Kinetic theory for non-equilibrium stationary states in long-range interacting systems

Abstract

We study long-range interacting systems perturbed by external stochastic
forces. Unlike the case of short-range systems, where stochastic forces usually
act locally on each particle, here we consider perturbations by external
stochastic fields. The system reaches stationary states where external forces
balance dissipation on average. These states do not respect detailed balance
and support non-vanishing fluxes of conserved quantities. We generalize the
kinetic theory of isolated long-range systems to describe the dynamics of this
non-equilibrium problem. The kinetic equation that we obtain applies to
plasmas, self-gravitating systems, and to a broad class of other systems. Our
theoretical results hold for homogeneous states, but may also be generalized to
apply to inhomogeneous states. We obtain an excellent agreement between our
theoretical predictions and numerical simulations. We discuss possible
applications to describe non-equilibrium phase transitions.

Full text

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arXiv:1111.6833v2 [cond-mat.stat-mech] 31 Jan 2012
Kinetic theory for non-equilibrium stationary states in
long-range interacting systems
Cesare Nardini1,2, Shamik Gupta1, Stefano Ruffo1,3, Thierry
Dauxois1and Freddy Bouchet1
1Laboratoire de Physique de l’Ecole Normale Sup´ erieure de Lyon, Universit´ e de Lyon and
CNRS, 46, all´ ee d’Italie, F-69007 Lyon, France
2Dipartimento di Fisica e Astronomia and CSDC, Universit` a di Firenze and INFN, via G.
Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
3Dipartimento di Energetica “Sergio Stecco” and CSDC, Universit` a di Firenze, CNISM
and INFN, via S. Marta 3, 50139 Firenze, Italy
E-mail: cesare.nardini@gmail.com,shamik.gupta@ens-lyon.fr,
stefano.ruffo@gmail.com,thierry.dauxois@ens-lyon.fr,freddy.bouchet@ens-lyon.fr
Abstract.
forces. Unlike the case of short-range systems, where stochastic forces usually act locally
on each particle, here we consider perturbations by external stochastic fields. The system
reaches stationary states where external forces balance dissipation on average. These states
do not respect detailed balance and support non-vanishing fluxes of conserved quantities.
We generalize the kinetic theory of isolated long-range systems to describe the dynamics
of this non-equilibrium problem. The kinetic equation that we obtain applies to plasmas,
self-gravitating systems, and to a broad class of other systems. Our theoretical results hold
for homogeneous states, but may also be generalized to apply to inhomogeneous states. We
obtain an excellent agreement between our theoretical predictions and numerical simulations.
We discuss possible applications to describe non-equilibrium phase transitions.
We study long-range interacting systems perturbed by external stochastic
PACS numbers: 05.20.Dd, 05.70.Ln, 05.40.-a

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1. Introduction
Most physical systems are out of equilibrium either because of coupling to thermal
baths at different temperatures or because of external forces that break detailed balance.
Studying non-equilibrium stationary states is an active area of research in modern statistical
mechanics. It is indeed a lasting challenge to achieve for non-equilibrium systems a level of
theoretical understanding similar to the one established for equilibrium systems [1–3].
In this Letter, we consider systems of particles interacting through two-body non-
integrable potentials, also called long-range interactions. Examples include plasmas and self-
gravitating systems (globular clusters, galaxies), where particles interact through repulsive
or attractive Coulomb and attractive Newton potential, respectively. Our work also applies
to a large class of models with non-integrable interactions, such as spins, vortices in two
dimensions, and others, which have been studied extensively in recent years [4–8].
Systems with non-integrable potentials are often forced through external stochastic
fields. For example, globular clusters are affected by the gravitational potential of their
galaxy, thereby producing a force that fluctuates along their physical trajectories.
addition, galaxies themselves feel the random potential of other surrounding galaxies, and
their halos are subjected to transient and periodic perturbations due, for example, to the
passing of dwarfs or to orbital decaying [9]. Plasmas may also be subjected to fluctuating
interactions imposed by environmental electric or magnetic fields [10].
situations often lead to a stationary state where the power injected by the external random
fields balances on average the dissipation. To the best of our knowledge, such non-equilibrium
stationary states in systems with non-integrable potentials have not been studied before, and
this work provides a first step in this direction.
