Group Theory in a Nutshell for Physicists

A. Zee Princeton University Press, 2016. $90.00 (632 pp.). ISBN 978-0-691-16269-0 M any books have been written about group theory's applications to physics. Some have an arid, mathematically rigorous style that often obscures physical insight. Other, less formal presentations usually cannot deliver the necessary knowhow for practical applications. In Group Theory in a Nutshell for Physicists, Anthony Zee, a physicist at the University of California, Santa Barbara, combines clarity of presentation with mathematical detail at a level of rigor acceptable to physicists. The result is a tour de force that guides readers through the universe of group theory and leads them to recent applications in particle physics, cosmology, and condensed matter.

The book is unique in its laid-back presentation. It is peppered with colorful stories about famous mathematicians and physicists and includes frequent interjections from fictitious characters. Particularly helpful are the mutterings of Dr. Feeling, who supplies intuitive understandings of formal definitions or theorems, and the observations by Confusio, who (not surprisingly) points out issues of possible confusion. The book is ideally suited to accompany a graduate course on symmetries in physics because of its pedagogical approach, the detail of its illustrative examples, and its many exercises. Readers need to be familiar with the basics of quantum mechanics, but little other advance knowledge is required since the 1 book starts with a brief review of linear algebra and a reminder of the properties of matrices.

After its mathematical refresher, the book turns to a detailed presentation of the representation theory of finite groups and the introduction of Lie groups. Early on, Zee introduces Lie algebras by way of three-dimensional rotations; the classification of those algebras by roots, weights, and Dynkin diagrams comes later. The book's mathematically detailed material is interspersed with group theoretical applications to physical systems. Given the author's distinguished career in particle physics, it is not surprising that most of the examples come from that field, but Zee occasionally ventures out to other areas with examples relevant to condensed-matter and atomic physics.

The book makes only a single mention of group theory applied to atomic nuclei (my field of expertise), and that appears as a footnote when Zee discusses the Elliott model. James Philip Elliott's application of SU( 3) is admittedly of less fundamental importance than the use of group theory in particle physics, but the mathematics behind Elliott's application is more sophisticated. After reading about how the finest minds in the particle-physics community struggled to get the eightfold way right, I can only admire Elliott's achievement even more, as he developed the SU(3) model of nuclei essentially by himself.

The book does not comprehensively discuss the representation theory of the symmetric group of permutations, and the author even advises readers to stay clear of the diagrammatic machinery of Young tableaux. That may be sensible advice when one is dealing with low-dimensional representations. However, as the dimension of the representation increases, as is the case, for example, in quantum many-body physics, Zee's treatment in terms of either totally symmetric or totally antisymmetric tensors rapidly becomes cumbersome, and Young tableaux are called for.

Eugene Wigner, who introduced group theory into quantum mechanics and is therefore one of the heroes of the book, famously wrote about the "unreasonable effectiveness" of mathematics. In the final chapters of his text, Zee forcefully makes the case for the unreasonable effectiveness of group theory and buttresses his case with many compelling examples. Group theory can generate everything from the Dirac equation for the electron to the equations that describe the expanding universe. Indeed, all known particles can be unified within the framework of the Lie group SU( 5).

With Group Theory in a Nutshell for Physicists, Zee convincingly demonstrates that group theory governs the physical universe, and he gives aspiring physicists the tools to understand its applications to their work.

Piet Van Isacker

LARGE HEAVY ION ACCELERATOR (GANIL) CAEN, FRANCE

Piet Van Isacker is a researcher with the French Alternative Energies and Atomic Energy Commission (CEA), working at France's Large Heavy Ion Accelerator (GANIL). In his research he applies group theoretical methods to study quantum mechanical many-body systems, in particular atomic nuclei.


Q A Wang

F Tsobnang

S Bangoup

F Dzangue

A Jeatsa

A Le Méhauté

Stochastic action principle and maximum entropy

Keywords: numbers : 05.45.-a, 05.70.Ln, 02.50.-r, 89.70.+c

A stochastic action principle for stochastic dynamics is revisited. We present first numerical diffusion experiments showing that the diffusion path probability depend exponentially on average Lagrangian action ∫ = b a Ldt A . This result is then used to derive an uncertainty measure defined in a way mimicking the heat or entropy in the first law of thermodynamics. It is shown that the path uncertainty (or path entropy) can be measured by the Shannon information and that the maximum entropy principle and the least action principle of classical mechanics can be unified into a concise form 0 = A δ , averaged over all possible paths of stochastic motion. It is argued that this action principle, hence the maximum entropy principle, is simply a consequence of the mechanical equilibrium condition extended to the case of stochastic dynamics.

1) Introduction

It is a long time conviction of scientists that the all systems in nature optimize certain mathematical measures in their motion. The search for such quantities has always been a major objective in the efforts to understand the laws of nature. One of these measures is the Lagrangian action considered as a most fundamental quantity in physics. The least action principle 1 [START_REF] De Maupertuis | Essai de cosmologie[END_REF] has been used to derive almost all the physical laws for regular dynamics (classical mechanics, optics, electricity, relativity, electromagnetism, wave motion, etc. [START_REF] Bangoup | Etude du principe de Maupertuis dans tous ses états[END_REF]).

This achievement explain the efforts to extend the principle to irregular dynamics such as equilibrium thermodynamics [START_REF] Broglie | La thermodynamique de la particule isolée[END_REF], irreversible process [START_REF] Onsager | Fluctuations and irreversible processes[END_REF], random dynamics [START_REF] Freidlin | Random perturbation of dynamical systems[END_REF] [START_REF] Eyink | Action principle in nonequilibrium statistical dynamics[END_REF], stochastic mechanics [START_REF] Guerra | Quantization of dynamical systems and stochastic control theory[END_REF] [START_REF] Pavon | Hamilton's principle in stochastic mechanics[END_REF], quantum theory [START_REF] Feynman | Quantum mechanics and path integrals[END_REF] and quantum gravity theory [START_REF] Hawking | [END_REF]. We notice that in most of these approaches, the randomness or the uncertainty (often measured by information or entropy) of the irregular dynamics is not considered in the optimization methods. For example, we often see expression such as R R δ δ = concerning the variation of a random variable R with an expectation R . This is incorrect because the variation of uncertainty aroused by the variation of the R may play important role in the dynamics.

Another most fundamental measure, called entropy, is frequently used in variational methods of thermodynamics and statistics. The word "entropy" has a well known definition given by Clausius in the equilibrium thermodynamics. But it is also used as a measure of uncertainty in stochastic dynamics. In this sense, it is also referred to as "information" or "informational entropy". In contrast to the action principle, entropy and its optimization have always been a source of controversies. It has been used in different even opposite variational methods based on different physical understanding of the optimization. For instance, there is the principle of maximum thermodynamic entropy in statistical thermodynamics [START_REF] Gibbs | Principes élémentaires de mécanique statistique[END_REF] [START_REF] Jaynes | The evolution of Carnot's principle, The opening talk at the EMBO Workshop on Maximum Entropy Methods in x-ray crystallographic and biological macromolecule structure determination[END_REF], the maximum information-entropy [START_REF] Tribus | Décisions Rationelles dans l'incertain[END_REF] [START_REF] Jaynes | Gibbs vs Boltzmann entropies[END_REF] in information theory, the principle of minimum entropy production [15] for certain nonequilibrium dynamics, and the principle of maximum entropy production for others [START_REF] Martyushev | Maximum entropy production principle in physics[END_REF] [17]. Certain interpretation of entropy and of its evolution was even thought to be in conflict with the mechanical laws [START_REF] Dorfmann | An introduction to Chaos in nonequilibrium statistical mechanics[END_REF]. Notice that these laws can be derived from least action principle. In fact, the definition of entropy is itself a great matter of investigation for both equilibrium and nonequilibrium systems since the proposition of Boltzmann and Gibbs entropy. Concerning the maximum entropy calculus, few people still contest the fact that the maximization of Shannon entropy yields the correct exponential distribution. But curously enough, few people are completely satisfied by the arguments of Jaynes and others [START_REF] Jaynes | The evolution of Carnot's principle, The opening talk at the EMBO Workshop on Maximum Entropy Methods in x-ray crystallographic and biological macromolecule structure determination[END_REF][13] [START_REF] Jaynes | Gibbs vs Boltzmann entropies[END_REF] supporting the maximum entropy principle by considering entropy as anthropomorphic quantity and the principle as only an inference method. This question will be revisited to the end of the present paper.

