Herbert 2016 Busemann

Athanase Papadopoulos

Marc Troyanov
email: marc.troyanov@epfl.ch

Marc Troyanov On

Hal

research institutions in France or abroad, or from public or private research centers

Keywords: Axioms of geometry, Minkowski spaces, Hilbert axioms, Busemann geometry, horospheres 53C70, 54E35, 53C23, 97E10

This paper is a commentary and a reading guide to three papers by Herbert Busemann, Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind. (On the geometries where circles of infinite radius are the shortest lines) (1932), Paschsches Axiom und Zweidimensionalität, (Pasch's Axiom and Two-Dimensionality) (1933) and Über Räume mit konvexen Kugeln und Parallelenaxiom. (On spaces with convex spheres and the parallel postulate) (1933). These are the first papers that Busemann wrote on the foundations of geometry and the axiomatic characterization of Minkowski spaces (finite-dimensional normed spaces). The subject of these papers followed Busemann for the rest of his life, and the three papers already contain several ideas and techniques that he developed later on, in his work on the subject which lasted several decades.

Introduction

The three papers [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF][START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF][START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF] by Busemann with which we are concerned in this note are among the first papers that he wrote. 1 The first one was published in 1932 and the two others in 1933. These papers concern the foundations of geometry, a field whose main object is to investigate the various existing geometries, in general from the axiomatic point of view. This concerns in particular the three "classical" geometries: Euclidean, hyperbolic and spherical, but projective and affine geometries were also extensively studied, since the 19th century. Furthermore, the axiomatization process, when Busemann started his work, had already been extended to other fields than geometry. One may mention arithmetic, but we also recall that Hilbert's Problem VI, from his famous 1900 twenty-three problems, asks for the axiomatization of mathematical physics.

The axiomatic method has a long history which starts in Greek Antiquity 2 with a culmination in Hilbert's Grundlagen der Geometrie [START_REF] Hilbert | Foundations of geometry[END_REF]. The first edition of this treatise appeared in 1899, and the seventh revised edition, to which Busemann refers, in 1930.

It is also worth mentioning that during the three decades that preceded Busemann's work, set theory and topology, which were closely related fields, experienced a rapid development, and several mathematicians, including Busemann himself, [START_REF] Alexandroff | Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen[END_REF] Busemann published a paper on analysis in 1930, Die Vollständigkeit der Minimalfolgen von Eigenwertproblemen, Nachrichten Göttingen 1930, 295-307. It seems he did not publish any paper in 1931.

2

It is well known that the Greeks started the axiomatic foundation of Euclidean geometry, but they also worked out the foundations of spherical geometry; this was done mainly by Theodosius and Menelaus. They also approached the foundations of projective geometry; one may think here of the work of Apollonius.

were familiar with them. 3 Busemann had also at his disposal the theory of metric spaces. We recall in this respect that the abstract notion of metric space as we understand it today first appeared in 1906, in Fréchet's thesis [START_REF] Fréchet | Sur quelques points de calcul fonctionnel, thèse de doctorat publ[END_REF]. At the time where Busemann started his work, this theory had already witnessed important developments, in particular by K. Menger who worked in Vienna. Busemann refers at several places to the work of Menger.

The papers [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF][START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF][START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF] by Busemann are important because the questions that Busemann considers and the ideas he expresses there are among those that followed him during the rest of his life. At the time where Busemann wrote them, he was working as an assistant (without salary) at the University of Göttingen, having obtained his doctorate in 1931 under the supervision of R. Courant. Hilbert, to whom we shall often refer in this note, was still teaching at that university, mostly on philosophy of mathematics, although he had officially retired in 1930.

The specific problem in which Busemann was interested in these three papers is that of characterizing Minkowski geometries. We recall that a Minkowski space is defined by a metric ρ on a finite-dimensional real vector space such that for any quadruple

X, Y, X ′ , Y ′ , if (Y ′ -X ′ ) = t(Y -X), then ρ(X ′ , Y ′ ) = |t|ρ(X, Y ), see [26, chap 1, §1]
. These spaces were introduced by Hermann Minkowski in the book [START_REF] Minkowski | Geometrie dre Zahlen[END_REF], to which Busemann refers. These spaces play a major role in Busemann's subsequent work. They are the infinitesimal Finsler geometries, that is, they play, in Finsler geometry, the role played by Euclidean spaces in Riemannian geometry. Developing a theory of Finsler spaces from a purely metric point of view, that is, without differentiability, was one of Busemann's favorite topics. Another reason for which Minkowski spaces are important in Busemann's work is that they constitute one of the two main classes of examples of metric spaces satisfying the conditions of Hilbert's Problem IV. 4 This problem, from the list of problems that Hilbert addressed at the Paris 1900 International Congress of Mathematicians, remained for several decades one of Busemann's favorite problems. In Hilbert's words, this is the problem of the straight line as the shortest distance between two points. This is usually interpreted as the question of characterizing the metrics on subsets of Euclidean (or projective) space such that the Euclidean (respectively projective) straight lines are the shortest lines. As a matter of fact, in his comments on that problem, Hilbert says much more. In particular he mentions twice Minkowski spaces. First, he considers them as one of the two main examples of spaces satisfying his problem, and then he asks for an axiomatic characterization of these spaces. Let us quote his words, from the English translation of his fourth problem [START_REF] Hilbert | Mathematische Probleme[END_REF]:

One finds that such a geometry really exists and is no other than that which Minkowski constructed in his book, Geometrie der Zahlen, 5 and made the basis of his arithmetical investigations. Minkowski's is therefore also a geometry standing next to the ordinary Euclidean geometry; it is essentially characterized by the following stipulations:

1. The points which are at equal distances from a fixed point O lie on a convex closed surface of the ordinary Euclidean space with O as a center. 2. Two segments 6 are said to be equal when one can be carried to the other by a translation of the ordinary Euclidean space.

3

We point out that Busemann, during his studies, followed the topology courses of P. S. Alexandroff, who visited Göttingen regularly. See [START_REF] Goenner | Documents concerning Busemann, from the library of[END_REF]. 4 The other class is that of Hilbert metrics.

5 [Hilbert's footnote] Leipzig, 1896.

6

In this paper, the word "segment" is used several times, in more than one sense. In each case the meaning should be clear from the context. In Minkowski's geometry the axiom of parallels also holds under the conditions that balls are strictly convex.

We note that in his formulation of the second property, Hilbert makes use of translations of the space. In Busemann's axiomatic characterization of Minkowski spaces, which is the subject of the two papers [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF] and [START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF], there is no mention of motion. We remind the reader that the question of including or not the motions in the axioms of geometry has been a delicate issue, for several centuries. It originates in Aristotle, who considered that motion should be avoided in the axioms of geometry, because it pertains to physics rather than mathematics, cf. e.g. Aristotle's Metaphysics A8, 989, b32-33 [START_REF]Aristotle, the Metaphysics[END_REF]. Euclid, in his list of axioms, followed Aristotle's point of view. The question of including or not motions in the foundations of geometry has been thoroughly discussed by several commentators of Euclid, in particular the Arabs; see e.g. [START_REF] Rashed | Thābit et la théorie des parallèles[END_REF], [START_REF] Rashed | Le mouvement en géométrie classique[END_REF] and the references there.