Unlike systems with short-range interactions, stochastic perturbations in long-range
interacting systems often act coherently on all particles and not independently on each
particle.Moreover, unlike short-range systems, it is not natural to consider long-range
systems as being coupled to thermal baths at the boundaries. Thus, the non-equilibrium
stationary states that we study are rather different from the ones in systems with short-
range interactions. These states do not verify detailed balance and support non-zero fluxes
of conserved quantities, which are basic ingredients of non-equilibrium stationary states.
Theoretical results on isolated systems with long-range interactions include the kinetic
theory description of relaxation towards equilibrium.
leads to the Lenard-Balescu equation or to the approximate Landau equation [11, 12].
These equations, or some of their approximations, are grouped as the collisional Boltzmann
equation in the astrophysical context.The main theoretical result of this Letter is a
generalization of the kinetic theory to describe non-equilibrium stationary states, valid for
small external perturbations and spatially homogeneous stationary states.
The non-equilibrium kinetic equation that we obtain describes the temporal evolution
In
These physical
In plasma physics, this approach
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of the one-particle distribution function. When the system is not far from equilibrium, it is
natural to expect that the system settles into a stationary state. We find that in such a state,
the one-particle momentum distribution is non-Gaussian. The kinetic equation describes the
evolution of the kinetic energy, and its prediction of the stationary state compares very well
with N-body numerical simulations.
2. Stochastically forced long-range interacting systems
Consider a system of N particles interacting through a long-ranged pair potential. The
Hamiltonian of the system is
H =
N
?
i=1
p2
2+
i
1
2N
N
?
i,j=1
v(qi− qj), (1)
where qi and pi are, respectively, the coordinate and the momentum of the i-th particle,
while v(q) is the two-body interaction potential. The particles are taken to be of unit mass.
In this paper, for simplicity, we consider qi’s to be scalar periodic variables of period 2π;
generalization to qi∈ Rn, with n = 1,2 or 3, is straightforward.
In self-gravitating systems, since the dynamics is dominated by collective effects, it is
natural and usual to rescale time in such a way that the parameter 1/N multiplies the
interaction potential [13]. In plasma physics, the typical number of particles with which one
particle interacts is given by the coupling parameter Γ = nλ3
and λDis the Debye length. It is then usual to rescale the time such that the inverse of a
power of Γ multiplies the interaction term [11]. These reasons justify the rescaling of the
potential energy by 1/N in Eq. (1), known as the Kac scaling in systems with long-range
interactions [14].
We perturb the system (1) by the action of the stochastic field F(qi,t). The resulting
equations of motion are
˙ qi=∂H
∂pi,
∂qi
where α is the friction constant, and F(q,t) is a statistically homogeneous Gaussian process
with zero mean and variance given by
D, where n is the number density
and˙ pi= −∂H
− αpi+√αF(qi,t), (2)
?F(q,t)F(q′,t′)? = C(|q − q′|)δ(t − t′).
The hypothesis that the Gaussian fields are statistically homogeneous, i.e., the correlation
function depends solely on |q − q′|, holds for any perturbation which does not break space
homogeneity. Such a hypothesis will also be essential for the following discussions where we
consider homogeneous stationary states. Now, C(q) represents correlation, and is therefore
a positive-definite function [15]. Its Fourier components are thus positive:
(3)
ck≡
1
2π
?2π
0
dq C(q)e−ikq> 0,C(q) = c0+ 2
∞
?
k=1
ckcos(kq). (4)
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It will be convenient to use the following equivalent Fourier representation of the Gaussian
field F(q,t):
F(q,t) =√c0X0+
∞
?
k=1
√2ck[cos(kq)Xk+ sin(kq)Yk],(5)
where Xkand Ykare independent scalar Gaussian white noises satisfying ?Xk(t)Xk′(t′)? =
δk,k′δ(t − t′), ?Yk(t)Yk′(t′)? = δk,k′δ(t − t′), and ?Xk(t)Yk′(t′)? = 0.
Using the It¯ o formula [16] to compute the time derivative of the energy density e = H/N
and averaging over noise realizations give
?de
where κ =?N
In the dynamics (2), fluctuations of intensive observables due to stochastic forcing are
of order√α, while those due to finite-size effects are of order 1/√N. Moreover, the typical
timescale associated with the effect of stochastic forces is 1/α (see, e.g., Eq. (6)), while the
one associated with relaxation to equilibrium due to finite-size effects is of order N, see [4,5].