In view of the fundamental character of entropy in stochastic dynamics, it seems that the associated variation approaches must be considered as first principles and cannot be derived from other ones (such as least action principle) for regular dynamics where uncertainty does not exist at all. However, a question we asked is whether we can formulate a more general variation principle covering both the optimization of action for regular dynamics and the optimization of information-entropy for stochastic dynamics. We can imagine a mechanical system originally obeying least action principle and then subject to a random perturbation which makes the movement stochastic. For this kind of systems, we have proposed a stochastic action principle [START_REF] Wang | Maximum path information and the principle of least action for chaotic system[END_REF][21] [START_REF] Wang | Non quantum uncertainty relations of stochastic dynamics[END_REF] which was originally a combination of maximum entropy principle (MEP) and least action principle on the basis of the following assumptions :

1) A random Hamiltonian system can have different paths between two points in both configuration space and phase space.

2) The paths are characterized uniquely by their action.

3) The path information is measured by Shannon entropy.

4) The path information is maximum for real process. This is in fact maximization of path entropy under the constraint associated with average action over paths (we assume the existence of this average measure). As expected, this variational principle leads to a path probability depending exponentially on the Lagrangian action of the paths and satisfying the Fokker-Planck equation of normal diffusion [START_REF] Wang | Maximum entropy change and least action principle for nonequilibrium systems[END_REF]. Some diffusion laws such as Fick's laws, Ohm's law, and Fourier's law can be derived from this probability distribution. We noticed that the above combination of two variation principles could be written in a concise form 0 = A δ [START_REF] Wang | Non quantum uncertainty relations of stochastic dynamics[END_REF], i.e., the variation of action averaged over all possible paths must vanish.

However, many disadvantages exist in the above formalism. The first one is that not all the above physical assumptions are obvious and convincing. For example, concerning the path probability, another point of view [START_REF] Evans | Detailed balance has a counterpart in non-equilibrium steady states[END_REF] says that the probability should depend on the average energy on the paths instead of their action. The second disadvantage of that formalism is we used the Shannon entropy as a starting hypothesis, which limits the validity of the formalism. One may think that the principle is probably no more valid if the path uncertainty cannot be measured by the Shannon formula. The third disadvantage is that MEP is already a starting hypothesis, while it was expected that the work might help to understand why entropy goes to maximum.

In this work, the reasoning is totally different even opposite. The only physical assumption we make is a stochastic action principle (SAP), i.e., 0 = A δ . The first and second assumptions mentioned above are not necessary because these properties will be extracted from experimental results. The third and fourth assumptions become purely the consequences of SAP. This work is limited to the classical mechanics of Hamiltonian systems for which the least action principle is well formulated. Neither relativistic nor quantum effects is considered.

2) Stochastic dynamics of particle diffusion

We consider a classical Hamiltonian systems moving, maybe randomly, in the configuration space between two points a and b. Its Hamiltonian is given by H=T+V and its

Lagrangian by

V T L - =
where T is the kinetic energy and V the potential one. The Lagrangian action on a given path is

∫ = b a Ldt A
as defined in the Lagrangian mechanics. These definitions need sufficiently smooth dynamics at smallest time scales of observation. In addition, if there are random noises perturbing the motion, the energy variation due to the external perturbation or internal fluctuation is negligible at a time scale τ which is nevertheless small with respect to the observation period. Hence

V T L - = and V T H + =
can exist, where T and V are kinetic and potential energies averaged over τ such as

∫ = τ τ 0 1 Tdt T .
It is known that if there is no random forces and if the duration of motion t ab = t b -t a from a to b is given, there is only one possible path between a and b. However, this uniqueness of transport path disappears if the motion is perturbed by random forces. An example is the case of particle diffusion in random media, where many paths between two given points are possible. This effect of noise can be easily demonstrated by a thought experiment in Figure 1.

See the caption for detailed description. In this experiment, it is expected that more a path is different from the least action path (straight line in the figure) between a and b, less there are particles traveling on that path, i.e., smaller is the probability that the path is taken by the Gaussian noise, uniform noise (with amplitudes uniformly distributed between -1 and 1) and truncated uniform noises (uniform noise with a cutoff of magnitude between -z and z where z<1, i.e., the probability is zero for the magnitude between -z and z).

3) A numerical experiment of particle diffusion and path probability

( ) dx x x N x dN x dp ) 2 exp( 2 1 ) ( ) ( 2 0 2 σ π σ - - = =
, where dN(x) is the particle number in the interval x-x+dx, N is the total number of falling particles and σ is the standard deviation (sd). The experiments show that the dp(x) is always Gaussian whatever the noise (uniform, Gaussian or other truncated uniform noises).

Dust particles a x 0 x

Air

h y 2a 2b
The observed distributions of particles are Gaussian for the three noises. The standard deviation of the distributions is uniquely determined by the nature of the noise (type, maximal magnitude, frequency etc.). This result was expected because of the finite variances of the used noises and of the central limit theorem saying that the attractor distribution is a normal (Gaussian) one if the noises (random variables) have finite variance.

What can we conclude from this experiment of falling particles which seems to be trivial?

First, let us suppose that the falling distance h is small so that the path y between a and any position x on the bottom can be considered as a straight line and the average velocity on y can be given by τ y where τ is the motion time from a to x (see Figure 2a). In this case, it is easy to show that the action A x from a to x is proportional to (x-x 0 ) 2 , i.e.,

τ τ τ τ τ τ τ τ τ τ 2 ) ( 2 2 2 2 2 2 ) ( 2 0 2 2 2 2 2 2 h m x x m h m y m mgh y m mgh y m V T A x - - = - = - = - = - = where τ 2 2 y m T = and 2 mgh V =
are the average kinetic and potential energy, respectively.

This analysis applies to any smooth motion provided h is small. Considering the observed Gaussian distribution of the falling particles in figure 2, we can write for small h

) exp( ) ( A x dN x η - ∝
where η is a constant. The probability that a particle takes the small straight path from a to x is proportional to the exponential of action A x . Now let us consider large h. In this case the paths may not be straight lines. But a curved path from a to x can be cut into small intervals at x 1, x 2, ... . The above analysis is still valid for each small segment. The probability that a particle takes the path to x is then equal to the product of the probabilities on every segment of that path from a to x and should be proportional to the exponential of the total action from a to x, i.e.,

( ) ( ) ( )

A A A p ax i i i i ax η η η - = ∑ - = ∏ - ∝ exp exp exp (1)
where A i is the action on the segment x i and A ax is the total action on a given path from a to x.

The constant η is a characteristic of the noise and should be the same for every segment. The conclusion of this section is the path probability depends exponentially on action as long as the particle distribution on the bottom of the box is Gaussian.

Concerning the exponential form of path probability, there is another proposal [START_REF] Evans | Detailed balance has a counterpart in non-equilibrium steady states[END_REF] ( )

k ab H k p γ - ∝ exp ) (
, i.e., the path probability depends exponentially on the negative average energy. According to this probability, the most probable path has minimum average energy, so that for vanishing noise (regular dynamics), this minimum energy path would be the unique one which must also follow the least action principle. Here we have a paradox because the real path given by least action principle is in general not the path of minimum average energy.

