication for (7) increases but remains lower than the one required using (6). For both methods, the consensus error is bounded and tends to zero and the number of communication decreases with this error.

|s i2 | < S max (|s i | ≤ [0, S max , 0] T ⇔ s i ≤ S max ), m (t) = [0, m 2 (t) , 0]

Results obtained with an unstable dynamics without perturbation

Perturbations

Figure 2 illustrates that using observer (7) lessens the number of messages required by observer (6) when perturbations are low. However, when the level of perturbations increases, the performance of observer (7) in terms of communication ratio become equivalent to observer (6) for time-varying perturbations.

CONCLUSION

This paper presents an event-triggered communication to reach consensus in multi-agent systems, with an improved agent state estimator to trigger communications using an estimate of the estimation errors. Convergence to consensus has been studied. Simulations have shown the effectiveness of the proposed estimators in presence of state perturbations.

An extensions of this work is the use of the new estimator to direct graph. Future works will adapt the method to time delay on the communication network and influence of the packet drops. T (I N ⊗ (c 1 L ⊗ (BF))) ∆ (A.4) as Tỹ = y, :

T (I N ⊗ (c 1 L ⊗ (BF))) ∆ = c1 k∈N 1 (BF(∆11-∆ 1k )) T . . . c1 k∈N N (BF(∆ N N -∆ N k )) T T
Remark ∆ ii = 0.F = -B T P and V12 expresses as

V12 ≤ c 1 N i=1   j∈Ni (y i -y j ) T -PBB T P k∈Ni (-∆ ik ) - j∈Ni (e i -e j ) T -PBB T P k∈Ni (-∆ ik )   (A.5)
Remark ∆ ji = y i j -y j so if can be shown j∈Ni (y i -y j ) T = z T i + j∈Ni ∆ T ji . Replacing this expression in(A.5) and defining M = PBB T P:

V12 ≤ c 1 N i=1   z T i M k∈Ni ∆ ik + j∈Ni (∆ ji ) T M k∈Ni ∆ ik -N i e T i M k∈Ni ∆ ik + j∈Ni e T j M k∈Ni ∆ ik   (A.6) Using |xy| ≤ 1 2bi x T x + bi 2 y T y, with b i > 0: V12 ≤ c1 N i=1 (zi-Niei) T M k∈N i ∆ ik + j∈N i Ni 1 2b i + b i 2 ×∆ T ij M∆ij +Ni j∈N i 1 2b i e T j Mej + b i 2 ∆ T ij M∆ij ≤ c1 N i=1 |zi-Niei| T M k∈N i |∆ ik |+ N i (N -1) 2b i e T i Mei + j∈N i Ni 1 2b i +bi ∆ T ij M∆ij (A.7)
For V1 , the expression becomes

V1 ≤ x T Lx + N i=1 δi -σz T i Θ i z i with δi =c1 |zi-Niei| T M k∈N i |∆ ik |+ N i (N -1) 2b i e T i Mei + 1 2b i +bi Ni j∈N i (∆ T ij M∆ij ) +δi. (A.8)
Then V1 ≤ 0 if, for i, j = 1 . . . N , the events are triggered when δi > σz T i Θz i . Remark 2. In order to reduce the number of broadcasted communications, a threshold η can be introduced so that δi > σz T i Θz i + η.

Upper bounding of

V2 V2 = 2x T (L ⊗ P) (1 N ⊗ m + s (t)) + x T Lx = 2x T Ls (t) + x T Lx because (L ⊗ P) (1 N ⊗ m) = ((L1 N ) ⊗ (Pm)) = 0 and L1 N = 0. Let V3 = 2x T Ls (t) and V4 = x T Lx. V3 = 2 N i=1 j∈Ni (x i -x j ) T Ps i V3 ≤ 2 N i=1 j∈Ni
x i -x j λ max (P) S max (A.9)

Bounding V4 requires first to note that:

L ⊗ PA + A T P + (LL) ⊗ -2cPBB T P = 1 λ 2 (L) (λ 2 (L) L) ⊗ PA + A T P + (LL) ⊗ (-2cM) ≤ 1 λ 2 (L) (LL) ⊗ PA + A T P -2M + 2 (1 -cλ 2 (L)) M ≤ 1 λ 2 (L) (LL) ⊗ (-2αP + 2 (1 -cλ 2 (L)) M) (A.10) Note: c 1 ≥ c ≥ 1 λ2(L) , so (1 -c 1 λ 2 (L)) ≤ 0. It can be shown that V4 = x T Lx is equal to: V4 = x T L ⊗ PA + A T P -(LL) ⊗ 2c 1 PBB T P x Using (A.10): V4 ≤ x T 1 λ 2 (L) (LL) ⊗ (-2αP + 2 (1 -c 1 λ 2 (L)) M) x ≤ 1 λ 2 (L) N i=1 j∈N i (xi-xj ) T (-2αP +2(1-c1λ2)M) j∈N i (xi-xj ) ≤ 1 λ 2 (L) N i=1 [N η -2αP + 2 (1 -c 1 λ 2 (L)) M N η] with η the positive constant threshold of Section A.1. V2 ≤ V3 + V4 so V2 ≤ N i=1 j∈N i (xi-xj ) T (-2αP+2(1-c1λ2(L))M) × j∈N i (xi-xj )+ j∈N i xi-xj λmax(P)Smax (A.11)
The condition of Theorem 1 is then:

S max ≤ N η λ max (P) λ 2 (L) -2αP + 2 (1 -c 1 λ 2 (L)) M 2 j∈Ni x i -x j λ max (P) S max ≤ j∈Ni x i -x j × -2αP + 2 (1 -c 1 λ 2 (L)) M j∈Ni x i -x j 1 λ 2 (L) 2 j∈Ni x i -x j λ max (P) S max ≤ - j∈Ni (x i -x j ) × (-2αP + 2 (1 -c 1 λ 2 (L)) M) j∈Ni (x i -x j ) 1 λ 2 (L)
Using it in (A.11), one obtains V3 ≤ 0. System converges to a bounding consensus.

Upper bound on V : Assume there is no perturbation. Study now the value of x i -x j when the conditions are satisfied. Let's first remark that x T Lx ≥ 0 so x T Lx ≤ λ max L x T x and x T Lx ≤ 0 so -x T Lx ≥ λ min -L x T x.

Using this, one obtains x T Lx ≤ -λmin(-L) λmax L

x T Lx. Define

β 2 = λmin(-L) λmax( L)
. With the triggering condition defined in Theorem 1, we obtain: .12) Solving this differential equation results in V (t) ≤ V (0) e -β2t + N η β2 and when t → ∞, V (t) ≤ N η β2 . We can express V (t) = 1 2 N i=1 k∈Ni (x i -x k )

V (t) ≤ x T Lx + N i=1 δ i -σz T i Θ i z i ≤ -β 2 V (t) + N η (A
T P (x i -x k ) and a bound on the difference between any two states i,j can be obtained. We can deduce λ min (P)

N i=1

x i -x j 2 ≤ N η β2 and so x i -x j 2 ≤ N η λmin(P)β2 .

Note that perturbation terms don't appear in terms δi and Θ i , but they will have an impact on error measurement, and then on the frequency of event triggering.

  0) = -4.72 0.67 -2.61 -1.96 3.03 -1.14 0.73 -0.99 -0.81 -0.63 -0.10 0.25 0.01 0.06 -0.05 (14) Robustness with respect to initial conditions is also tested by adding a random Gaussien noise (µ b , σ 2 b ) with µ b = 0, σ 2 b = 5 to the values of the initial states known by the agents. The total number of messages broadcasted during a simulation is denoted N m . With the simulation duration T = 5 s, N m is upper bounded by N m = N T /dt.

  T where m 2 (t) can be either constant (see e.g. Figure 2 (b)) or represent a Gaussian noise (0, σ 2 b ) (Figure 2 (a)).

Fig. 1 .

 1 Fig. 1. Comparison between observer (6) and observer (7) with known Initial States (I.S.) and perturbed Estimation of the Initial States (E.I.S.).

Fig. 2 .