In the following, we analyze the dynamics (2) in the joint limit N → ∞ and α → 0.
While the first limit is physically motivated on grounds that most long-range systems indeed
contain a large number of particles, the second one allows us to study non-equilibrium
stationary states for small external forcing. Moreover, for small α, we will be able to develop
a complete kinetic theory for the dynamics.
For simplicity, we discuss in this Letter the continuum limit Nα ≫ 1, when stochastic
effects are predominant with respect to finite-size effects. Generalization to other cases (Nα
of order one, or, Nα ≪ 1) is straightforward, as discussed in the conclusion.
dt
?
i/(2N) is the kinetic energy density. The average kinetic energy density
in the stationary state is thus ?κ?ss= C(0)/4.
+ ?2ακ? =α
2C(0), (6)
i=1p2
3. Kinetic theory
A natural framework to study the dynamics (2) is the kinetic theory. We now describe
the theoretical approach to derive this theory, while some of the technical results will be
explicitly obtained in a longer paper [17]. The central result is the kinetic equation (10)
below, which describes the time evolution of the single-particle distribution function.
We consider the Fokker-Planck equation associated with the equations of motion (2).
It describes the evolution of the N-particle distribution function fN(q1,...,qN,p1,...,pN,t)
(after averaging over the noise realization, fNis the probability density to observe the system
with coordinates and momenta around the values {qi,pi}1≤i≤Nat time t). This equation can
be derived by standard methods [16]. We get
∂fN
∂t
=
N
?
i=1
?
−pi∂fN
∂qi
+∂(αpifN)
∂pi
?
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+
1
2N
N
?
i,j=1
∂v(qi− qj)
∂qi
?∂
∂pi
−
∂
∂pj
?
fN+α
2
N
?
i,j=1
C(qi− qj)∂2fN
∂pi∂pj.(7)
We have proved by analyzing the so-called potential conditions [18] for this Fokker-Planck
equation that a sufficient condition for the stochastic process (2) to verify detailed balance
is that the Gaussian noise is white in space, that is, ck= c for all k. This condition is not
satisfied for a generic correlation function C. Steady states are then true non-equilibrium
ones, with non-vanishing currents and a balance between external forces and dissipation.
Similar to the Liouville equation for Hamiltonian systems, the N-particle Fokker-
Planck equation is a very detailed description of the system.
we want to describe the evolution of the one-particle distribution function f(z1,t) =
? ?N
In plasmas and self-gravitating systems, due to the long-range nature of the interactions,
the one-particle distribution function is not affected by the two-particle distribution function
at leading order in 1/N, and therefore, its evolution is described at leading order by the
Vlasov equation. Finite-size effects however induce weak correlations whose effects on the
long-time evolution of the one-particle distribution can be computed self-consistently in the
framework of the kinetic theory by using perturbation theory. A complete treatment of the
problem leads to the Lenard-Balescu equation [11,12]. In a similar way, for our problem, the
evolution will be described at leading order by the Vlasov equations due to the long-range
nature of the interactions. Weak stochastic forces lead to weak correlations that affect the
long-time evolution. This case can be treated by following a generalized kinetic approach,
as we now describe.
Substituting in the N-particle Fokker-Planck equation (7) the reduced distribution
function fs(z1,...,zs,t) =? ?N
reduced distribution functions into connected and non-connected parts, e.g., f2(z1,z2,t) =
f(z1,t)f(z2,t) + αg(z1,z2,t), and then neglect the effect of the connected part of the three-
particle correlation on the evolution of the two-particle correlation function. This scheme
is consistent at leading order in the small parameter α, and is the simplest closure scheme
for the hierarchy. For simplicity, we moreover assume that the system is homogeneous: f
depends on p, and g depends on |q1− q2|, p1and p2, only. The first two equations of the
hierarchy are then
Using kinetic theory,
i=2dzi fN(z1,...,zN,t) (we use the notation zi = (qi,pi) whenever convenient). Note
that the normalization is?dz f(z,t) = 1.
i=s+1dzifN(z1,...,zN,t), and using standard techniques [19],
we get a hierarchy of equations, similar to the BBGKY hierarchy. We split the
∂f
∂t− α∂
∂p[pf] −α
2C(0)∂2f
∂p2= α∂
∂p
?
dqdp2v′(q)g(q,p,p2,t),(8)
and
∂g
∂t+
?
p1∂g
∂q1
−∂f
∂p
????
p1
?
dq3dp3v′(q1− q3)g(q3− q2,p3,p2,t)
?