4) An action principle for stochastic dynamics

Recently, the following stochastic action principle (SAP) was postulated [START_REF] Wang | Maximum path information and the principle of least action for chaotic system[END_REF][22] :

0 = A δ (2)
where

A p r D A ab δ δ ∫ = ) (
is the average of the variation A δ over all the paths. It can be written as follows

ab ab ab ab S A p A r D A p r D A p r D A δ η δ δ δ δ δ 1 ) ( ) ( ) ( - = ∫ - ∫ = ∫ = (3)
where

∫ = A p r D A ab ) (
is the ensemble average of action A, and ab S δ is defined by

( ) p A r D A A S ab ab δ η δ δ η δ ∫ = - = ) ( . (4)
Eq.( 4) makes it possible to derive S ab directly from probability distribution if the latter is known. Let us consider the dynamics in the section 3 that has the exponential path probability ( )

A Z p ab η - = exp 1 ( 5
)
where

( ) ∫ - = A r D Z η exp ) (
is the partition function of the distribution. A trivial calculation tell us that ab S δ is a variation of the path entropy S ab given by Shannon formula

∫ - = p p r D S ab ab ab ln ) ( . (6)
Eq.( 4) is a definition of entropy or information as a measure of uncertainty of random variable (action in the present case) [START_REF] Wang | Some invariant probability and entropy as a measure of uncertainty[END_REF]. It mimics the first law of thermodynamics

dW dU dQ + = where E p E U i i i ∑ = =
is the average energy, E i is the energy of the state i with probability p i , dW is the work of the forces ) (

j i i i j q E p F ∂ ∂ ∑ - =
and q j is some extensive variables such as volume, surface, magnetic moment etc. The work can be written as

∑ - = - = ∑ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∑ - = i i i i j j j i i i dE dE p dq q E p dW . So the first law becomes dE E d dQ - =
. We see that by Eq.( 4) a "heat" Q is defined as the measure the randomness of action (or of any other random variables in general [START_REF] Wang | Some invariant probability and entropy as a measure of uncertainty[END_REF]). In Eq.( 6), this heat" is related to the Shannon entropy since the probability is exponential. If the probability is not exponential, the functional of the entropy is probably different from the Shannon one, as discussed in [START_REF] Wang | Some invariant probability and entropy as a measure of uncertainty[END_REF].

With the help of Eqs.( 2) and ( 5), it is easy to verify that

A p p ab ab δ η δ - = (7)
and

A p p ab ab δ η δ 2 2 - = . (8)
From Eqs.( 7) and ( 8), the maximum condition of p ab , i.e., 0 = p ab δ and 0

2 < p ab δ , is transformed into 0 = A δ and 0 2 > A δ
if the constant η is positive, that is the least action path is the most probable path. On the contrary, if η is negative, we get 0 = A δ and 0 2 < A δ , the most probable path is a maximum action one.

In our previous work, we have proved that the probability distribution of Eq.( 5) satisfied the Fokker-Planck equation in configuration space. It is easy to see that [START_REF] Wang | Maximum path information and the principle of least action for chaotic system[END_REF], in the case of free particle, Eq.( 5) gives us the transition probability of Brownian motion with 0 2 1 > = mD η where m is the mass and D the diffusion constant of the Brownian particle [START_REF] Kubo | Statistical physics II, Nonequilibrium statistical mechanics[END_REF].

5) Return to the regular least action principle

The stochastic action principle Eq.( 2) should recover the usual least action principle 0 = A δ when the stochastic dynamics tends to regular dynamics with vanishing noise. To show this, let us put the probability Eq.( 5) into Eq.( 6), a straightforward calculation leads to

A Z Sab η + =ln . (9)
In regular dynamics, p ab =1 for the path of optimal (maximal or minimal or stationary) action A 0 and p ab =0 for other paths having different actions, so that 0 = ab S from Eq.( 6). We have only one path, the integral in the partition function gives

( ) ( ) 0 exp exp ) ( A A q D Z η η - = ∫ - = .
Eq.( 9) yields 0 A A= . On the other hand, we have

( ) 0 = - = A A Sab δ δ η δ . Thus our principle 0 = A δ implies 0 0 = = A A δ δ or, more generally, 0 = A δ
. This is the usual action principle.

6) Stochastic action principle and maximum entropy

Eq.( 3) tells us that the SAP given by Eq.( 2) implies

0 ) ( = -A Sab η δ . ( 10
)
meaning that the quantity ( )

A Sab γ
should be optimized. If we add the normalization condition, the SAP becomes:

0 )] 1 ) ( ( [ = ∫ - + - p q D A S ab ab α η δ (11)
which is just the usual Jaynes principle of maximum entropy. Hence Eq.( 2) is equivalent to the Jaynes principle applied t path entropy.

Is Eq.( 2) simply a concise mathematical form of Jaynes principle associated to average action? Or is there something of fundamental which may help us to understand why entropy gets to maximum for stable or stationary distribution?

From section 4, we understand that, in the case of equilibrium system, the variation dEi is a work dW. However, in the case of regular mechanics, dW=0 is the condition of equilibrium meaning that the sum of all the forces acting on the system should be zero and the net torque taken with respect to any axis must also vanish. So it seems reasonable to take 0 = dEi as an equilibrium condition for stochastic equilibrium. In other words, when a random motion is in (global) equilibrium, the total work

∑ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∑ - = j j j i i i dq q E p
] 1 [ = + - α η δ U S
where S is the thermodynamic entropy. This analysis seems to say that the maximum entropy (maximum randomness) is required by the mechanical equilibrium condition in stochastic situation.

Remember that dEi can also be written as a variation of free energy . The stochastic equilibrium condition can be put into 0 = dF .

Coming back to our SAP in Eq.(2), the system is in nonequilibrium motion. If there is no noise, the true path satisfies

0 = A δ and 0 = ∂ ∂ - ∂ ∂ ∂ ∂ r L r L t j j
. When there is noise perturbation, we have [START_REF] Wang | Non quantum uncertainty relations of stochastic dynamics[END_REF] [START_REF] Jaynes | The evolution of Carnot's principle, The opening talk at the EMBO Workshop on Maximum Entropy Methods in x-ray crystallographic and biological macromolecule structure determination[END_REF] where 0 [START_REF] Tribus | Décisions Rationelles dans l'incertain[END_REF] where

0 ) ( = ∑ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - ∂ ∂ ∂ ∂ ∫ = ∫ j j j j b a j ab dr dt r L r L t p r D dA
≠ = ∂ ∂ - ∂ ∂ ∂ ∂ f r L r L t j j
[ ] ∑ = ∑ = ∫ ∫ j j j ab j j j j ab dW p r D dr f p r D dW ) ( ) (
is the ensemble average (over all paths) of the time average

∫ = ∫ = b a j ab b a j j ab j dt W d t dt r d f t dW 1 1 and r d f dW j j j =
is the work of random force over the variation (deformation) r d j of a given path. Eq.( 13) means

0 = dW ( 14
)
since t ab is arbitrary. Eq.( 14) implies that the average work of the random forces at any moment over any time interval and over arbitrary path deformation must vanish. This condition can be satisfied only when the motion is totally random, a state at which the system does not have privileged degrees of freedom without constraints. Indeed, it is easy to show that the maximum entropy with only the normalization as constraint yields totally equiprobable paths. This argument also holds for equilibrium systems. The vanishing work 0 = =dW dE i needs that, if there is no other constraint than the normalization, no degree of freedom is privileged, i.e., all microstates of the equilibrium state should be equiprobable. This is the state which has the maximum randomness and uncertainty.

To summarize this section, the optimization of both equilibrium entropy and nonequilibrium path entropy is simply the requirement of the mechanical equilibrium conditions in the case of stochastic motion. There is no mystery in that. Entropy or dynamical randomness (uncertainty) must take the largest value for the system to reach a state where the total virtual work of the random forces should vanish. Entropy is not necessarily anthropomorphic quantity as claimed by Jaynes [START_REF] Jaynes | Gibbs vs Boltzmann entropies[END_REF] to be able to take maximum for correct inference. Entropy is nothing but a measure of physical uncertainty of stochastic situation.

Hence maximum entropy is not merely an inference principle. It is a law of physics. This is a major result of the present work.