 2 Fig. 2. Comparison between observer (6) (Dash line) and observer (7) (Thick line).

A. 2

 2 Estimation of ∆ikIt can be shown that the time derivative of the component ∆ ik of ∆ is defined as :∆ik = (A + c 1 N i BF) ∆ ik + c 1 BF j∈Ni (∆ ji -∆ jk ) using (7). Then ∆ satisfies: ∆ = [I N 2 ⊗ A + c 1 (N ⊗ I N ⊗ (BF))] ∆ +c 1 Ã ⊗ (BF) ∆ -c 1 Ā ⊗ (BF) ∆ (A.13) ∆ = Z∆ (A.14) with Z = Ãc + Ñ, Z ∈ R N N n , Ãc = I N 2 ⊗ A + c 1 (N ⊗ I N ⊗ (BF)), Ñ = Ã ⊗ (BF) -Ā ⊗ (BF).The solution of the differential equation ∆ = Z∆ is of the form ∆ (t) = exp (Z (t

  -t k )) ∆ (t k ) where t k is the last communication time in the network N . Remark ∆ ij (t) = E ij exp Z t -t i k ∆ t i k and remind E + ij the pseudo-inverse matrix of E ij such as E + ij E ij = I N 2 n . So define ∆i

ij (t) = E ij exp Z t -t i k E + ij E ij ∆ t i k . ∆i ij (t) = Zij t -t i j,k ∆ ij t i j,k (A.15) with Zij (t) = E ij exp (Zt) E + ij .

(a) with Gaussian m (t) (b) with constant m (t)

ACKNOWLEDGEMENTS

We thanks Direction Generale de l'Armement (DGA) for a financial support to this study.

Appendix A.

A.1 Proof consensus convergence

Define x = x T 1 . . . x T N T the global state vector. The system gathering the dynamics of all the agents is then

Define the Lyapunov function : V = x T Lx, with L = L⊗P and L a Laplacian matrix, so L1 N = 0. In this section, the graph is undirect so L is symmetric.

Upper bounds on V1 and on V2 are derived in the two following sections.

Upper bounds on V1 : Let

Let us introduce the two following terms V11 = 2x T L Āx + B 1 y and V12 = 2x T L B∆. The expression of V11 is the one in Garcia et al. (2014c).

From Garcia et al. (2014c) one obtains V11 = x T Lx + N i=1 δ i -z T i Θ i z i .


M Alwan

N Candoni

Ph Dumas

H R Klein
email: klein@cinam.univ-mrs.fr

Statistical evidence of strain induced breaking of metallic point contacts

Keywords: 73, 63, Rt electronic transport in nanocontacts -81, 07, Lk fabrication of nanocontacts

A Scanning Tunneling Microscopy in Break Junction regime and a Mechanically Controllable Break Junction are used to acquire thousands of conductance-elongation curves by stretching until breaking and re-connecting Au junctions. From a robust statistical analysis performed on large sets of experiments , parameters such as lifetime, elongation and occurrence probabilities are extracted. The analysis of results obtained for different stretching speeds of the electrodes indicates that the breaking mechanism of di-and mono-atomic junction is identical, and that the junctions undergo atomic rearrangement during their stretching and at the moment of breaking.

Introduction

During the last decades, electronic properties of atomic sized metallic contacts have interested several groups employing different techniques [START_REF] Agraït | Quantum properties of atomic-sized conductors[END_REF]. First studies were carried out using Scanning Tunneling Microscopy in Break junction regime (STM-BJ) [START_REF] Gimzewski | Transition from the tunneling regime to point contact studied using scanning tunneling microscopy[END_REF]. The second most used technique is the Mechanically Controllable Break Junction (MCBJ) developed by Muller et al. [START_REF] Muller | Conductance and supercurrent discontinuities in atomic-scale metallic constrictions of variable width[END_REF] in 1992 and based on an earlier design by Moreland and Ekin. [START_REF] Moreland | Electron tunneling experiments using nb-sn "break" junctions[END_REF] Both STM-BJ and MCBJ techniques are based on the stretching of an Au wire. During stretching, the Au wire gets thinner, and finally breaks leading to two electrodes. Before breaking, at a given bias voltage, the current flowing through the junction is measured as a function of the relative elongation of Au wire. As it is expected in diffusive regime, the current first decreases linearly while the elongation of the wire. However, in the ultimate stage before breaking, while the diameter of the Au wire consists only in a few atoms, the electronic transport becomes ballistic with the conductance given by:

G = 2e 2 h N n=1
T n

Where N is the number of available channels for the electrons in the junction, e the electron charge, h the Planck's constant and T n the transmission probability of the channels. In junctions of atomic dimensions, conduction channels are directly related to wave functions of the metal [START_REF] Scheer | The signature of chemical valence in the electrical conduction through a single atom contact[END_REF]. In the particular case of gold, the conductance of a single atom-junction is due to one conductance channel with transmission probability close to unity [START_REF] Vega | Different wavelength oscillations in the conductance of 5d metal atomic chains[END_REF]. Thus, its conductance is close to 2e 2 /h, the conductance quantum G 0 .