+ {1 ↔ 2}
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= C(q1− q2)∂f
∂p
????
p1
∂f
∂p
????
p2
,(9)
where the symbol {1 ↔ 2} means an expression obtained from the bracketed one by
exchanging 1 and 2, while the prime denotes differentiation.
To obtain from these equations a single kinetic equation for the distribution function
f, we have to solve Eq. (9) for g as a function of f and plug the result into the right hand
side of Eq. (8). From these two equations, we readily see that the two-particle correlation g
evolves over a timescale of order one, whereas the one-particle distribution function f(p,t)
evolves over a timescale of order 1/α. We use this timescale separation, and compute the
long-time limit of g from Eq. (9) by assuming f to be constant. This procedure is equivalent
to making the Bogoliubov’s hypothesis for deriving the kinetic theory of isolated long-range
systems. For the timescale separation to be valid, it is also required that the one-particle
distribution function f(p,t) is a stable solution of the Vlasov equation at all times.
The solution of equations of the type (9) is quite technical (see the long appendix in
Nicholson’s book [11]). Equation (9) differs from the corresponding equation for an isolated
long-range system in that the term on the right hand side is different in the two cases, and
cannot be solved by methods known in the literature. The main technical achievement that
aided this work is to be able to solve Eq. (9). In a future paper [17], we will give the details
on the solving procedure. In brief, the method relies on making a parallel between the
Lyapunov equations for infinite-dimensional Ornstein-Uhlenbeck processes and their general
solutions, and Eq. (9). Using this method, we get the desired kinetic equation:
∂f
∂t− α∂(pf)
∂p
− α∂
∂p
?
D[f]∂f
∂p
?
= 0,(10)
where
D[f](p) =1
2C(0) + 2π
∞
?
k=1
vkck
?∗
dp1
?
1
|ǫ(k,kp)|2+
1
|ǫ(k,kp1)|2
?
1
p1− p
∂f
∂p
????
p1
.
(11)
Here, vkis the k-th Fourier coefficient of the pair potential v(q), the quantity ckis defined
in Eq. (4), while?∗indicates the Cauchy integral, and the dielectric function ǫ is
ǫ(k,ω) = lim
η→0+
?
1 − 2πivkk
?
dp
1
−i(ω + iη) + ikp
∂f
∂p
?
.(12)
The kinetic equation (10) is the central result of the Letter.
This kinetic equation has the form of a non-linear Fokker-Planck equation, since the
diffusion coefficient D[f](p) itself is a function of the unknown distribution function f. As
Eq. (11) shows, this coefficient has two parts, namely, (i) a linear part, C(0)/2, which is due
to the mean-field effect of the stochastic forces, and (ii) a non-linear part due to correlations
induced in the system by the stochastic forces. The contributions of different modes of the
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stochastic force are independent of each other, that is, contributions proportional to ckdo
not couple with vk′ with k ?= k′.
For consistency, the prediction of the evolution of the kinetic energy from the kinetic
equation has to agree with Eq. (6). We have checked this by using Eqs. (10) and (11),
and proving that the integrals in the non-linear part of the diffusion coefficient give no
contribution to the kinetic energy.
As already mentioned, and is evident from Eq. (10), the time scale for the kinetic
evolution is 1/α. This has been checked by performing direct numerical simulations, see
Fig. 1 and the next section. Thus, α can be eliminated from the kinetic equation (10) by
a redefinition of time. Therefore, even for vanishingly small value of α, if a stationary
distribution exists, it will be at distance of order one from a Gaussian momentum
distribution.
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
0.1 1 10
<p4>
αt
N=104, α=10-3
N=103, α=10-3
N=103, α=10-2
Kinetic theory
1
4
8
0.1 1 10 100 1000
<p4>
t
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1 1 10
Kinetic energy density <κ>(t)
αt
N=104, α=10-3
N=103, α=10-3
N=103, α=10-2
Kinetic theory
0.3
0.6
0.9
0.1 1 10 100 1000
t
?κ?
?κ?