7) Concluding remarks

We have presented numerical experiments showing the path probability distribution of some stochastic dynamics depends exponentially on Lagrangian action. On this basis, a stochastic action principle (SAP) formulated for Hamiltonian system perturbed by random forces is revisited. By using a new definition of statistical uncertainty measure which mimics the heat in the first law of equilibrium thermodynamics, it is shown that, if the path probability is exponential of action, the measure of path uncertainty we defined is just Shannon information entropy. It is also shown that the SAP yields both the Jaynes principle of maximum entropy and the conventional least action principle for vanishing noise. It is argued that the maximum entropy is the requirement of the conventional mechanical equilibrium condition for the motion of random systems to be stabilized, which means the total virtual work of random forces should vanish at any moment within any arbitrary time interval. This implies, in equilibrium case, 0 = dE i , and in nonequilibrium case, 0 = = dW dA . In both cases, the randomness of the motion must be at maximum in order that all degrees of freedom are equally probable if there is no constraint. By these arguments, we try to give the maximum entropy principle, considered by many as only an inference principle, the status of a fundamental physical law.

Figure 1 A

 1 Figure 1

  each path? If it exists, how does it change from path to path? What are the quantities associated with the paths which determines the change in path probability? To answer these questions, we have carried out numerical experiments (Figure2) showing the dust particles fall from a small hole a on the top of a two dimensional experimental box to the bottom of the box. A noise is introduced to perturb symmetrically in the direction of x the falling particles. We have used three kind of noises:

Figure 2 2a:

 2 Figure 2

We continue to use this term "least action principle" here considering its popularity in the scientific community, although we know nowadays that the term "optimal action" is more suitable because the action of a mechanical system can have a maximum, or a minimum, or a stationary for real paths[START_REF] Gray | Progress in Classical and Quantum Variational Principles[END_REF].


est destinée au dépôt

  Nathalie Luchaire. Exploring the Microvine developmental and physiological adaptation mechanisms to elevated temperature : characterization of the impact on carbon balance and its involvement in early reproductive development failure. Agricultural sciences. Montpellier SupAgro, 2015. English. NNT : 2015NSAM0011. tel-01404852


Giovanni Neglia
email: giovanni.neglia@inria.fr

Sara Alouf
email: sara.alouf@inria.fr

Abdulhalim Dandoush
email: dandoush@esme.fr

Sebastien Simoens
email: sebastien.simoens@transport.alstom.com

Pierre Dersin
email: pierre.dersin@transport.alstom.com

Alina Tuholukova
email: alina.tuholukova@inria.fr

Jérôme Billion
email: jerome.billion@transport.alstom.com

Pascal Derouet
email: pascal.derouet@transport.alstom.com

Perfor- Mance

la diffusion de documents scientifiques de niveau

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

  the next tick of ZC clock before processing can start LOC processing and EOA generation take a tick of ZC clock Offset Delivery delay EOA waits until the next tick of CC clock before being processed EOA may wait an additional tick if there are tasks with higher priority LOC EOA CC ZC CC and ZC servers work on the basis of ticks. CC generates LOC every third tick of its clocks.The figure explains the procedure of one LOC-EOA exchange.

Delay Model

Loss Model ( ,

EB rate

We derived general formula for the EB rate that requires to provide the loss and delay model

Probability that the k-th EOA arrives too late to deactivate the timer of LOC 1 Probability that the k-th LOC-EOA exchange is lost.

As the train moves, it regularly loses communication due to handover phases. These handover phases depend on train's mobility.

It is hard to derive a closed-form expression for EB rate in presence of handovers (case 2).

We evaluate the EB rate with Monte-Carlo simulations. ✔ Too loose bounds justify our analysis.

Computational costs do not depend on the loss probability value. Thus it is efficient even when emergency brakes are rare (as they should be).


M Bergmann
email: michel.bergmann@inria.fr

A Iollo

Bioinspired Swimming Simulations

Keywords: Bioinspired Swimming, Level Set, Skeleton, Optimal transportation, Cartesian mesh, Penalization

We present a method to simulate the flow past bioinspired swimmers starting from pictures of an actual fish. The overall approach requires i) a skeleton graph generation to get a level-set function from pictures; ii) optimal transportation to obtain the velocity on the body surface; iii) flow simulations realized with a Cartesian method based on penalization. This technique can be used to automate modeling swimming motion from data collected by biologists. We illustrate this paradigm by simulating the swimming of a mackerel fish.

Introduction

Autonomous swimming robots can address in the future needs such as search and rescue, environmental monitoring and emergency response in rapidly unfolding scenarios. Fishes have the ability to evolve in complex and unpredictable environments that may abruptly change. They are capable of maneuvering in ways that can hardly be achieved by present engineering devices. In this sense, the design of such devices can significantly benefit from bioinspired principles.

However, quantitative evaluation of the geometrical and dynamical characteristics of swimming in experiments is challenging [START_REF] Triantafyllou | Hydrodynamics of flishlike swimming[END_REF][START_REF] Wen | Understanding undulatory locomotion in fishes using an inertia-compen sated flapping foil robotic device[END_REF]. In particular, precise control of the kinematics and dynamics of the experimental model is difficult; harder still is the accurate measurement of forces and power of a freely swimming model. In this regard, numerical models are well suited for such investigation.

Three-dimensional numerical simulation of swimming has only become viable in the last decade [START_REF] Mittal | A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries[END_REF][START_REF] Anup | A new mathematical formulation and fast algorithm for fully resolved simulation of self-propulsion[END_REF][START_REF] Curet | A versatile implicit iterative approach for fully resolved simulation of self-propulsion[END_REF][START_REF] Hieber | An immersed boundary method for smoothed particle hydrodynamics of self-propelled swimmers[END_REF][START_REF] Gazzola | Simulations of single and multiple swimmers with nondivergence free deforming geometries[END_REF][START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF] with the progress of both numerical schemes and computers architectures. Nonetheless, many of the studies presented in the literature typically deal with prescribed swimmer geome-tries. Only few start from actual fish geometries, like for example Patankar's group [START_REF] Anup | A new mathematical formulation and fast algorithm for fully resolved simulation of self-propulsion[END_REF][START_REF] Curet | A versatile implicit iterative approach for fully resolved simulation of self-propulsion[END_REF][START_REF] Bhalla | A unified mathematical framework and an adaptive numerical method for fluid-structure interaction with rigid, deforming, and elastic bodies[END_REF]] and Mittal's group [START_REF] Mittal | A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries[END_REF][START_REF] Bozkurttas | Low-dimensional models and performance scaling of a highly deformable fish pectoral fin[END_REF][START_REF] Dong | Computational modelling and analysis of the hydrodynamics of a highly deformable fish pectoral fin[END_REF] .

The aim of this paper is to model a three-dimensional swimmer starting from actual fish pictures. The first step is to build the three-dimensional fish profile starting from two-dimensional data retrieved from one or several subsequent pictures of a fish. To this end, a skeleton technique [START_REF] Galusinski | Skeleton and level set for channel construction and flow simulation[END_REF] is used to reconstruct the three-dimensional level set function describing the body surface. Then the skeleton can be deformed using an appropriate swimming law to obtain a sequence of level set functions corresponding to snapshots of the body surface uniformly taken at different instants. Otherwise, if a sequence of pictures of swimming is available, the body surface can be reconstructed with the same technique for each snapshot. The level set functions give at each instant an implicit description of the surface. This is quite different from usual swimming simulations where the deforming body is fully described in a Lagrangian framework.

With skeleton deformation or with a sequence of actual pictures, we typically can reconstruct only a small number of the snapshots necessary to simulate a swimming stroke. This is due to the fact that the time scale of the simulation is significantly smaller than the time step between two subsequent reconstructed three-dimensional snapshots. Moreover, the surface deformation velocity is required to set the boundary conditions of the flow problem. For this reason it is necessary to build intermediate level-set functions and to compute the deformation velocity field between subsequent reconstructed fish snapshots. Optimal transportation [START_REF] Villani | Optimal Transport, old and new[END_REF] is well suited to achieve this goal providing an objective model to compute intermediate geometries and deformation velocities.