The question of describing axioms for Minkowski geometries remained central for Busemann. It is investigated in his later papers, e.g. [START_REF] Busemann | The foundations of Minkowskian geometry[END_REF][START_REF] Busemann | Minkowskian geometry, convexity conditions and the parallel axiom[END_REF], and in his books [START_REF] Busemann | The geometry of geodesics[END_REF] and [START_REF] Busemann | Recent synthetic differential geometry[END_REF]. Besides the question of characterizing Minkowski spaces, Busemann made an extensive investigation of the geometry of these spaces, cf. [START_REF] Busemann | Transitive geodesics in Minkowski planes[END_REF][START_REF] Busemann | Minkowskian geometry, convexity conditions and the parallel axiom[END_REF]. There is more about these results in the paper [START_REF] Berestovskii | Busemann's results, ideas, questions and locally compact homogeneous geodesic spaces[END_REF] in the present volume.

We shall comment more on Hilbert's Problem IV. In particular, we shall point out that several directions that Hilbert indicated, in the comments that accompany this problem, concerning the solution. Busemann remained interested in that problem during several decades; cf. [START_REF] Papadopoulos | Hilbert's fourth problem[END_REF] for a review of his work on this problem. He started by following Hilbert's line of thought, but he then brought a certain number of fundamental ideas that were completely novel. His program combines the then recent developments of metric geometry, set theory and topology, together with a knowledge of the classical bases of axiomatic and projective geometry. At several occasions, Busemann commented on his metric approach to the axioms. In the first two paragraphs of [START_REF] Busemann | Metric Methods in Finsler Spaces and in the Foundations of Geometry[END_REF]Chap. 3], he writes:

The axiom that there is only one straight line through two given points is fundamental in the foundation of geometry, in projective geometry and in non-euclidean geometry. At the stage where this axiom is formulated a metric is generally not yet defined; therefore it is meaningless to speak of the straight line as a geodesic.

The early geometers frequently tried to introduce the straight lines as shortest connections, 7 but without success since there was no abstract definition of a metric.

Hilbert suggested in his comments on Problem IV the inclusion of metric elements as part of the axiomatics. More precisely, he asks whether the triangle inequality could be introduced as an axiom in the foundation of geometry. He writes in [START_REF] Hilbert | Mathematische Probleme[END_REF]:

We are asking, then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and, in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.

Regarding this passage, let us recall that the triangle inequality is a theorem in Euclidean geometry; see the end of the Appendix for a short discussion.

7

For instance, Legendre in his Elements de géométrie gives the following definition: La ligne droite est une ligne indéfinie qui est le plus court chemin entre deux quelconques de ses points (The straight line is an undefined line that is the shortest path between any two of its points).

We mentioned that the paper [START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF] concerns the axioms of two-dimensional geometry. We note in this respect that it may happen that in some results of Busemann, the case of dimension two requires a special statement (cf. for example the results of the paper [START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF] that are recalled in §5 below). Since we are talking about dimension, we note right away that there exist axiomatic definitions of dimension (in the style of Hilbert's Foundations), but that Busemann in his work uses the notion of topological dimension, for which he refers to Alexandroff [START_REF] Alexandroff | Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen[END_REF].

The plan of the rest of this note is the following.

In §2, we introduce some basic notions that are used by Busemann, in particular the notion of line space.

In §3, we comment on the content of the paper [START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF] where Busemann proves that a line space is 2-dimensional if and only if Pasch's Axiom holds.

In §4, we comment on the content of Busemann's paper [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF]. The main result in this paper is a characterization of Minkowski spaces by the fact that horospheres, with the induced metric, are themselves line spaces.

In §5, we comment on the content of Busemann's paper [START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF]. In this paper, a characterization of Minkowski spaces is given in terms of a parallelism relation satisfied by the space together with a convexity property of its spheres. This property of convexity of spheres will play a prominent role in the later works of Busemann.

In an appendix at the end of the present note, we have included, for the convenience of the reader, the list of Hilbert's axioms of Euclidean geometry to which Busemann refers.

The notion of line space

In the three papers, Busemann uses the following notion that he attributes to Menger.

Definition 2.1. A line space is a set equipped with a non-negative distance function ρ(X, Y ), called a metric, satisfying the following 6 axioms.

i. ρ(X, Y ) = ρ(Y, X). ii. ρ(X, Y ) = 0 if and only if X = Y . iii. ρ(X, Y ) + ρ(Y, X) ≥ ρ(X, Z).
iv. Every sequence which is bounded with respect to this metric has an accumulation point. v. For every pair of points (X, Y ) there exists a unique point U such that

ρ(X, U ) = ρ(U, Y ) and ρ(X, U ) + ρ(U, Y ) = ρ(X, Y ).
vi. For every pair of points (X, Y ) there are exactly two points V and W such that

ρ(X, Y ) = ρ(Y, V ) = ρ(W, X), ρ(X, Y ) + ρ(Y, V ) = ρ(X, V ), ρ(Y, X) + ρ(X, W ) = ρ(Y, W ). X Y U V W
The first three axioms are the usual axioms of a metric space. Axiom iv is a formulation of the Bolzano-Weiertrass property for sequences in R n . It seems that the introduction of this statement in the axioms of a geometry is a novelty.

Axiom v is about the existence and uniqueness of midpoints and Axiom vi is about the existence and uniqueness of doubled points.

Busemann notes that, using techniques from metric geometry as in Menger's paper [START_REF] Menger | Untersuchungen über allgemeine Metrik[END_REF], one may prove that any pair of distinct points X and Y in a line space belongs to a unique geodesic that is isometric to R. He attributes this fact to Biedermann. Busemann also notes that in a line space, Hilbert's linear axioms are satisfied:8 I, 1,2,3 (first half); II, 1,2,3, III, 1,2,3 and V.1.

In his later works, Busemann calls such a space a straight line space (e.g. in [START_REF] Busemann | Metric Methods in Finsler Spaces and in the Foundations of Geometry[END_REF]) or simply a straight space (e.g. in [START_REF] Busemann | The geometry of geodesics[END_REF]).

The paper on Pasch's axiom and 2-dimensionality

In the short paper [START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF], Busemann recalls Pasch's axiom 9 which states that for any three non-collinear points A, B, C in a plane, if a line meets the segment AB, then it meets the union of the segments AC and CB. He proves that under some topological conditions, Pasch's axiom is equivalent to the fact that the space is homeomorphic to the plane. More precisely, Busemann shows that if the lines (in the sense of axiomatic geometry) are the geodesics for a metric satisfying the axioms of a line space as defined in § 2, then the fact that Pasch's axiom holds is equivalent to the fact that the space is homeomorphic to the plane. The proof is sketched in [START_REF] Troyanov | On Pasch's Axiom and Desargues' Theorem in Busemann's work[END_REF].

Note that in a line space, one can easily define the congruence of segments by stating that AB ≡ A ′ B ′ if ρ(A, B) = ρ(A ′ , B ′ ), and this congruence satisfies Hilbert's axioms (III.1) to (III.3).