Therefore, in MCBJ or STM-BJ experiments, a gold monoatomic junction is characterized by a constant conductance of 1G 0 during the stetching of the junction, i.e. a ballistic plateau at 1G 0 . It is noteworthy that the length of this plateau gives a direct access to the lifetime of the mono-atomic junction. Similarly, the ballistic plateau observed at 2G 0 is distinctive of a junction containing two atoms in its section.

Such junctions are often employed to bind molecules displaying terminations with a high affinity to metals such as thiols or amino groups [START_REF] Chen | Measurement of single-molecule conductance[END_REF]. In this purpose, the mono-atomic junction is stretched until it breaks, leading to two Au nano electrodes. Then in the presence of molecules, these ones can graft on the electrodes and the conductance of the Au-molecule-Au junction are measured, giving access to single molecule conductance [START_REF] Leoni | Tools for conductance measurements of metallic and molecular nanowires[END_REF]. However such studies require stabilizing junctions for a period as long as possible. Thus, the understanding of the Au junction stability and breaking mechanism is crucial.

Tsutsui et al have studied the dependence of the lifetime of monoatomic gold contacts on the stretching speed of the electrodes [START_REF] Tsutsui | Formation and self-breaking mechanism of stable atom-sized junctions[END_REF]. They showed that for stretching speed higher than 1nm.s -1 , lifetime is limited by the strain applied to the junction. Similar results were obtained for molecular junction by Huang et al [START_REF] Huang | Single molecule junctions formed via au thiol contact: Stability and breakdown mechanism[END_REF]. For lower stretching speed, lifetime is rather limited by thermal motion, leading to discrete conductance fluctuations depending on the temperature of the experiment [START_REF] Klein | Conductance fluctuations in gold point contacts: an atomistic picture[END_REF].

Literature generally focuses on the ultimate ballistic plateau at 1G 0 , whereas the properties of the junction may depend on its earlier configuration. Therefore, in this work we explore as much ballistic plateaus at 1G 0 as at 2G 0 . Our study concerns the behavior of mono-and di-atomic junctions associated to 1 and 2G 0 conductances, respectively, at different stretching speeds in ambient conditions. Our specificity is the comparison of results obtained with two complementary techniques STM- BJ and MCBJ, developed in the laboratory, in order to avoid artifacts in these delicate measurements.

A great deal of effort is made in this paper to investigate experimental results through an objective statistical approach in order to highlight geometry and mechanical properties of Au junction. This is made possible by the ability to repeat the atomic contact experiments thousands of times using break junction setups (MCBJ or STM-BJ).

Experiments

Set-up

In STM-BJ, Au junctions are obtained by indenting an Au face with a gold tip. Surfaces are prepared by electron beam evaporation under ultra high vacuum conditions of a 100nm layer of Au with a preferential [111] orientation [START_REF] Klein | Self-assembled monolayers of decanethiol on au(111)/mica[END_REF], while gold tips are electrochemically etched from 0.25 mm gold wire (99.99%, Goodfellow).

In our home-made MCBJ (sketched in figure 1), a thin Au wire (diameter 250µm) is glued onto a phosphorous bronze bending beam with two droplets of epoxy (Stycast 2850FT, curing agent 24LV). The basic principle consists in stretching the wire by bending the substrate with a mechanical actuator, until the junction breaks and gives two separate electrodes. Relaxing the force on the elastic substrate by back-bending allows the junction to be closed again. The mechanical actuator is composed of a stepping motor and a piezoelectric transducer (PZT) in series, which permits to control the process by ramping the zpiezovoltage. Because the wire is fixed, the vertical movement of the PZT is demagnified, allowing accurate control of wire stretching (elongation). The relative elongation can reach a resolution in the range of 5 picometers. Moreover, the MCBJ exhibits an outstanding mechanical stability (<5pm/s drift of the electrodes after breaking at room temperature for our setup). This stability results from the reduction of the mechanical loop which connects one electrode to the other, from centimeters, in the case of an STM scanner, to approximatively 0.2 mm in our MCBJ. The drawback is that the demagnification factor depends on the geometrical parameters of the sample, mainly on the distance between the glue droplets. It is thus difficult to obtain an absolute calibration of the relative displacement of the electrodes. An accurate calibration method using the Gundlach oscillations in the tunnel current has been proposed by Kolesnychenko et al [START_REF] Yu | Calibration of the distance between electrodes of mechanically controlled break junctions using field emission resonance[END_REF]. Unfortunately this method is not suitable for operation in ambient conditions. To circumvent this potential calibration issue, we have combined the MCBJ measurements with STM-BJ ones, where displacements are easily calibrated.