Optimal transportation is an old topic first introduced by Monge in 1781 [START_REF] Monge | Memoire sur la theorie des des deblais et des remblais[END_REF]. It consists in a distance minimization problem that allows to define a mapping between an initial and a final mass distribution. This problem has attracted considerable attention these last years especially from a numerical point of view [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF][START_REF] Iollo | A lagrangian scheme for the solution of the optimal mass transfer problem[END_REF][START_REF] Saumier | An efficient numerical algorithm for the l2 optimal transport problem with periodic densities[END_REF][START_REF] Lombardi | Eulerian models and algorithms for unbalanced optimal transport[END_REF][START_REF] Hug | Multi-physics optimal transportation and image interpolation[END_REF]. Here we exploit the scheme introduced in [START_REF] Bouharguane | Numerical solution of the monge-kantorovich problem by density lift-up continuation[END_REF] to define a non-linear interpolation mapping between geometries at different instants and to compute the corresponding deformation velocities.

In the following we describe the numerical scheme, the swimmer kinematic modeling and show validations with examples of applications.

Numerical modeling

Flow problem

The flow configuration is given in figure 1 where Ω s is the domain defined by the body, Ω f is the domain filled by the fluid, and Ω = Ω f ∪ Ω s is the whole domain. The boundary between Ω f and Ω s is noted Γ s .

The flow is modeled using the incompressible Navier-Stokes equations written in the fluid domain Ω f as follows

Ω f (χ = 0) Ω s (χ = 1) Γ s u s u, p ∂Ω
ρ ∂u ∂t + (u • ∇)u = -∇p + ∇ • 2µD(u) + ρg in Ω f , (1a)
∇ • u = 0 in Ω f . (1b)
where D(u) = ∇u+∇ T u

2

, u = (u, v, w) is the three-dimensional velocity field, p is the pressure field, µ is the dynamic viscosity, ρ is the density and g is the gravity. Unsteady boundary conditions are imposed on the body boundary

u(x, t) = u s (x, t) on Γ s , (1c)
and on the external boundary

u(x, t) = u ext (x, t) on ∂Ω. (1d)
The dynamics of the boundary Γ s that represents the swimmer will be described later.

Classically, system (1) is discretized using a body fitted mesh such that the boundary condition on the body boundary can be easily and accurately taken into account. However, these kind of meshes are difficult to generate in complex unsteady geometries in three-dimensions as large mesh deformations are required when the body is displaced or is changing topology.

In this study we will use a fixed Cartesian grid. The body boundary is defined and tracked with a level set function [START_REF] Sethian | Level Set Methods and Fast Marching Methods[END_REF] and the boundary condition on Γ s is taken in account in the Navier-Stokes equation using a penalty term [START_REF] Angot | A penalization method to take into account obstacles in a incompressible flow[END_REF].

We thus consider the penalized Navier-Stokes equations in the whole domain Ω (fluid and body):

∂u ∂t + (u • ∇)u = - 1 ρ ∇p + 1 ρ ∇ • 2µD(u) + g + χλ(u s -u) in Ω, (2a)
∇ • u = 0 in Ω, (2b)
where boundary conditions on the body boundary are implicitly imposed through the penalty term χλ(u su). In the penalty term, χ is the characteristic function (χ(x) = 1 if x ∈ Ω s and χ(x) = 0 elsewhere) and λ is a large penalty parameter. The value of λ depends on the numerical scheme used to discretize the penalty term in the momentum equation. If this term is solved in an explicit way, λ ≤ ρ/∆t, where ∆t is the numerical time step. Otherwise, with an implicit discretization of the penalty term, no restrictions are imposed for the value of λ. It has been proven [START_REF] Van Yen | Approximation of the Laplace and Stokes Operators with Dirichlet Boundary Conditions Through Volume Penalization: A Spectral Viewpoint[END_REF] that the solution of the system (2) tends to the solution of the system (1) as 1/ √ λ and that there exist optimal λ with respect to the space discretization. We chose arbitrarily λ = 10 8 .

In what follows, the characteristic function will be computed from the signed distance function φ, so that χ = H(φ) where H denotes the Heaviside function.

The system (2) is solved in time using a projection method [START_REF] Chorin | Numerical solution of the Navier-Stokes equations[END_REF][START_REF] Temam | Sur l'approximation de la solution des equations de navierstokes par la methode des pas fractionnaires ii[END_REF] and is discretized in space on a uniform Cartesian mesh using second order centered finite differences for the diffusive term and third-order upwind finite differences for the convective term.

The modeling, the details of the numerical method, as well as some validations of our approach can be found in [START_REF] Bergmann | An accurate cartesian method for incompressible flows with moving boundaries[END_REF][START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF]. Other immersed boundary methods based on Lagrange multipliers can impose the non slip condition on the immersed surface in a different way [START_REF] Glowinski | A fictitious domain approach for the direct simulation of incompressible fluid flow past moving rigid bodies: Application to particulate flow[END_REF][START_REF] Taira | The immersed boundary method: a projection approach[END_REF][START_REF] Kallemov | An immersed boundary method for rigid bodies[END_REF]. We have proven that the present scheme is second-order accurate in space for velocity fields (in both L 2 and L ∞ norms), and first-order accurate for the pressure field [START_REF] Bergmann | An accurate cartesian method for incompressible flows with moving boundaries[END_REF]. This makes our approach suitable to perform accurate simulations, at least for moderate Reynolds numbers. The resolution of system (2) requires knowledge of both the unsteady characteristic function χ and the body velocity u s . The following sections will explain how to compute these quantities.

The velocity field in a self propelled body can be decomposed such that

u s (x, t) = u(x, t) + u θ (x, t) + u(x, t) (3)
where u(x, t), u(x, t) and u θ (x, t) are the deformation, the translation and the rotation velocities, respectively. While the deformation velocity is imposed (by muscles), the translation and rotation velocities are the results of the hydrodynamic forces and torques applied by the fluid onto the body. These latter are computed integrating the Newtons laws. Once the body geometry is obtained (or imposed), we can deform the backbone (the midline) following swimming laws observed in nature. Considering that the backbone of the steady body is 0 ≤ x ≤ ℓ, y = 0 and z = 0, we can apply a swimming law in the plan (x, z) that remains the backbone length to be constant. The swimming law observed in nature for most of fishes is [START_REF] Barrett | Drag reduction in fish-like locomotion[END_REF]:

z(x, t) = a(x) sin(2π(x/λ + f t)), (4a)
where the envelop is

a(x) = c 0 + c 1 x + c 2 x 2 . ( 4b
)
This swimming law is defined with a constant phase velocity c p = λf , where λ and f are the wave length and the frequency of the oscillations. The parameters c 0 , c 1 and c 2 are adjusted to reach a maximal tail amplitude A/2 that is an important parameter for the swimming [START_REF] Lighthill | Aquatic animal propulsion of high hydrodynamical efficiency[END_REF]. It has been shown [START_REF] Videler | Fast continuous swimming of two pelagic predators, saithe (pollachius virens) and mackerel (scomber scombrus): a kinematic analysis[END_REF][START_REF] Yu | Three-dimensional numerical simulation of hydrodynamic interactions between pectoral-fin vortices and body undulation in a swimming fish[END_REF] that carangiform swimming mode is close to c 0 = 0.02, c 1 = -0.12 and c 2 = 0.2 for a unit length fish with 0 ≤ x ≤ 1. These parameters can be rescaled when modifying the fish length.

Generation of level set functions from pictures: the skeleton approach

Numerical simulations of fish-like swimmers [START_REF] Bergmann | Modeling and simulation of fish-like swimming[END_REF][START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF] or jellyfishes [START_REF] Bergmann | An accurate cartesian method for incompressible flows with moving boundaries[END_REF] are possible with the scheme present above. The swimmer geometries were prescribed to mimic realistic fishes or jellyfishes. The motion of these swimmers was described in a purely Lagrangian way by tracking markers distributed on the swimmer boundary. The objective is now to simulate bioinspired swimmers using geometries and possibly deformation laws obtained by pictures or sequences of pictures. As an example we consider the definition of a swimmer starting from the mackerel fish presented in figure 2 (left column) [START_REF] Hovnanian | Methode de Frontieres Immergees pour la Mecanique des Fluides[END_REF].

As mentioned in §2, our numerical approach requires the knowledge of the level set function (distance function) associated with a body to get a second order scheme in space. A characteristic function (mask function) is sufficient for a first order penalty method [START_REF] Bergmann | Modeling and simulation of fish-like swimming[END_REF].