In the last part of the paper, Busemann proposes the following notion of angle congruence in a 2-dimensional line space. Let g and h be two rays with common origin O and g ′ , h ′ two other rays with common origin O ′ . Consider the points G and H on g and h respectively satisfying ρ(O, G) = ρ(O, H) = 1 and similarly the point

G ′ and H ′ on g ′ ,h ′ such that ρ(O ′ , G ′ ) = ρ(O ′ , H ′ ) = 1.
Then the angles (g, h) and (g ′ , h ′ ) are said to be congruent if ρ(G, H) = ρ(G ′ , H ′ ). This angle congruence is clearly an equivalence relation and Hilbert's axiom (III.4) is satisfied.

In general however, Axiom (III.5) is not satisfied and therefore the first theorem on triangle congruence in Hilbert's Foundations fails for a general line space. We recall that this theorem states the following (sometimes called the SAS -for sideangle-side -triangle congruence criterion).

If ABC and

A ′ B ′ C ′ are two triangles such that A ′ B ′ ≡ AB, A ′ C ′ ≡ AC and B ′ A ′ C ′ ≡ BAC, then triangles ABC and A ′ B ′ C ′ are congruent, in particular we also have B ′ C ′ ≡ BC.
Busemann concludes his paper by stating that it is well known that one cannot say more in general, even with a quadratic metric. Busemann probably means here that in a two-dimensional Riemannian manifold (say complete, simply connected and without conjugate points), although the notion of angle is well founded, the theorem on triangles congruence generally fails.

Geometries where Circles with Infinite Radius are the Shortest Curves

Let us now discuss the content of the paper On the Geometries where Circles with Infinite Radius are the Shortest Curves [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF]. Busemann starts with the following definition: Definition 4.1. A line space satisfies the limit circle axiom if in addition to (i)-(vi) above, it also satisfies the following condition: vii. For any pair of distinct points X, Y and any sequence {P n } such that ρ(P n , X) → ∞ and (ρ(P n , X)ρ(P n , Y )) → 0 we also have (ρ(P n , X)ρ(P n , U )) → 0 and (ρ(P n , X) -

ρ(P n , V )) → 0,
where U is the midpoint of X and Y and V is the doubling point, that is Y is the midpoint of X and V .

Remaks. 1.) Condition vii is equivalent to the requirement that (ρ(P n , X)ρ(P n , Z)) → 0 for any point Z on the line through X and Y . In Euclidean geometry, this holds if the sequence P n diverges to infinity on a line othogonal to the line through X and Y .

2.) This condition also implies that the horospheres in our line space are totally geodesic.

The goal of the paper is to prove the following Theorem 1. A 3-dimensional line space satisfying the limit circle axiom is isometric to a Minkowski space. Furthermore, the metric balls are strictly convex and the spheres have a unique tangent plane at any point.

Busemann also proves the converse: every Minkowski space with the above properties is a line space satisfying the limit circle axiom.

Let us describe in some detail the content of the paper. The article contains 5 sections comprising a total of 41 facts which are either technical lemmas or propositions having some interest in themselves.

In Section 1, Busemann describes the segments and lines in a line space and spells their basic properties. The results can be summarized by saying that the usual incidence axioms for lines such as those listed in Hilbert's Grundlage der Geometrie are satisfied.

Let us note Fact 3, which states that the join of a point P and a segment AB is homeomorphic to a triangle in R 2 if A, B and P are not on a line.

In Section 2, Busemann introduces the notion of limit sphere in a general line space. By definition, a limit sphere is the limit of a sequence of spheres through a point P with centers Q n lying on a fixed ray and such that ρ(P, Q n ) → ∞. The limit spheres are also called horoshperes. 10 Note that Busemann introduced the notion of horosphere in a general metric space, as a level set of a Busemann functions in [10, p. 102 and p.132], see also [START_REF] Papadopoulos | Metric spaces, convexity and nonpositive curvature[END_REF]. Recall that for any geodesic ray γ, the associated Busemann function β γ is defined as

β γ (X) = lim t→∞ (ρ(X, γ(t)) -t).
The equivalence between the notions of horosphere and level set of some Busemann function follows from Fact 7.

In Section 2 of the paper [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF], Busemann establishes the following properties of horospheres:

i.) Two horospheres associated to the same ray are equidistant. 10 It is also worth recalling that the notion of limit sphere was introduced by Lobachevsky, in the setting of his work on hyperbolic geometry. We refer the reader to the English edition of the Pangeometry [START_REF] Lobachevsky | Pangéométrie ou précis de géométrie fondée sur une théorie générale et rigoureuse des parallèles[END_REF] (p. 8). We also note that Lobachevsky, in some other memoirs, used the word horisphere (see e.g. [START_REF] Lobachevsky | Geometrische Untersuchungen zur theorie der Parallellinien (Geometrical researches on the theory of parallels)[END_REF]), which was later on transformed into horosphere. Lobachevsky proved that in hyperbolic space, horospheres are equipped with a Euclidean structure. His proof is based on the fact that Euclid's axioms are satisfied on horospheres, for a appropriate notion of line, angle, etc.

ii.) Horospheres are convex in the sense that every point not on a horosphere has a unique nearest point on that horosphere. iii.) Horospheres in an n-dimensional line space are locally homeomorphic to standard spheres; therefore their topological dimension is n -1.

In Section 3, Busemann considers horospheres in a line space satisfying the limit circle axiom. He first observes that the horospheres in such a space are flat subspaces. In particular they are themselves line spaces (of dimension one less). Furthermore, any horosphere separates the ambient space into two convex regions.

Busemann then specializes his investigation to dimension 3. He calls plane a horosphere in a 3-dimensional line space satisfying the limit circle axiom. Any ray defining the horosphere is said to represent the normal direction of that plane.

Busemann then shows that there is a unique plane through three non aligned points (Facts 17 and 25). He also proves that given a point P not on a plane there is a unique footpoint (nearest point) for P on the plane.

Note that this property implies that at any point on a sphere, there is a unique plane tangent to that sphere (where by "tangent" one means that the intersection of the plane and the sphere is a singelton).

In Section 4, Busemann discusses the geometry of 2-dimensional line spaces satisfying the limit circle axiom. He first observes that the horospheres in such a space are exactly the lines. In the rest of the section he shows that such a space is homeomorphic to R 2 and Pash's axiom is satisfied. The arguments are similar to those in [START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF], see Section 3. It is also easy to show that in such a space the parallel postulate (as formulated in Hilbert's Axiom IV) holds.

In Section 5, Busemann recapitulates the previous discussion in the following claim: For a 3-dimensional line space satisfying the limit circle axiom, all the projective axioms of a 3-space have been verified. This means that all Hilbert's axioms of incidence and order, that is, groups (I) and (II) are satisfied. This also implies that Desargues' theorem holds in a suitable sense. Since the parallel postulate also holds, X can be mapped homeomorphically to R 3 in such a way that the geodesics in this plane are mapped to straight lines in R 3 , see [START_REF] Troyanov | On Pasch's Axiom and Desargues' Theorem in Busemann's work[END_REF] for a discussion of Desargues' theorem and its relation with such a mapping.