For the two setups, experiments are carried out in air using a bias voltage of the electrodes of 130 mV. The current is measured using a DLPCA-200 I/V converter (Femmto Messtechnik). Conductance versus elongation traces are recorded using a dedicated software described in detail elsewhere [START_REF] Leoni | Are conductance plateaus independent events in atomic point contact measurements? a statistical approach[END_REF]. Briefly, while the MCBJ or STM-BJ regulation loop is maintained off, the two electrodes are brought into contact prior to breaking off the junction by pulling the electrodes away with variable stretching speeds. Conductance data are simultaneously acquired at a sampling frequency of 10 kHz. The regulation loop is then activated for approximatively 1s and the cycle is repeated automatically thousands of times.

Methods

The recorded curves show ballistic plateaus in the transition between contact and vacuum tunneling. The quantization of conductance appears at approximately integer multiples of the Landauer conductance quantum,G 0 = 2e 2 /h as expected. However, some plateaus in the conductance curves cannot be unambiguously identified to integer multiples of the conductance quantum, nG 0 . In an attempt for an objective analysis of the data, we identify conductance plateaus by the abrupt changes of conductance, which separate them. The plateaus shorter than three consecutive acquisition points (0.2 ms) are discarded. From the acquisition of thousands of individual conductance curves of the same junction, histograms of conductance values are constructed, as shown on figure 2, More advanced statistics are performed after acquisition. The algorithms developed for these tasks are described elsewhere [START_REF] Leoni | Are conductance plateaus independent events in atomic point contact measurements? a statistical approach[END_REF]. They allow the extraction of occurences, average conductances and lifetimes of conductance plateaus. The same algorithms are used to analyse STM-BJ and MCBJ experiments.

Results and discussion

We have tested , using STM-BJ and MCBJ, stretching speeds of the electrodes in the range from 0.5 to 1000 nm.s -1 . In the recorded conductance-elongation curve, we focus on plateaus G 0 2G 0 Fig. 3. Histogram of lifetimes of G0 and 2G0 plateaus recorded using the MCBJ at a stretching speed of 142 nm.s -1 (bias 130 mV). Both histograms are obtained from sets of 75000 elongation curves, and can be described as an exponential decay law with similar time constants. STM-BJ experiments exhibit a similar behaviour.

at 1G 0 ( 0.8G 0 < G < 1.2G 0 ) and 2G 0 ( 1.6G 0 < G < 2.4G 0 ), which are attributed to mono-and di-atomic sections of the junction, respectively.

Lifetimes of mono-and di-atomic junctions

As conductance measurement is acquired at a constant stretching speed, the elongation of the contacts leading to ballistic plateaus at 1G 0 and 2G 0 is directly proportionnal to junctions' lifetime, reflecting their stability. From sets of thousands conductance curves, we can plot histograms of lifetimes of monoand di-atomic junction lifetimes. As shown in figure 3, the occurrence of plateaus lifetime typically leads to an exponential decay law given by the general expression

p(t) = exp(-t/τ ) τ
where τ corresponds to the mean value of lifetime. This exponential decrease is a clear indication of the random nature of the breaking process, as we have already reported in [START_REF] Leoni | Are conductance plateaus independent events in atomic point contact measurements? a statistical approach[END_REF]. Without any further investigations, this random process suggests that mono-and di-atomic junctions' breaking are driven either by thermal motion, by a random external perturbations or by the constraint applied to the junction, which would provide the system enough energy to break. The dependence of the lifetime of ballistic plateaus at 1 and 2 G 0 (τ 1 and τ 2 respectively) on the stretching speed shown in figure 4,indicate that the breaking is due to the strain applied to the junction. Indeed the strain increase with the stretching speed, and thus the lifetime decreases. Noteworthy, τ 1 and τ 2 exhibits similar values, and the same dependence on the stretching speed.