The process is synthesized in figure 2. We start with pictures of a mackerel fish. Top, bottom and side views are necessary. To rebuild fish in 3D from only three pictures (front, side, top), an underlying assumption is made: the whole cross-section of the fish along its length is geometrically similar to the fish on front view (front picture). A dilation takes place in width and another takes place in height. In the tail region of the fish, two connected components can appear on the cross section, each connected component will be geometrically similar to the front view. The scaling factors will be extracted from the thickness of the fish (from the side view) and the width (from the top view), according to the longitudinal position of the transverse cut. Segmentations are first performed to extract the boundaries (black lines in center column). The distance function (right column) can be computed by solving Eikonal equation. Then the skeleton is computed (red lines in the center column) as being the center points of the inscribed circles. The natural tool to identify this information is the centerline or skeleton of an object as recalled in [START_REF] Galusinski | Skeleton and level set for channel construction and flow simulation[END_REF]. It identifies the locus of points equidistant from further points of the boundary. The special feature here is that the relevant skeleton is not constructed from the Euclidean distance. In order to identify the thicknesses and the widths of the fish, the distance is measured in a fixed direction (depending on the height or along the width) of the side view or top view. The skeleton is then simplified compared to the skeleton associated with the Euclidean distance. Thus, the top view will have a skeleton made of a line from the head to the tail of the fish, with a single point of the skeleton on each transverse cut.

The skeleton of the profile view for the vertical distance will consist of a central line splitting in the fish's tail. Finally, for each transverse cut of the profile, the number of connected components of the profile view is the number (1 or 2) of points of the skeleton of this cut. This property is true thanks to the choice of the distance and is not true with Euclidian distance. Initial rescaling of pictures has been performed so that the fish length is the same for top and side views, and height is the same for top and side views. A simple 3D reconstruction of the fish could then take place, but the goal is to deform the fish, which will have the effect of losing the notion of directional distance, more complex work is to be implemented. First, the skeleton of the top view (for the vertical distance) S v is distorted by law (4). the concept of direc- tional distance is no longer suitable for reconstruction, then the chosen approach is to distort the classic skeleton associated with Euclidean distance, relying on distorted S v . This new skeleton allows to reconstruct, for the Euclidean distance, the deformed fish seen from top view. Then, the 3D reconstruction of the fish can be implemented by harnessing both concepts skeleton as shown in figure 4 (right).

It is thus possible to generate a series of level set functions corresponding to different time t for the swimming law [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF]. In what follows, we considered the periodical swimming law (4) and uniformly extract N snapshots for the level set function over one period (one swimming stroke). Finally, the level set functions have been modified (transported) with a rigid velocity so that we do not add any linear or angular translations during deformation. Linear and angular translations have only to be the results of the hydrodynamic forces and torques.

Note that in what follows, the level set are chosen to be the signed distance functions (positive inside the body).

As mentioned, we have access to N (here N = 20) snapshots. We define by {ϕ k } k=0,...,N -1 the level set function over a swimming period T referred to the center of mass of each image. Here, k denotes the time t = k T i/N for the i th swimming period. The number of available snapshots is of course inessential, but it is typically small compared to the number of snapshots necessary for a numerical simulation. It is then necessary to build missing level set function corresponding to the Naiver-Stokes time discretization. Moreover, it is also necessary to compute the fish velocity for the penalty method. This velocity can be decomposed into a deformation velocity, to be computed from the subsequent N level set functions, and a rigid velocity computed from hydrodynamics effects. The deformation velocity is derived from an optimal transportation model.

Optimal transportation is a model to compute a mapping between two densities. The model assumes that the mass is transported along straight lines with constant initial velocity. The initial velocity is found such that to overall distance measured in a given norm is minimum. The formulation of the L 2 optimal transportation problem is illustrated with respect to generic densities ̺ 0 and ̺ 1 . The main step is to compute the velocity field transforming ̺ 0 into ̺ 1 . Then it is possible to compute any desired intermediate densities for time t = r T i/N with 0 < r < 1.

Let ̺ 0 (ξ), ̺ 1 (x) be two smooth enough non-negative scalar density functions with compact support Ω 0 and Ω 1 , where ξ, x ∈ R d and d is the space dimension. We assume that

Ω0 ̺ 0 (ξ)dξ = Ω1 ̺ 1 (x)dx.
Let X : Ω 0 → Ω 1 a smooth one-to-one map such that X(ξ) realizes the transfer of ̺ 0 onto ̺ 1 , i.e., a map that satisfies the following Jacobian equation:

̺ 0 (ξ) = det(∇X(ξ))̺ 1 (X(ξ)).
This equation is under determined with respect to X(ξ) and a solution is selected among all possible maps by introducing the following L 2 Kantorovich-Wasserstein distance: inf

X Ω0 ̺ 0 (ξ)|X(ξ) -ξ| 2 dξ.
The L 2 Monge-Kantorovich problem (MKP) corresponds to finding a map X o such that this infimum is achieved. It has been proved that this problem admits a unique solution [START_REF] Brenier | Polar factorization and monotone rearrangement of vectorvalued functions[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Villani | Optimal Transport, old and new[END_REF], which is the gradient of a.e. convex function

Ψ o : Ω 0 → R: X o (ξ) = ∇Ψ o (ξ).
The main idea of solution relies on the assumption that we dispose of a mapping at iteration n that is a perturbation of the optimal mapping X o (ξ) [START_REF] Bouharguane | Numerical solution of the monge-kantorovich problem by density lift-up continuation[END_REF]. We derive a linear perturbation equation that is used to iteratively improve the initial guess. Let us assume that the mapping obtained at iteration n is

X n (ξ) = ∇ ξ Ψ o + ∇ ξ Ψ n ǫ .
where the error Ψ n ǫ satisfies Ψ n ǫ 2 ≈ ǫ. We define ̺ n 0 (ξ) as the initial density at iteration n that mapped by X n (ξ) gives the exact final density ̺ 1 (x). Then, taking a first-order Taylor expansion, we have

̺ n 0 (ξ) := ̺ 1 (X n (ξ)) det (∇ ξ X n (ξ)) = ̺ 1 (X o ) det (∇ ξ X o ) =̺0(ξ) + det (∇ ξ X o )̺ 1 (X o )Tr (∇ ξ X o ) -1 ∇ 2 ξ Ψ n ǫ + det (∇ ξ X o )∇ x ̺ 1 (X o ) • ∇ ξ Ψ n ǫ + o(ǫ),
where Tr denotes the matrix trace operator. At first order in ǫ, we have

̺ n 0 (ξ) -̺ 0 (ξ) det (∇ ξ X o ) ≈ ̺ 1 (X o )Tr (∇ ξ X o ) -1 ∇ 2 ξ Ψ n ǫ + ∇ x ̺ 1 (X o ) • ∇ ξ Ψ n ǫ (5)
The mapping update is then found by the iteration

X n+1 = X n -α ∇Ψ n ǫ ,
that converges to X o as a geometric series, for α ∈ [0, 1], in the linearized approximation [START_REF] Loeper | Numerical solution of the monge-ampere equation by a newton's algorithm[END_REF][START_REF] Saumier | An efficient numerical algorithm for the l2 optimal transport problem with periodic densities[END_REF].

If the optimal solution and the approximated mapping are a perturbation of identity, then equation ( 5), with

(∇ ξ X o ) -1 = Id and Tr ∇ 2 ξ Ψ n ǫ = ∆ ξ Ψ n ǫ , reduces to ̺ n 0 (ξ) -̺ 0 (ξ) ≈ ∇ ξ • (̺ 1 (X n (ξ))∇ ξ Ψ n ǫ ) (6)
that is equivalent to a semidiscretization in time of the continuity equation written in Eulerian form.