Busemann then observes that the spheres in X are strictly convex and at each point they admit a unique tangent plane. He then shows (Fact 39) that any pair of spheres in X are related by a similarity transformation, and from this he concludes that the line space is a Minkowski space (Fact 39). This completes the proof of the main Theorem.

Busemann then proceeds to prove the converse, namely that Minkowski metrics on R 3 with strictly convex spheres and with a unique tangent plane at each point are line spaces satisfying the limit circle axiom (Fact 41). He concludes the paper by recalling that it is then sufficient to add a single axiom to conclude that the line space is Euclidean. He states the additional axiom as a condition involving a configuration of 6 points proposed by O. Veblen. Note that many other equivalent conditions were proposed by a number of mathematicians. The most elegant and famous among them is probably the parallelogram law published in 1935 by P. Jordan and J. von Neumann in [START_REF] Jordan | On Inner Product in Linear Metric Spaces[END_REF].

Spaces with Convex Spheres and the Parallel Postulate

In the fifth postulate of Euclid's Elements and in Hilbert's axiom (IV), the notion of parallelism between two lines refers to the fact that the two lines are coplanar and disjoint. In the paper On spaces with convex spheres and the parallel postulate [START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF], Busemann introduces a different notion of parallelism, which he will also adopt (with a small variation) in his later work, in particular, in his book The Geometry of geodesics. The definition is the following: Definition 5.1 (Space satisfying the parallel postulate). A line space is said to satisfy the parallel postulate if whenever g is a straight line, P a point not on g, and X n a sequence of points on g converging to infinity, the straight lines P X n always converge to the same straight line (depending only on g and P .) Using this notion, Busemann gives a characterization of Minkowski spaces that uses spheres, rather than horospheres as in the previous paper. More precisely, he first obtains the following result, valid in dimension ≥ 3: Theorem 2. If in a line space of dimension ≥ 3 the distance spheres are convex and differentiable and the parallel axiom holds, then the space can be mapped topologically onto a Euclidean space in such a way that the geodesics are mapped to straight lines and the metric is Minkowskian.

The differentiability of a sphere, in Busemann's metric setting, means the existence and uniqueness of "as few tangent lines as possible" at each point of the sphere. This is made precise in §3 of his paper.

The case of dimension 2 requires different hypotheses. In Busemann's terminology, in a line space, the distance function is said to be convex if the distance function from an arbitrary fixed point to the points moving on an arbitrary line is convex. In the same manner, by taking the α-th power of this function, one defines the notion of a line space in which the α-th power of the distance function is convex.

In the case of dimension two, Busemann obtains the following.

Theorem 3. In a two-dimensional line space in which the parallel postulate holds, if for some α ≥ 1 the α-th power of the distance function is convex and differentiable then the same conclusion as in Theorem 2 holds.

The outline of the paper is the following. The paper is divided into 5 sections. Section 1 concerns the notion of parallelism in line spaces. Busemann introduces the notion of asymptotic lines, and he makes a thorough investigation of this notion, making the relations with limit spheres and their orthogonal trajectories. Busemann refers to the book [START_REF] Cartan | Leçons sur la géométrie des espaces de Riemann[END_REF] by É. Cartan in which the latter studied such a relation in Riemannian spaces of nonpositive curvature. In fact, the notion of asymptoticity of lines, in the setting of surfaces embedded in 3-space, was already studied by Hadamard in his paper [START_REF] Hadamard | Les surfaces à courbures opposées et leurs lignes géodésiques[END_REF]. The study made by Busemann in his paper is a generalization of those made by Hadamard and Cartan to this setting of abstract line spaces. Using this notion of parallelism, Busemann states his parallel postulate in the setting of line spaces. The definition involves 5 statements concerning asymptotic lines. Dimension 2 requires a special study, and for this case Busemann uses the result of his paper [START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF].

Section 2 concerns the convexity of spheres and its consequences. Busemann makes the relation of this notion with the existence of midpoints, and he notes that this was already noticed by Minkowski in his Geometrie der Zahlen [START_REF] Minkowski | Geometrie dre Zahlen[END_REF]. Busemann refers again to Cartan's book [START_REF] Cartan | Leçons sur la géométrie des espaces de Riemann[END_REF] for the case of Riemannian manifolds. In this general setting, Busemann gives a formulation using the convexity of the distance function and the uniqueness of foots of perpendiculars. A thorough investigation of the notion of perpendicularity in this non-Riemannian setting is needed. In particular, the notion of perpendicularity is extended to a pair of lines which do not meet: a line g is said to be perpendicular to a line h if there is a parallel to g which is perpendicular to h. Again, in the case of dimension two, a special study is made. All these ideas will play an essential role in the later work of Busemann.

Section 3 concerns the characterization of Minkowski spaces of dimension > 2. In this section, Busemann defines the notion of differentiablity of spheres in this metric setting. This involves the introduction of metric notions of hypersurfaces, midsurfaces and hyperspheres, the use of the parallel postulate, and of several topological lemmas.

In Section 4, Busemann studies the question of embedding a 2-dimensional line space into a 3-dimensional one. He recalls that Hilbert, in his Grundlagen, already considered the question of the embedding of a plane satisfying all the axioms of order and intersections in a three-dimensional projective space. He showed that the embeddability is equivalent to the fact that Desargues property is satisfied. This is one of the starting points for Busemann's work on what he called Desarguesian spaces. In the paper under review, Busemann adresses a similar question. The space is equipped with a metric and with a certain system of curves, called lines. Busemann proves that if in this space Desargues theorem holds, then the twodimensional space can be embedded in a three-dimensional space such that these lines are sent to lines of the projective space. The proof uses a construction by Hessenberg [START_REF] Hessenberg | Grundlagen der Geometrie[END_REF].

Section 5 concerns the characterization of 2-dimensional Minkowski spaces. The embedding of §4 is used. The convexity property of the α-th power of the distance function to which we referred in Theorem 3 above is equivalent to the fact that in that space, the spheres are convex. Busemann uses this fact to show the main theorem in dimension two, namely, if in a 2-dimensional line space the parallel postulate holds and if there exists α > 1 such that the α-th power of all distance functions are convex and differentiable, then the metric is Minkowskian.

The paper ends with a discussion of the convexity of ellipses and hyperbolas, with their natural definition in a general 2-dimensional line space. This is naturally motivated by the result on the convexity of spheres. Busemann shows in particular that if in such a space the branches of any hyperbola are convex, then for any two points, the perpendicular bisectors exist and are lines. Then he proves that if in any two-dimensional line space the branches of any hyperbola are convex curves, then the metric is either Euclidean of hyperbolic.

Finally, let us note that some of the results of the three papers [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF][START_REF] Busemann | Paschsches Axiom und Zweidimensionalität[END_REF][START_REF] Busemann | Über Räume mit konvexen Kugeln und Parallelenaxiom[END_REF] were also reproved in the book [START_REF] Busemann | The geometry of geodesics[END_REF] Appendix: Hilbert's axioms

In this section, we reproduce, for the readers' convenience, Hilbert's axioms from his Foundation of Geometry. We are following the second English edition [START_REF] Hilbert | Foundations of geometry[END_REF], which is based on the tenth German edition. This edition differs marginally from the seventh edition, to which Busemann refers. Besides the axioms themselves, Hilbert's treatise also contains the consequences of the axioms as well as numerous comments enriching the axiomatic system.