This result is in excellent quantitative agreement with data for ballistic plateaus at 1 G 0 previously published by Tsustui [START_REF] Tsutsui | Formation and self-breaking mechanism of stable atom-sized junctions[END_REF] and Huang [START_REF] Huang | Single molecule junctions formed via au thiol contact: Stability and breakdown mechanism[END_REF], using an MCBJ and a STM-BJ respectively.

We can imagine that the junction breaks like a constrained rope, one strands (conduction channel) after the other at a given lengthening [START_REF] Rubio-Bollinger | Mechanical properties and formation mechanisms of a wire of single gold atoms[END_REF]. In such a scenario, a diatomic junction displays two parallel conduction channels for the electrons flowing trough the junction. One of these channels breaks under strain, leading to a mono-atomic junction. We would then expect τ 1 to be much shorter than τ 2 because the last channel has already been stretched in the di-atomic junction before it breaks. However, τ 1 and τ 2 show quite close values and the same dependence on v, suggesting that the breaking mechanism is not sequential, and has the same origin for mono-and di-atomic junctions. G 0 2G 0 Fig. 5. Plateaus occurence probabilities at 1 and 2 G0 vs stretching speed. These values are obtained from sets of 75000 conductance curves for each stretching speed using MCBJ and STM-BJ at a bias of 130mV. Increasing the stretching speed (and thus the strain applied to the junction), lead to a decrease of the occurence probabilities of conductance plateaus.

Occurence of mono-and di-atomic junctions

Our advanced statistical analysis allows extracting not only lifetimes but also occurence probabilities of ballistic plateaus at 1 and 2 G 0 , P 1 and P 2 respectively. The dependence of these probabilities on the stretching speed is shown on figure 5. This curve shows that the ocurences of ballistic plateaus dramatically decrease while the stretching speed (and thus the constraint) increases.

In addition, on figure 6, the dependence of the product of the probabilities P 1 .P 2 on P 1&2 , the joint probability to observe conductance plateaus at 1 and 2 G 0 on the same conductance curve, is plotted . These probabilities show very close values for all the stretching speed explored in this study with the STM-BJ and MCBJ setups. It points out that the occurences of ballistic plateaus are statistically independent events. In the case of sequential breaking of di-and mono-atomic junctions, plateaus at 1 and 2 G 0 must appear together in conductancetime curves leading to correlated occuring probabilities. The observed statistical independence means that the plateau at 1 G 0 exists wether the plateau at 2 G 0 exists or not, and viceversa, as shown by the combination of the probabilities P 1 and P 1&2 . Using these values we can calculate that 50% of the 1 G 0 events are subsequent to 2 G 0 events at the stretching speed of 142 nm.s -1 , to be compared to 23 % at 284 nm.s -1 and 60 % at 23.7 nm.s -1 for instance.

Consequently, the di-atomic junction is certainly not composed of two conduction channels in parallel, which would break sequentially. But rather a more complicated atomic structure implying a width of two atoms in its thinner part, as confirmed by tight-binding molecular dynamic [START_REF] Coura | On the structural and stability features of linear atomic suspended chains formed from gold nanowires stretching[END_REF][START_REF] Dreher | Structure and conductance histogram of atomic-sized au contacts[END_REF][START_REF] Leoni | Are conductance plateaus independent events in atomic point contact measurements? a statistical approach[END_REF]. These simulations show that the junction displays an asymetric structure during stretching, meaning that the strain applied to the atomic bonds is not homogeneous. Fig. 6. Product of the probabilities to observe a plateau at G0 (P1) times the probability to observe a plateau at 2G0 (P2) vs the joint probability P 1&2 to observe both plateaus during the same experiment. This graph gathers the results obtained from the STM-BJ and MCBJ experiments for the whole stretching speed range of the present study. Dashed line : P1.P2 = P 1&2