Choice of the density functions

Since the level set functions corresponding to the N fish snapshots are not strictly positive, they are not admissible candidates for optimal transportation. A regularized mask function (Heaviside of the level set function) is also not satisfying since the weight of the thin caudal fin in the mapping is very limited and the mapping tends to smear it in comparison to the whole body. We thus considered Gaussian density functions defined from the level sets {ϕ k } k=0,...,N -1 , with a support that is approximatively equal to ℓ/10 (ℓ being the length of the fish) with maximum near the interface but inside the body. In particular, we take

̺ 0 = G(ξ, k) = exp (ϕ k (ξ, t)-ϕ b ) 2 γ + ε, with γ = 2 • 10 -5 , ϕ b = 1.5 • 10 -3
and the regularization parameter ε = 10 -3 . In a similar way, we take ̺ 1 = G(ξ, k + 1). Using these definitions of the densities to be transported, each point of the body surface has a comparable weight in the transportation. We denote by G 0 the value of the Gaussian function corresponding to the swimmer surface. The density G(ξ, k) is presented in figure 5 where the external white line is the body boundary G 0 .

Computation of intermediate images

Starting from the optimal mapping, the computation of intermediate images is straightforward. Let X(ξ, τ ) = ξ + τ (X o -ξ) and

̺(ξ, τ ) = ̺ 0 (ξ) det (∇ ξ X(ξ, τ ))
We have ̺(ξ, 0) = ̺ 0 (ξ), ̺(ξ, 1) = ̺ 1 (X o (ξ)) and for 0 < τ < 1 we obtain the intermediate images. As for the initial deformation velocity, given the time step ∆t s between two images (here ∆t s = T /20), the deformation velocity from ̺ 0 to ̺ 1 is

u 0 = X o -ξ ∆t s (7)
The index 0 highlights the fact that this velocity has to be mapped to the actual position of the swimmer in space, since the level set snapshots are taken with respect to a center of mass fixed in space. Figure 6 shows the reconstruction the intermediate density for τ = 0.5 between densities ̺ 0 and ̺ 1 .

The same approach can be applied to the whole series of level sets {ϕ i } N i=0 , thus computing the corresponding deformation velocities and intermediate densities. Starting from these densities we can compute a characteristic function and a sign distance function using the fast marching approach [START_REF] Sethian | A fast marching level set method for monotonically advancing fronts[END_REF][START_REF] Sethian | Level Set Methods and Fast Marching Methods[END_REF] or by solving an Hamilton Jacobi equation [START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase flow[END_REF][START_REF] Luddens | Enablers for high-order level set methods in fluid mechanics[END_REF]. In what follows, we recall that ϕ(ξ, t) are the level set function of the deformed profiles with fixed center of mass and u 0 (ξ, t) is the deformation velocity.

5 Dynamics of the swimmer

Computation of the swimmer velocity and position

Since we are interested in a self propelled swimmer, the position of the swimmer is determined with the level set function φ(ξ, t) that is an image of the level set function ϕ(ξ, t) obtained by the transformation characterizing the rigid motion (translation plus rotation), X r (ξ, t). We have

φ(ξ, t) = ϕ(X r (ξ, t)).
The characteristic function χ required in the penalty method is

χ(ξ, t) = H(φ(ξ, t) -G 0 ),
where the constant G 0 is defined in §4.1. We can then impose χ(ξ, t) = 1 inside the whole body, and compute the signed distance function for second order penalty method [START_REF] Bergmann | An accurate cartesian method for incompressible flows with moving boundaries[END_REF].

In the same way, the velocity of the swimmer is u 0 (ξ, t) = u(ξ, t)+u θ (ξ, t)+ u 0 (ξ, t), where u 0 (ξ, t) is computed from [START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF]. Since the position of the swimmer is modified, the velocity has to act onto the modified position and we have:

u s (ξ, t) = u 0 (X r (ξ, t)).
The transformation X r (ξ, t) as well as the translation velocity u(ξ, t) and rotation velocity u θ (ξ, t) are evaluated using the Newton's laws computing the forces and the torques exerted by the fluid onto the body.

Computation of the forces and the torques

The forces and the torques exerted by the fluid onto the body could be computed integrating the stress tensor T(u, p) = -pI + µ(∇u + ∇u T ) on the body boundary. Let n be the unit outward normal vector to the interface Γ, and r = xx G with x G the mass center, the force and the torques write:

F = - Γs T(u, p) n dx, (8a)
M = - Γs r ∧ T(u, p) n dx. (8b)
However, since the body boundary is defined in an implicit way as being the zero level of the sign distance function, integration of equations ( 8) is thus not straightforward. Thus we turn to moment balance as follows.

We consider an arbitrarily subdomain Ω f delimited by the boundaries Γ s and Γ f surrounding the body with domain Ω s such that the forces and torques write [START_REF] Noca | On the evaluation of time-dependent fluid-dynamic forces on bluff bodies[END_REF]:

F = - d dt Ω f (t) u dV + Γ f (t) (T + (u -u f ) ⊗ u) n dS - Γs(t) ((u -u s ) ⊗ u) n dS. (9a)
M = - d dt Ω f (t) r i ∧ u dV + Γ f (t) r ∧ (T + (u -u f ) ⊗ u) n dS - Γs(t) r ∧ ((u -u s ) ⊗ u) n dS. (9b)
where u s and u f are respectively the velocities of boundaries Γ s and Γ f , that might in general be different from that of the fluid. Of course, in many applications (like for swimmers), the integral over the Γ s boundary vanish due to the fact that we have u = u s . In the following simulations, Γ f coincides with the external boundary of the domain where homogeneous boundary conditions are applied, and thus u f = 0.

6 Validation and application to actual pictures 6.1 Validation

Validation for free swimmer defined by markers

The numerical solver has been already intensively validated for several bodies where the geometry is defined by markers and the body followed in a Lagrangian way [START_REF] Bergmann | Modeling and simulation of fish-like swimming[END_REF][START_REF] Bergmann | An accurate cartesian method for incompressible flows with moving boundaries[END_REF][START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF]. We now present a validation test case for a free swimming threedimensional eel. This test case has been set up using conforming mesh by Kern and Koumoutsakos [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] from original two-dimensional simulations done by Carling et al. [START_REF] Carling | Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional Navier-Stokes equations and Newton's laws of motion[END_REF]. This three-dimensional eel has also been studied numerically in [START_REF] Pal | A Forced Damped Oscillation Framework for Undulatory Swimming Provides New Insights into How Propulsion Arises in Active and Passive Swimming[END_REF] and [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF] using immersed boundary leading to a good benchmarking test. We prescribe the deformation kinematics of the eel midline while the translation and rotation velocities are computed from fluid-structure interactions. Both the geometry profile and the deformation kinematics of the eel are fully described in [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF]. The curvature of the backbone is prescribed along the arc length s of the midline of length ℓ as

κ(s, t) = ∂α ∂t = K(s) sin(2π(f t -τ (s))) tanh(πf t). (10)
The amplitude enveloppe K(s) is defined by a cubic spline that takes the values K 0 , K 1 , K 2 and K 3 at s = 0, ℓ/3, 2ℓ/3 and ℓ, respectively, τ (s) = sτ tail /ℓ, f is the frequency of the traveling wave along the body and α is the angle made by the tangent vector to the midline and the axis defined by the initial body midline at rest. The Cartesian coordinates can then be easily obtained.

The transient deformation kinematics during the initial period 1/f is modeled by tanh(πf t). The computational domain is 8ℓ × 4ℓ × ℓ which is uniformly discretized by 1024 × 512 × 128 grid points leading to 128 mesh points along the body length. We have shown [START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF] that 128 points along the swimmer represents the best compromise between accuracy and computational costs with our numerical method. We chose the same physical constants and deformation kinematics parameters given in [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] to maximize the eel velocity, i.e. K 0 = 1.51, K 1 = 0.48, K 2 = 5.74, K 3 = 2.73, and τ tail = 1.44. The frequency is set to f = 1 Hz and the body length is ℓ = 1 m. The adaptative time step is chosen with a classical CFL condition with cf l = 0.2. The temporal evolutions of the swimming velocities, normalized by the wave velocity ℓf for the eel test case presented in [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] with respect to the time, which is normalized by 1/f . linear and angular swimming velocities for the eel test case presented in [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] are plotted in Figure 7. The results are normalized using the body length ℓ and the stroke frequency f , see [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF]. The linear velocities represent thus body length per stroke. These velocities are computed using Newtons laws from forces and torques computed from [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF]. Despite the normalized axial swimming velocity at steady state slightly exceeds the ones obtained in [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] and [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF] (0.51 vs. 0.47), we note good agreements between the overall behavior of normalized linear and angular velocities, especially with [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF] where the same transient is used. Note that, as in [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF], the V -velocity does not oscillate with zero mean value due to the initial stroke. Figure 8 shows the isovorticity surface with magnitude 4. The overall behavior is consistent with previous studies [START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] and [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF].