The goal of this system of axioms is to lay down foundations for Euclidean geometry in 3-space. Three undefined notions (objects) are assumed: points (denoted by capital Latin letters), lines (denoted by small Latin letters) and planes(denoted by small Greek letters). Some undefined relations are also assumed; these are the incidence relations (a point is on a line, a line is contained in a plane etc.), the order relation (a point lies between two others) and the congruence relation formalizing the idea that two segments or two angles can be superposed.

The axioms are divided into five groups describing the basic properties of incidence, order and congruence. The parallel postulate and a continuity (or completeness) axiom are also assumed. IV. The parallel postulate (IV.) Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a. V. Axioms of continuity (V.1) Archimedes' Axiom: If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B. (V.2) Completeness' Axiom: An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and from V.1 is impossible.

On the triangle inequality. Before ending this paper, we aim at clarifying a point. In his comment to his IVth problem, Hilbert suggests to consider the triangle inequality as an Axiom and not as a Theorem (see Hilbert's quotation on page 3 in the present paper). However the triangle inequality is not explicitly stated as a theorem in Hilbert's foundations.

To understand in what sense Hilbert considers the triangle inequality as a theorem, let us recall that in Euclid's Elements, Proposition 20 of Book 1 says that In any triangle the sum of any two sides is greater than the remaining one. The proof derives from Proposition 5: On an isosceles triangle the angles at the base are equal to one another and Proposition 19: In any triangle the greater angle is subtended by the greater side.

In his Foundations, more precisely, in the comments following the congruence axioms, Hilbert discusses the notions of angle and segment comparison. Axiom (III.3) allows also an obvious definition of segment addition. With this material, Hilbert proves the equality of the base angles in an isosceles triangle as Theorem 11 and he proves that in every triangle the greater angle lies opposite to the greater side as Theorem 23. One may then prove the triangle inequality exactly as in Euclid. This justifies Hilbert's claim that it is a proposition that in every triangle the sum of two sides is greater than the third and gives a meaning to his suggestion to assume this fact as an axiom instead.

Let us end with two short comments. First we stress that in the above proof the angle congruence Axiom plays an important role. Thus the triangle inequality in classical Euclidean geometry appears as a consequence of some properties of angles. Finally let us mention that although Hilbert did not explicitly state the triangle inequality in the Foundations, he wrote a paper in 1902 about the role of the theorem on the base angles in an isosceles triangle [START_REF] Hilbert | Ueber den Satz von der Gleichheit der Basiswinkel[END_REF]. This paper is reproduced in appendix II of [START_REF] Hilbert | Foundations of geometry[END_REF]. There, the triangle inequality is explicitly stated as a proposition.

I.

  Axioms of incidence (I.1) For every two points A, B there exists a line that contains each of the points A, B. (I.2) For any two distinct points A, B there exists no more than one line that contains each of the points A, B. (I.3) There exist at least two points on a line. There exist at least three points that do not lie on a line. (I.4) For any three points A, B, C that do not lie on the same line there exits a plane α that contains each of the points A, B, C. For every plane there exists a point which it contains. (I.5) For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C. (I.6) If two points A, B of a line a lie in a plane α, then every point of a lies in α. (I.7) If two planes α, β have a point A in common, then they have at least one more point B in common. (I.8) There exist at least four points not lying in a plane. II. Axioms of order (II.1) If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line, and B lies also between C and A. (II.2) For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B. (II.3) Of any three points on a line, there exists no more than one that lies between the other two. (II.4) Pasch's Axiom: Let A, B, C be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through either a point of the segment BC or a point of the segment AC. III. Axioms of congruence (III.1) If A, B are two points on a line a, and if A ′ is a point on the same or another line a ′ , then it is possible to find a point B ′ on a given side of the line a ′ so that the segment AB is congruent to the segment A ′ B ′ . We write AB ≡ A ′ B ′ . (III.2) If two segments are congruent to a third one they are congruent to each other. (III.3) On the line a let AB and BC be two segments which except for B have no point in common. Furthermore, on the same or another line a ′ let A ′ B ′ and B ′ C ′ be two segments having no point other than B ′ in common. In that case if AB ≡ A ′ B ′ and BC ≡ B ′ C ′ , we have AC ≡ A ′ C ′ . The next two axioms concern the congruence of planar angles and are of little interest in Minkowski geometry. (III.4) 11 Given an angle in a plane α and a half-line in another plane β there exists exactly two half-lines in β generating angles congruent with the given angle. (III.5) If ABC and A ′ B ′ C ′ are two triangles such that A ′ B ′ ≡ AB, A ′ C ′ ≡ AC and B ′ A ′ C ′ ≡ BAC, then we also have ABC ≡ A ′ B ′ C ′ and ACB ≡ A ′ C ′ B ′ . 11 We have summarized Axiom III.4 -which is long in Hilbert's version -while keeping the meaning.

The numbering here refers to that of Hilbert's axioms in the appendix of this article.

It may be useful to recall that Pasch's axiom was known and used by Euclid's commentators long before Pasch; see e.g.[START_REF] Rashed | Thābit et la théorie des parallèles[END_REF].

Acknowledgement The authors would like to thank Valerii Berestovskii for reading this paper and sending corrections. The first author acknowledges support from the ANR program FINSLER.


S Joubaud

N B Garnier

F Douarche

A Petrosyan

S Ciliberto

Experimental study of work fluctuations in a harmonic oscillator

Keywords: fluctuation theorems, work, out of equilibrium théorèmes de fluctuation, travail, hors-équilibre

The work fluctuations of a harmonic oscillator in contact with a thermostat and driven out of equilibrium by an external force are studied experimentally. For the work both the transient and stationary state fluctuation theorems hold. The finite time corrections are very different from those of a first order Langevin equation. The heat and work fluctuations are studied when a periodic forcing is applied to the oscillator. The importance of the choice of the "good work" to compute the free energy from the Jarzinsky equality is discussed.

Résumé

Etude expérimentale des fluctuations du travail dans un oscillateur harmonique. Nous étudions expérimentalement les fluctuations du travail d'une force externe sur un oscillateur harmonique en contact avec un bain thermique. Nous trouvons que les théorèmes de fluctuations pour les états transitoires et stationnaires sont vérifiés. Toutefois les corrections de temps fini pour l'état stationnaire sont très différentes de celles calculées dans le cadre de l'équation de Langevin du premier ordre. Nous étudions aussi le forçage périodique de l'oscillateur. Enfin nous discutons l'importance du choix du "bon travail" pour calculer l'énergie libre en partant de l'égalité de Jarzinsky.