Breaking mechanism

During stretchnig, the junction stiffness increases with the accumulated elastic energy. Then, the only way to relax a part of this excess of energy is a plastic deformation, by the rupture of the most strained bond. This occurs for given elongations, independent of the stretching speed, 0.1-0.15 nm in our study. Several experimental studies at low temperature [START_REF] Yanson | Formation and manipulation of a metallic wire of single gold atoms[END_REF] and in ambient conditions [START_REF] Huang | Single molecule junctions formed via au thiol contact: Stability and breakdown mechanism[END_REF] reported an elongation at rupture of 0.1 to 0.25 nm corresponding to inter-atomic distance between gold atoms , in good agreement with molecular dynamics simulations. Thus, the junction has to be considered more as a contact than as a chain of atoms. G. Rubio-Bollinger estimated a rupture force of 1.5 ± 0.5 nN [START_REF] Rubio | Atomic-sized metallic contacts: Mechanical properties and electronic transport[END_REF]. By considering the junction as a spring, the ration between this force and elongation at rupture leads to a spring constant of 10-15 N.m -1 , and an energy release at rupture of 0.2-0.7 eV. Numerical simulations of junction breaking are in excellent agreement with experimental values [START_REF] Vélez | When do nanowires break? A model for the theoretical study of the long-term stability of monoatomic nanowires[END_REF].

Our study points out that similar results are obtained both for mono-and di-atomic junctions, indicating that their breaking mechanism is identical. As the junction undergoes plastic deformation, the corresponding energy release can lead either to atomic reconfiguration of the junction or to breaking of the remaining bonds. Hence the conductance elongation curve, after a 2 G 0 conductance plateau, will display either a rupture or a conductance plateau at 1 G 0 corresponding to a junction with a structure not correlated with the previous one. This fully explains the fact that 1 and 2 G 0 conductance plateaus are statistical independent events.

Conclusion

A MCBJ and a STM-BJ have been used to acquire thousands of conductance -elongation curves by stretching until breaking and re-connecting Au junctions. Data on ballistic plateaus at 1G 0 and 2G 0 are extracted and statistically analysed, in order to explain the breaking of mono-and di-atomic Au junctions in ambient conditions.

Our study relies on a robust statistical analysis performed on large sets of experiments. Moreover, it combines results obtained with two complementary techniques : STM-BJ and MCBJ. The excellent agreement between the results, strengthen the analysis and ensures the lack of artifacts in lifetime measurements related either to external perturbations (vibrations, thermal drift) in STM-BJ experiments, or calibration issues in MCBJ ones.

The dependence of the lifetime of gold junctions on the stretching speed of the electrodes showed that for stretching speed higher than 1nm.s -1 , lifetime is limited by the strain applied to the junction. This is in agreement with previously published works focusing on mono-atomic junctions. The present study extends this result to di-atomic junctions indicating that their breaking mechanism is identical. Moreover, the analysis of occurrence probabilities of the ballistic plateaus gives an experimental evidence that junctions undergo atomic rearrangement during their stretching and at the moment of breaking.

To our knowledge, it is the first time that a statistical approach provides such complementary informations on breaking mechanism of Au nano-junctions.

Fig. 1 .

 1 Fig. 1. Schematics of our home made MCBJ. Top : The basic principle consists in stretching the wire by bending the substrate with a mechanical actuator , until the junction breaks and gives two separate electrodes. Relaxing the force on the elastic substrate by back-bending allows the junction to be closed again. The mechanical actuator is composed of a stepping motor and a piezoelectric transducer (PZT) in series. Down : A thin Au wire (diameter 125µm) is glued onto a phosphorous bronze bending beam with two droplets of epoxy.

Fig. 2 .

 2 Fig. 2. Typical histogram of conductance constructed from a set of 9000 conductance vs elongation traces recorded using the STMBJ at a stretching speed of 142 nm.s -1 (bias 130 mV). Two conductance vs elongation traces are also shown. On these traces conductance plateaus are separated by abrupt changes of conductance marked by the dashed arrows.

Fig. 4 .

 4 Fig. 4. Lifetimes of ballistic plateaus at 1 and 2 G0vs stretching speed of the electrodes obtained with MCBJ (circles and squares) and STM-BJ (up and down triangles) at a bias of 130 mV. Values calculated from 75000 experiments. Horizontal error bars on MCBJ points represents the uncertainty in the calibration of the relative displacement of the electrodes.