Validation for free swimmer defined by pictures

Our goal is then to validate the proposed method using optimal transporta-Figure 8: Three-dimensional isovorticity structure with magnitude 4 colored by z-velocity field, w. The same representation parameters as in [START_REF] Pal | Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion[END_REF] is used. tion comparing it with a well defined test case. All the following numerical simulations are performed using Cartesian mesh with ∆x = ∆y = ∆z = ℓ/128 on 256 CPUs.

In this section we do not consider an actual sequence of fish snapshots. We consider a fish model where the geometry is known and defined by markers. The geometry is however chosen to fit as best as possible the mackerel fish, except the lunate tail. Figure 9 presents the fish geometry for this test case as well as the markers (note that we actually use two times more markers in each directions).

Our goal is then to compare the two numerical models for the fish just defined:

• Case I ("Lagrangian"): the level set functions and the deformation velocities are given at each Navier-Stokes time step by following the Lagrangian markers on the body surface;

• Case II ("Eulerian"): we suppose that we only know 20 snapshots of the level set function over one stroke period. We have thus to compute all the missing level set as well as the deformation velocity using optimal transportation.

In both cases the length fish is ℓ = 10 cm and the swimming law (4) is defined with frequency f = 4 Hz and c 0 = -0.002, c 1 = -0.12, c 2 = 2 and λ = ℓ. These parameters are close to those observed in nature [START_REF] Yu | Three-dimensional numerical simulation of hydrodynamic interactions between pectoral-fin vortices and body undulation in a swimming fish[END_REF]. We defined the swimming law in the plan (x, z) leading to associated deformation velocities fields ( u 0 , w 0 ) computed from [START_REF] Bergmann | Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer[END_REF]. A comparison of the deformation velocity fields u 0 and w 0 obtained in cases I ("exact") and II (approximated using optimal transportation) is presented in figure 10. The Lagrangian velocity is only defined inside the body while the Eulerian one is defined in the whole domain where optimal transportation is computed. A very good agreement is obtained for the velocity w 0 : the Eulerian deformation velocity field and the Lagrangian one continuously match at the body surface. Some difference are however visible for the u 0 component. Due to the fish configuration, the velocity u 0 can be considered to be the velocity that is tangential to the body. Since optimal transportation is irrotational by definition, it is not able to model any skin rotation. The tangential velocity can not be computed if no other constraints are taken into account, such as done for example in [START_REF] Lombardi | Eulerian models and algorithms for unbalanced optimal transport[END_REF][START_REF] Hug | Multi-physics optimal transportation and image interpolation[END_REF] using a rigid motion constraint or a divergence-free constraint.

Theses Lagrangian and Eulerian velocities are then used to simulate the self propelled swimming in the both cases. According to equation (3), the swimming velocities of the center of mass x G are u G s (t) = u s (x G , t). These self propelled velocities are presented in figure 11. As expected, the swimming velocity w G s shows good agreement between cases I and II. The velocity u G s is influenced by the differences observed for the deformation velocity u 0 in figure 10. However, considering all the modeling hypothesis, the Eulerian forward swimming velocity correctly approaches the actual Lagrangian forward velocity from which it has been derived.

Mackerel fish swimming from pictures

We have applied the method described above to recover the geometry of a Mackerel, to deform its skeleton by a swimming law and to simulate swimming. Figure 12 shows the wake flow generating by the fish built from pictures. We chose the same parameters as before for the swimming law, i.e. the fish length is 10 cm and the swimming frequency is f = 4 Hz. The swimming velocity of the ("exact", presented inside the fish) and II (approximated using optimal transportation, represented outside the fish). mackerel fish is slightly higher than the previous test cases (u G s ≈ -0.15 m/s). This difference can be explained by a different shape, and moreover by the lunate tail for the real fish.

Conclusions

In this work we have described a procedure to define a bioinspired geometry for a swimmer starting either from one image of a fish or from a sequence of images. This approach can be seen as a step closer in converting observed swimming pictures to a realistic numerical model, reducing arbitrary assumptions and time spent in post-processing images for data fitting to get body kinematics. Using this procedure it is possible to recover an Eulerian deformation velocity on the base of a limited number of modeling hypothesis, namely those at the base of L 2 optimal transportation. We have shown an in silico example where we compare the swimming velocity of a swimmer whose geometry and deformation law is completely determined a priori, to that obtained by the identification method we propose. The results in terms of deformation velocity show that the irrotational hypothesis at the base of optimal transportation induces an error in the tangential velocity of the body. This error coupled to the discretization inaccuracies due to the fully Eulerian treatment of the geometry obtained from Lagrangian data reflects in an overestimation of the actual velocity of the swimmer of less than 10%. As an example of an actual application we have modeled a swimmer using pictures of a Mackerel. Future improvements of the results may be obtained by constrained optimal transportation.

Figure 1 :

 1 Figure 1: Sketch of the general flow configuration.

Figure 2 :

 2 Figure 2: Skeleton steps: pictures (left), segmentation with skeleton (center) and associated level set functions (right).

  For a given cut, let us denote a a point of the skeleton of the profile view and let b be the point of the skeleton of the top view. Denote d a the vertical distance from a to the boundary of the fish and d b the horizontal distance from b to the boundary of the fish. The connected component of the cut fish associated to (a, b) can be rebuilt by a scaling (based on factors d a and d b ) of the front view and translated according to the position (a, b). It also becomes possible to mirror the top view of the fish (or front view) by rectifying the skeleton in a line to correct the defects of shooting fish in the picture, see figure 3.

Figure 3 :

 3 Figure 3: Re-centering (top view).

Figure 4 :

 4 Figure 4: Three-dimensional profiles defined by the zero isocontour of the level set function of the actual fish. Undeformed (left) and deformed (right).

Figure 5 :

 5 Figure 5: Gaussian density used in the plan y = 0. The external white contour corresponds to the fish boundary.

Figure 6 :

 6 Figure 6: Construction of an image for τ = 0.5∆t s (red) between ̺ 0 (green) and ̺ 1 (blue) in the plan y = 0.

Figure 7 :

 7 Figure7: Evolution of the linear (U black line, V dashed line) and angular (ω) swimming velocities, normalized by the wave velocity ℓf for the eel test case presented in[START_REF] Kern | Simulations of optimized anguilliform swimming[END_REF] with respect to the time, which is normalized by 1/f .

Figure 9 :

 9 Figure 9: Geometry profile used to validate the method.

  (a) Deformation velocity field (W = w 0 ) in the z-direction, i.e. normal to the swimming direction (b) Deformation velocity field (U = u 0 ) in the x-direction, i.e. in the swimming direction

Figure 10 :

 10 Figure10: Comparison of the deformation velocity fields obtained in cases I ("exact", presented inside the fish) and II (approximated using optimal transportation, represented outside the fish).

Figure 11 :

 11 Figure 11: Comparison of the self propelled velocities obtained in cases I and II. The swimming velocity is u G s and the lateral velocity is w G s .

Figure 12 :

 12 Figure 12: Isovorticity representation of the wake generated by the mackerel fish starting form actual pictures.

Computation of intermediate level-sets and deformation velocities: optimal transportation

Acknowledgements

The authors thank C. Galusinski and C. Nguyen for contributions on fish construction. This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the Investments for the future Programme IdEx Bordeaux-CPU (ANR-10-IDEX-03-02). The simulations presented in this paper were carried out using the PLAFRIM experimental parallel testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil Régional d'Aquitaine, FeDER, Université de Bordeaux and CNRS (see https://plafrim.bordeaux.inria.fr/).