Introduction

Thermal fluctuations play a very important role in the out of equilibrium dynamics of small systems such those that one can find in nanotechnologies and biophysics. Indeed in these systems the amount of injected energy is often comparable to that of thermal fluctuations, which may produce unwanted and unexpected behaviors. For example very large fluctuations may force the system to move in the opposite direction of the one imposed by the external forces. In the same way the heat exchanges with the thermal bath are such that the heat may instantaneously flow from a cold source to a hot one. Of course these are not violations of the second principle of thermodynamics which fixes only the mean values and it does not make any statement for fluctuations. The recent fluctuation theorems FTs [1]- [7] allow one to compute the probability of these negative events for the work and for the heat flux. It is interesting to notice that the fluctuations of the work done by the external Email address: sergio.ciliberto@ens-lyon.fr (S. Ciliberto). forces to drive the system between two equilibrium states A and B allows one to compute, in some cases, the free energy difference ∆F between A and B, using either the Jarzinsky equality (JE) [9,8] or the Crooks relation (CR) [10]. Indeed the JE and CR take advantage of these work fluctuations and relate ∆F to the probability distribution function (PDF) of the work performed on the system to drive it from A to B along any path γ(either reversible or irreversible) in the system parameter space. In this paper we will study the possibility of using these theorems on real experimental data. Indeed the proofs of FTs and JE are based on a certain number of hypothesis; experimenting on real devices is useful not only to check those hypothesis, but also to check whether the predicted effects are observable or remain only a theoretical tool. There are not many experimental tests of FTs. Some of them are performed in dynamical systems [11] in which the interpretation of the results is very difficult. Other experiments are performed on stochastic systems, one on a Brownian particle in a moving optical trap [12] and another on electrical circuits driven out of equilibrium by injecting in it a small current [13].

Preprint submitted to Elsevier Science 27th March 2007

The last two systems are described by first order Langevin equations and the results agree with the predictions of ref. [7,6].In this paper we study the out-of-equilibrium fluctuations of a harmonic oscillator whose damping is mainly produced by the viscosity of the surrounding fluid, which acts as a thermal bath of temperature T . The test using a harmonic oscillator is particularly important because the harmonic oscillator is the basis of many physical processes [14,15]. The paper is organized as follow. In the next section we will briefly describe the experimental set-up. The fluctuation theorem will be studied is section three. In section 4 we will describe the results for the work and heat for a the stationary state fluctuation theorem when the oscillator is submitted to a periodic driving.

In section 5 the JE and Crooks are used to compute the free energy. The importance of the choice of the "good work" for the JE is pointed out too. We show indeed that a definition of work which satisfies FT cannot be used to compute the free energy with JE. Finally we conclude in section 6.

Experimental set-up

We recall here only the main features of the experimental set-up, more details can be found in ref. [16,18]. The oscillator is a torsion pendulum composed by a brass wire and a glass mirror glued in the middle of this wire. It is enclosed in a cell filled by a water-glycerol solution at 60% concentration. The motion of this pendulum can be described by a second order Langevin equation:

I eff d 2 θ dt 2 + ν dθ dt + C θ = M + √ 2kBT ν η, (1)
where θ is the angular displacement of the pendulum, I eff is the total moment of inertia of the displaced masses, ν is the oscillator viscous damping, C is the elastic torsional stiffness of the wire, M is the external torque, kB the Boltzmann constant and η the noise, delta-correlated in time. In our system the measured parameters are the stiffness C = 4.5 × 10 -4 N m rad -1 , the resonant frequency fo = C/I eff /(2π) = 217Hz and the relaxation time τα = 2I eff /ν = 9.5ms. The external torque M is applied by means of a tiny electric current J flowing in a coil glued behind the mirror. The coil is inside a static magnetic field, hence M ∝ J. The measurement of θ is performed by a differential interferometer, which uses two laser beams impinging on the pendulum mirror [16,18]. The measurement noise is two orders of magnitude smaller than the thermal fluctuations of the pendulum. θ(t) is acquired with a resolution of 24 bits at a sampling rate of 8192Hz, which is about 40 times fo. The calibration accuracy of the apparatus, tested at M = 0 using the the Fluctuation Dissipation Theorem, is better than 3%(see [18]).

Fluctuation theorems

It is important to make a distinction between Stationary State Fluctuation Theorem (SSFT) and and Transient Fluctuation Theorem (TFT) (as defined in ref. [7,6]). TFT implies that the initial state when the force is applied to the system is an equilibrium one. In contrast SSFT concerns out of equilibrium stationary states where all transients have already relaxed. To study SSFT and TFT we apply to the oscillator a time dependent torque M (t) as depicted in Fig. 1a, and we consider the work Wτ done by M (t) over a time τ :

Wτ = 1 kB T t i +τ t i [M (t) -M (ti)] dθ dt dt . (2)
By definition, the TFT implies that ti = 0 in eq.2 whereas ti ≥ 3τα for SSFT. As a second choice for M (t), the linear ramp with a rising time τr is replaced by a sinusoidal forcing; this leads to a new form of stationary state which has never been considered in the context of FT. We examine first the linear forcing M (t) = Mot/τr (Fig. 1a)), with Mo = 10.4 pN.m and τr = 0.1 s = 10.7 τα. The response of the oscillator to this excitation is comparable to the thermal noise amplitude, as can be seen in Fig. 1b) where θ(t) is plotted during the same time interval of Fig. 1a). Because of thermal noise the power Wτ injected into the system (eq.2) is itself a strongly fluctuating quantity. The FTs state [7] that the probability density functions (PDF) P (Wτ ) of Wτ satisfies:

ln P (Wτ ) P (-Wτ ) = Σ(τ ) Wτ (3)
where for TFT

Σ(τ ) = 1, ∀τ (4)
and for SSFT

Σ(τ ) → 1 for τ → ∞, (5)
(see refs. [6,7] for a clear discussion on the difference between TFT and SSFT). [15,19]) with no adjustable parameter).

Transient fluctuation theorem

We consider first the TFT. The probability density functions P (Wτ ) of Wτ are plotted in Fig. 2a) for different values of τ . We see that the PDFs are Gaussian for all τ and the mean value of Wτ is a few kBT . We also notice that the probability of having negative values of W is rather high for the small τ . The function

S(Wτ ) ≡ ln P (Wτ ) P (-Wτ ) (6)
is plotted in fig. 2b). It is a linear function of Wτ for any τ , that is S(Wτ ) = Σ(τ ) Wτ . Within experimental error, we measure the slope Σ(τ ) = 1. Thus for our harmonic oscillator the TFT is verified for any time τ . This was expected [2,6], and a derivation of this generic result for a second order Langevin dynamics is given in ref. [15,19].

Stationary state fluctuation theorem

We now consider the SSFT with ti ≥ 3τα in eq.2. We find that the PDFs of Wτ , plotted in Fig. 3a), are Gaussian with many negative values of Wτ for short τ . The function S(Wτ ), plotted in Fig. 3b), is still a linear function of Wτ , but, in contrast to TFT, the slope Σ(τ ) depends on τ . In Fig. 3c) the measured values of Σ(τ ) are plotted as a function of τ . The function Σ(τ ) → 1 for τ ≫ τα. Thus SSFT is verified only for large τ . The finite time corrections of SSFT, which present oscillations whose frequency is close to f0, agree quite well with the theoretical prediction computed for a second order Langevin dynamics [15]. We stress that the finite time correction is in this case very different from that computed in ref. [6,7] for the first order Langevin equation. The results of Figs. 2,3 have been checked for several Mo/τr without noticing any difference. The errors bars in the figures are within the size of the symbols, and they come only from the calibration errors of the harmonic oscillator parameters, and statistics of realizations (typically 5 × 10 5 cycles have been used).