(a) (b)
Figure 1. (a) Kinetic energy density ?κ? and (b) ?p4? as a function of αt, for the values
C(0) = 1.5 and c1= 0.75. The data for different N and α values are obtained from numerical
simulations of the stochastically forced HMF model, and involve averaging over 50 histories
for N = 104and 103histories for N = 103. The data collapse implies that α is the timescale
of relaxation to the stationary state. The inset shows the data without time rescaling by α.
While a linear Fokker-Planck equation with non-degenerate diffusion coefficient can be
proven to converge to a unique stationary distribution [18], this is not true in general for
non-linear Fokker-Planck equations such as Eq. (10). We expect that if the system is not
too far from equilibrium, the kinetic equation will have a unique stationary state. Far from
equilibrium, the kinetic equation could lead to very interesting dynamical phenomena, such
as bistability, limit cycle or more complex behaviors. The main issue is then the analysis of
the evolution of the kinetic equation. Although some methods to study this type of equations
exist [20], in order to provide some preliminary answers, we have devised a numerical iterative
7

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scheme to compute some of the stationary states of the kinetic equation (10). We now
describe the scheme.
A linear Fokker-Planck equation whose diffusion coefficient D(p) is strictly positive
admits a unique stationary state
?p
For a given distribution fn(p), we compute the diffusion coefficient Dn(p) from Eq. (11), and
then fn+1using Dnand Eq. (13). This procedure defines an iterative scheme. Whenever
convergent, this scheme leads to a stationary state of Eq. (10). Each iteration involves
integrations, so we expect the method to be robust enough when starting not too far from
an actual stationary state. However, we have no detailed mathematical analysis yet.
In the next section, we discuss numerical results on N-particle simulations, and the
computation of stationary states from the iterative method mentioned above.
fss(p) = Aexp
?
−
0
dp′
p′
D(p′)
?
.(13)
4. Stochastically forced HMF model
Until now, we have presented our theoretical analysis for a general two-particle interaction
v(q). In order to perform simple numerical simulations, we now consider the case of the
stochastically forced attractive Hamiltonian mean-field (HMF) model, which corresponds to
the choice v(q) = 1 − cosq.
The HMF model serves as a paradigm to study long-range interacting systems, and
describes particles moving on a circle under deterministic Hamiltonian dynamics [21,22].
This model has been studied a lot in recent times. It displays many features of generic
long-range interacting systems, such as the existence of quasistationary states [4, 22].
In equilibrium, the system displays a second-order phase transition from a high-energy
homogeneous phase to a low-energy inhomogeneous phase at the energy density ec= 3/4.
Since the Fourier transform of the HMF interparticle potential is, for k ?= 0, vk =
−[δk,1+ δk,−1]/2, where δk,iis the Kronecker delta, we see from the kinetic equation (10) that
only the stochastic force with wave number k = 1 contributes to the non-linear part of the
diffusion coefficient; all the other stochastic forces give only a mean-field contribution through
the term C(0). Thus, the two parameters that dictate the evolution of the stochastically
forced HMF model are C(0) and c1. From (6), we know that C(0) = 4?κ?ssis proportional to
the kinetic energy in the final stationary state. Moreover, Eq. (4) implies that c1≤ C(0)/2.
If c1= 0, the kinetic equation reduces to a linear Fokker-Planck equation with diffusion
coefficient C(0)/2. This equation also describes the HMF model coupled to a Langevin
thermostat, studied in [23,24]. As the kinetic equations are the same, the dynamics coincide
at leading order in α. However, we know that at higher orders, detailed balance is broken
in our case, whereas it holds for the Langevin dynamics.
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In the case c1 = 0, the homogeneous stationary states of the kinetic equation have
Gaussian momentum distribution f(p). As has been studied thoroughly in the context of
canonical equilibrium of the HMF model, these states are stable for kinetic energies greater
than 1/4, i.e., for C(0) > 1.
For values of C(0) and c1 such that C(0) > 1 and c1 ≪ C(0), we then expect the
stationary states to be close to homogeneous states with Gaussian momentum, so that the
numerical iterative scheme to locate stationary states of the kinetic equation is expected to
converge for well-chosen initial conditions. We have checked the convergence for the set of
values of c1used in the simulations reported in the paper.