Periodic forcing

We describe in this section the results of the periodic forcing. In this case M (t) = Mo sin ω d t and the work expression (eq.2) is replaced by

Wn = Wτ=τ n = 1 kB T t i +τ t i M (t) dθ dt dt , (7)
where τ is a multiple of the period of the driving (τ = 2nπ/ω d with n integer). The starting phase tiω d is averaged over all possible ti to increase statistics. This periodic forcing is a stationary state that has never been studied in the details in the context of FT. As already done for the first order Langevin equation [6,7,17], we want to consider here also the heat fluctuations, that is the energy dissipated by the oscillator towards the heat bath. Multiplying eq. ( 1) by dθ dt and then integrating between ti and ti + τ , we obtain exactly the first law of thermodynamics. The change in the internal energy ∆Uτ over a time τ is:

∆Uτ = U (ti + τ ) -U (ti) = Qτ + Wτ ( 8
)
where Qτ is the heat given to the system. Equivalently, (-Qτ ) is the heat dissipated by the system. For a harmonic oscillator described by a second order Langevin equation (eq.1) the internal energy U (t) and the heat Qτ have the following expressions:

U (t) = 1 kB T 1 2 I eff dθ(t) dt 2 + 1 2 Cθ(t) 2 (9)
Qτ = ∆Uτ -Wτ = - 1 kB T t i +τ t i ν dθ dt (t ′ ) 2 dt ′ + 1 kB T t i +τ t i η(t ′ ) dθ dt (t ′ )dt ′ . ( 10
)
The first term in eq.10 corresponds to the viscous dissipation and is always positive, whereas the second term can be interpreted as the work of the thermal noise which have a fluctuating sign. We rescale the work Wτ (the heat Qτ ) by the average work Wτ (the average heat Qτ ) and define: wτ = Wτ Wτ (qτ = Qτ Qτ ). Averages are time-averages, and they are proportional to τ on the stationary state under consideration.

As an example we consider a measurement performed at Mo = 0.78pN m and at ω d /(2π) = 64Hz. The PDFs of wτ , ∆Uτ and qτ are plotted in Fig. 4 for different values of n. The average of ∆Uτ is clearly zero because the time τ is a multiple of the period of the forcing. The PDFs of the work (fig. 4a) are Gaussian for any n whereas the PDFs of heat fluctuations qτ have exponential tails(fig. 4c). These exponential PDFs can be understood taking into account that, from eq. ( 10), -Qτ = Wτ -∆Uτ and that ∆Uτ has an exponential PDF independant of n (fig. 4b). In the case of the periodic forcing the PDF of qτ can be exactly computed (see ref. [19]):

P (Qτ ) = 1 4 exp σ 2 W 2 e Qτ -Qτ erfc Qτ -Qτ + σ 2 W 2σ 2 W + e -(Qτ -Qτ ) erfc -Qτ + Qτ + σ 2 W 2σ 2 W , (11)
where erfc stands for the complementary Erf function and σ 2 W is the work variance. In Fig. 4c, we have plotted the analytical PDF from eq. ( 11) together with the experimental ones, using the values of σ 2 W and Qτ from the experiment. The agreement is perfect for all values of n and with no adjustable parameter, using eq. (11).

To quantify the symmetry of the PDF around the origin, we define the function S as:

S(eτ ) ≡ 1 Eτ ln P (eτ ) P (-eτ ) (12)
( f ) where eτ stands for either wτ or qτ and Eτ stands for either Wτ or Qτ . The question we ask is whether:

lim τ →∞ S(eτ ) = eτ (13)
as required by SSFT. S(qτ ) is plotted in Fig. 4d) for different values of n ; three regions appear: (i) For large fluctuations qτ , S(qτ ) equals 2. When τ tends to infinity, this region spans from qτ = 3 to infinity.

(ii) For small fluctuations qτ , S(qτ ) is a linear function of qτ . We then define Σq(n) as the slope of the function S(qτ ), i.e. S(qτ ) = Σq(n) qτ . This slope is plotted in Fig. 4e where we see that it tends to 1 when τ is increased. So, SSFT holds in this region II which spans from qτ = 0 up to qτ = 1 for large τ . (iii) A smooth connection between the two behaviors. The PDF of the work being Gaussian, the functions S(wτ ) are proportional to wτ for any τ , i.e. S(wτ ) = Σw(n) wτ (ref. [15]). Σw(n) is plotted in Fig. 4e) and we observe that it matches experimentally Σq(n), for all values of n. So the finite time corrections to the FT for the heat are the same than the ones of FT for work [15]:

Σw(n) = Σq(n) = 1 + K/n + 1/n O e -τn/τα , where K is a constant.
We find that that the scenario plotted in fig. 4a)-e) is the same for any driving frequency ω d [15,19]. However the finite time corrections of Σw(n) are a function of ω d . The function Σw(n), measured at ω d /2π = 64Hz, is compared with that measured at ω d /2π = 256Hz in in fig. 4f). The convergence rate is quite different in the two cases, which agree with the theoretical predictions for a second order Langevin equation (see ref. [15,19]).

We also notice that in the case of the sinusoidal forcing, the convergence is very slow in Figs. 4e) and 4f), we see that it takes several dozens of excitation period (500 ms for ω d /2π = 64Hz) to get Σ(n) = 1 within one percent. On the contrary for a ramp forcing, this is achieved in about (20 ms), i.e. a few τα (see Fig. 3c).

In this section we have seen that also in the case of the sinusoidal forcing the agreement between the computed and measured finite time corrections is very good. These results prove not only that FTs asymptotically hold for any kind of forcing, but also that finite time corrections strongly depend on the specific dynamics. The detailed theoretical analysis of this behavior for a second order Langevin equation can be found in ref. [19].

TFT and the Jarzynski equality the Crooks relation

In this section we want to discuss the relationship between TFT and JE. The Jarzynski equality (JE) relates the free energy difference ∆F between two equilibrium states A and B of a system in contact with a heat reservoir to the PDF of the work performed on the system to drive it from A to B along any path γ in the system parameter space. Actually thermal fluctuations are here very useful because they allow the system to explore all possible paths from A to B. Specifically, let us consider the work done by the external torque on the harmonic oscillator described in section 3. The external torque M (t) drives the system from M (0) = 0 (equilibrium state A) to M (t > τr) = Mo (equilibrium state B). Jarzynski defines the work performed by M (t) to drive the system from A to B as

W J = - τr 0 Ṁ θ dt = -M θ τr 0 -W cl , (14)
where

W cl = - τr 0 M θ dt = -Wτ r (15)
is the classical work such that -W cl /(kBT ) is equal to Wτ r defined in eq.2. Here we define W cl such that it has the same sign of W J . Thus W J and W cl are related but they are not the same and they differ by a boundary therm, which makes an important difference in the fluctuations of these two quantities. This difference between the W and W cl , has been already pointed out in ref. [20]. One can consider an ensemble of realizations of the "switching process" from A to B with initial conditions all starting in the same initial equilibrium state. Then the work may be computed for each trajectory in the ensemble. The JE states that [9] ∆F = - where • denotes the ensemble average, β -1 = kBT . In other words exp [-βW diss ] = 1, since we can always write W = ∆F + W diss where W diss is the dissipated work. In our experiment we can also check the CR which is related to the JE and which gives useful and complementary information on the dissipated work. Crooks considers the forward work W f to drive the system from A to B and the backward work W b to drive it from B to A. If the work pdfs during the forward and backward processes are P f (W J ) and P b (-W J ), one has [10,8]

1 β log exp [-βW J ] , (16) (a)
P f (W J ) P b (-W J ) = exp (β[W J -∆F ]) = exp [βW diss ]. (17)
A simple calculation from Eq. ( 17) leads to Eq. ( 16). However, from an experimental point of view this relation is extremely useful because one immediately sees that the crossing point of the two pdfs, that is the point where P f (W J ) = P b (-W J ), is precisely ∆F . Thus one has another mean to check the computed free energy by looking at the pdfs crossing point W×.

Many details of the test of the JE and CR in the harmonic oscillator can be found in ref. [18]. Here we want only to stress the difference between the fluctuations of the classical and the Jarzinsky works. This can be seen in fig. 5 where we plot the the pdfs of W J (figs. b). The stiffness C of the oscillator was 7.5 10 -4 N m rad -1 in this experiment. In Figs. 5, bullets are the experimental data and the continuous lines their fitted Gaussian pdfs. Comparing the pdfs of W J and W cl for the same Mo we see that the variance of W cl is much larger than that of W J . Indeed W cl presents both positive and negative values and, as we have seen in sect.3, P (-W cl ) satisfies TFT. Thus it is clear that if W J satisfies eq.16 this equation cannot be satisfied by W cl . Indeed using the measured values of W J in eq.16 one finds β∆F = -23.3 ± 0.4 for Mo = 11.9 and β∆F = -6.3 ± 0.3 for Mo = 6.1. These values correspond to ∆F = -M 2 o 2C for the driven harmonic oscillator. In Fig. 5, the pdfs P f (W ) and P b (-W ) cross at βW ≃ -23.5 (fig. 5a) and in βW ≃ -6.1 (5c), which again correspond to the expected values. Thus we see that JE and CR can be safely applied on experimental data to measure ∆F if the work W J defined in eq.14 is used. In ref. [18] it has been shown that this result is true independently of the ratio τr/τα and of the maximum amplitude of Mo. This has been checked at the largest Mo and the shortest rising time τr allowed by our apparatus.

The fact that the JE and CR are satisfied only by W J but not by W cl can be easily understood by noticing that a variable that satisfies TFT cannot be used in eq.16. Indeed one may rewrite eq.16 as:

exp -β∆F = < exp(-β W ) >= ∞ -∞ exp(-β W )P ( W )d W ( 18
)
where W is an energy injected into the system on a time τ . If P ( W ) satisfies TFT then from eq.3 and eq.4 :

P ( W ) = exp(-β W )P (-W ), ∀τ

. Replacing this identity in eq.18 one finds: showing that if W satisfies TFT then cannot be used to compute ∆F . Thus one concludes that as W cl satisfies TFT it cannot satisfy the JE as it shown in eq.19. This observation points out the importance of the choice of the "good variables" in order to use the JE to compute the free energy difference in a more complex system. Several other limitations for a safe application of JE and CR on real data have been already discussed in ref. [18].

Conclusions

In conclusion we have applied the FTs and JE to the work fluctuations of an oscillator driven out of equilibrium by an external force. The experimental results agree with theoretical predictions both for TFT and SSFT (eqs.3,4 and 5). Indeed we observe that in the transient case when the system starts in equilibrium (TFT) the FT (eq.3) holds for any time, whereas when the system is in a stationary out of equilibrium regime (SSFT), eq.3 holds only asymptotically. The SFFT presents a complex convergence to the asymptotic behavior which strongly depends on the form of the driving. The exact formula of this convergence can be computed using several experimental evidences of the statistics of the fluctuation. The details of derivation of the finite time correction and of the PDF for the heat will be the subject of a long paper [19]. We have also discussed the importance of the choice of the variable in order to safely apply JE. The results reported in this paper are useful for many applications going from biological systems to nanotechnology, where the harmonic oscillator is the simplest building block.

Figure 1 .

 1 Figure 1. a) Typical driving torque applied to the oscillator; b) Response of the oscillator to the external torque (gray line). The dark line represents the mean response θ(t) to the applied torque M(t).

Figure 2 .

 2 Figure 2. TFT. a) P (Wτ ) for TFT for various τ /τα: 0.31 (•), 1.015 (2), 2.09 (⋄) and 4.97 (×). Continuous lines are Gaussian fits. b) TFT; S(Wτ ) computed with the PDF of a). The straight continuous line has slope 1, i.e., Σ(τ ) = 1, ∀τ .

Figure 3 .

 3 Figure 3. SSFT with a ramp forcing. a) PDF of Wτ for various τ /τα: 0.019 (•), 0.31 (2), 2.09 (⋄) and 4.97 (×). b) Corresponding functions S(Wτ ). c) The slope Σ(τ ) of S(Wτ ) is plotted versus τ (2: experimental values; continuous line: theoretical prediction from refs. [15,19]) with no adjustable parameter).

Figure 4 .

 4 Figure 4. Sinusoidal forcing. a) PDFs of wτ . b) PDFs of ∆Uτ . c) PDFs of qτ . d) Functions S(qτ ). In all plots, the integration time τ is a multiple of the period of forcing, τ = 2nπ/ω d , with n = 7 (•), n = 15 (2), n = 25 (⋄) and n = 50 (×). Continuous lines are theoretical predictions with no adjustable parameter. e) The slope Σq(n) of S(qτ ) for qτ < 1, plotted as a function of n (•).The slope Σw (n) of S(wτ ) plotted as a function of n(2). Continuous line is theoretical prediction. f) The slopes Σ(n), plotted as a function of n for two different driving frequencies ω d = 64 Hz (2) and 256 Hz (•); continuous lines are theoretical predictions from ref.[15,19] with no adjustable parameter.

Figure 5 .

 5 Figure 5. Mo = 11.9pN m (a) P f (W ) and P b (-W ), (b) P f (W cl ) and P b (-W cl ); Mo = 6.1pN m: (c) P f (W ) and P b (-W ), (d) P f (W cl ) and P b (-W cl ) (experimental forward and backward pdfs are represented by • and 2 respectively, whereas the continuous lines are Gaussian fits)

  5 a,c) and W cl (figs.5 b,d) for the forward and backward processes measured for two values of Mo specifically Mo = 11.9pN m (figs.5a,b) and Mo = 6.1pN m (figs.5a,

  β W ) P (-W ) exp(β W ) d W = 1 (19)

Acknowledgments This work has been partially supported by EEC contract DYGLAGEMM and ANR-05-BLAN-0105-01.