To check the theory, we have performed numerical simulations of the stochastically
forced HMF model. In Fig. 1, we show the evolution of the kinetic energy and ?p4? =
(1/N)?N
state, computed using the iterative scheme. In both cases, we observe a very good agreement
between the theory and simulations.
For a more accurate comparison, we have obtained the stationary momentum
distribution from both N-body simulations and the numerical iterative scheme.
comparison between the two is shown in Fig. 2, left panel, where we also show the Gaussian
distribution with the same kinetic energy. The agreement between theory and simulations
is excellent.
i=1p4
i, and compare them with theoretical predictions. In the latter case, we have
compared the long-time asymptotic value with the kinetic theory prediction for the stationary
The
0
0.25
0.5
-3 0 3
Distribution f(p)
Momentum p
Gaussian
N-body simulation
Kinetic theory
0.4
0.8
1.2
1.6
2
2.4
-11 0 11
Diffusion constant D[f](p)
Momentum p
C(0)=1.5, c1=0
C(0)=1.5, c1=0.75
C(0)=1.5, c1=0.3
Figure 2. The figure on the left shows the stationary momentum distribution f(p) for
α = 0.01, C(0) = 1.5, and c1 = 0.75. The data denoted by crosses are results of N-
body simulations of the stochastically forced HMF model with N = 10000, while the black
broken line refers to the theoretical prediction from the kinetic theory. For comparison, the
red continuous line shows the Gaussian distribution with the same kinetic energy (stationary
state at C(0) = 1.5, c1= 0). The figure on the right shows the diffusion coefficient D[f](p)
for the stationary momentum distribution f(p) for different values of C(0) and c1.
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Page 10

5. Conclusions
In this work, we studied the effect of external stochastic fields on Hamiltonian long-range
interacting systems by generalizing the kinetic theory of isolated long-range systems. Our
theoretical results are general, being applicable to any long-range inter-particle potential,
space dimensions and boundary conditions. In this paper, we demonstrated an excellent
agreement between the theory and numerical simulations for one representative case.
Here, we discussed the kinetic theory in the limit Nα ≫ 1. The extension to general
values of Nα is straightforward: Because of the linearity of the equations of the BBGKY
hierarchy, the finite-N and stochastic effects give independent contributions. The kinetic
equation at leading order of both stochastic and finite-size effects is
∂f
∂t= Lα[f] + LN[f],
where Lαis the operator described in Eq. (10) and LN(of order 1/N) is the Lenard-Balescu
operator [11]. For instance, in the case Nα ≪ 1 and in dimensions greater than one, the
operator LN is responsible for the relaxation to Boltzmann equilibrium after a timescale
of order N, whereas the smaller effect of Lαselects the actual temperature after a longer
timescale of order 1/α.
We note that an equivalent approach to derive the kinetic theory is to write an evolution
equation for the noise-averaged empirical density ρ(p,q,t) = (1/N)?N
the resulting equation as a multiplicative term. This equation can be treated perturbatively,
and may be shown to lead to the kinetic equation (10).
Let us mention some open issues. For technical simplicity, we assumed a homogeneous
state in our approach. Recently, Heyvaerts [25] has generalized the Lenard-Balescu equation
to some non-homogeneous cases; his approach could be used to generalize the theory
developed here to inhomogeneous states. There is no difficulty in principle, although actual
computation could be more involved.
An interesting follow up of this work is to study the dynamics of the kinetic
equation (10), both analytically and numerically. This may unveil very interesting behaviors,
such as bistability or limit cycles. Bistability was observed in two-dimensional turbulence
with stochastic forcing [26], in a framework which has deep connection with the one studied
in this Letter. One of the motivations for this work was to make a first step in formulating
a kinetic theory for the point vortex model and the Euler equations in two-dimensional
turbulence [8]. This subject will be the topic of further investigations.
(14)
i=1?δ(qi(t)−q)δ(pi(t)−
p)?, by analogy with the Klimontovich approach for isolated systems. The noise appears in
6. Acknowledgments
C. N. acknowledges the EGIDE scholarship funded by Minist` ere des Affaires´Etrang` eres. S.
G. and S. R. acknowledge the contract LORIS (ANR-10-CEXC-010-01). F. B. acknowledges
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Page 11

the ANR program STATOCEAN (ANR-09-SYSC-014). Numerical simulations were done at
PSMN, ENS-Lyon. We thank Hugo Touchette for comments on the manuscript.
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