Eric Bringuier
email: eric.bringuier@univ-paris-diderot.fr

Disorder-induced quantum-to-classical transition, or how the world becomes classical

Decoherence theory explains how quantum mechanics gives rise to classical mechanics through the entanglement of a quantum system's evolution with the degrees of freedom of the environment. The present article explores another pathway from the quantum to the classical behaviour. We consider a spinless particle interacting with a disordered landscape of potential energy. The matterwave evolution is handled within time-dependent quantum statistical mechanics, in which the wave function is replaced by a Wigner function defined in position-momentum space. Upon zooming out to scales exceeding the correlation length of the disorder, it is found that the description only involves the state populations as defined in classical statistical physics. Quantum coherence effects are significant only over smaller spatial scales, where they give rise to a noise superimposing on the classical description. The waning of coherence, which reflects the emergence of classicality, is due to the multiple scattering of matter waves; and the framework may be viewed as a stochastic wave mechanics.

1 The quantum-to-classical transition in decoherence theory state, namely 2 -1/2 (ψ 1 + ψ 2 ) if the overlap is negligible, of state operator 2 -1 [|ψ 1 )(ψ 1 | + |ψ 2 )(ψ 2 | + |ψ 1 )(ψ 2 | + |ψ 2 )(ψ 1 |], at zero time. The cross terms embody the system's ability to exhibit interference effects. Owing to environmental influences, such a superposition state is not stable over time, and it decays into a statistical mixture 2 -1 [|ψ 1 )(ψ 1 | (t) + |ψ 2 )(ψ 2 | (t) ] at a long enough time t > 0. The coherences, defined as the off-diagonal elements of the matrix of the state operator [6,7], are destroyed, hence the name ' decoherence '. Decoherence theory determines the basis with respect to which offdiagonal elements disappear as t → +∞, and the typical time scale of their disappearance. It is the interaction Hamiltonian that primarily determines the basis and the time scale, together with the system's Hamiltonian. As the interaction is often dependent on the position variable, a Schrödinger-cat-like superposition of macroscopically distant wave packets ψ 1 and ψ 2 rapidly decoheres into a classical mixture, whereby the interference effect is effectively suppressed. The formalism of decoherence theory rests upon the fact that the environmental degrees of freedom are unobserved in most practical cases, owing to their huge number and/or uncontrollable nature. By tracing out those degrees of freedom, a reduced state operator is obtained. That operator is the repository of the information allowing to predict (statistically) the behaviour of the system embedded in its environment.

For a long time, it was believed that quantum mechanics only holds at the microscopic level. The emergence of classicality in quantum theory has been a long-standing enigma. A clue was offered by decoherence theory initiated in the 1970s and 1980s in the works of Zeh and Zurek [1][2][START_REF] Strunz | Coherent Evolution in Noisy Environments[END_REF][4][5]. One could then understand, within the framework of quantum theory, why the macroworld appears to be classical. For instance, a macroscopic object is found in an approximate position of its centre of mass, but never in a Schrödinger-cat-like superposition of two or more macroscopically distinct positions. Explaining such an effective superselection rule was one motivation of decoherence theory. Besides, the derivation of classical concepts in a purely quantum framework removes many oddities of the Copenhaguen interpretation. The new theory rests upon the unavoidable coupling of the quantum system to the environment hosting it. The latter, for instance a huge number of air molecules and/or photons, scatters the guest system, the dynamics of which is entangled with that of the many-degree-of-freedom environment. This strongly affects the measurement of variables attached to the system itself. Even though, classically speaking, the influence of the environment on the system (for instance, the damping of its momentum) is negligible, interference effects with respect to certain physical quantities (especially position) become prohibitively difficult to observe in most cases. Consider two states of a particle, ψ 1 and ψ 2 , such as Gaussian wave packets of welldefined centroids. They may superpose to give a new This paper investigates another approach to the quantum-to-classical transition. The system studied herein is a spinless particle and the environment hosting it is very simple. It is a static potential-energy landscape of a disordered nature. By ' disordered ' we mean a random scalar field which may have many instantiations (' realisations ') distributed according to a probability law. The behaviour of the guest particle will be obtained by averaging over all realisations of the energyscape. Our goal is to recover the classical-mechanical behaviour of the guest particle subjected to a non-random potentialenergy field besides the random field. We will examine how quantum mechanics turns into classical-statistical mechanics. It is only possible to recover statistical physics, rather than Newton's deterministic dynamics, because the information encoded in Schrödinger's state vector ψ or von Neumann's state operator |ψ)(ψ| is inherently statistical. Quotation from Einstein and Infeld [8] : ' Quantum physics formulates laws governing crowds and not individuals. Not properties but probabilities are described, not laws disclosing the future of systems are formulated, but laws governing the changes in time of the probabilities and relating to great congregations of individuals '. Likewise, Schrödinger wrote: ' We never experiment with just one electron or atom or (small) molecule. In thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences ' [9]. degrees of freedom, haphazardly distributed throughout a volume Ω. This is pictured in figure 2 which shows two instantiations of the random field U 2 . In the example of an electron in a weakly ionised gas, the scatterers are the neutral atoms or molecules of the gas, which in a first approximation are taken as infinitely heavy and therefore stay at rest. Both fields U 1 and U 2 are time-independent (static); but eventually this assumption can be relaxed. There are two other differences between the two fields: (i) U 1 is taken to be smooth whereas U 2 exhibits smallscale irregularities, or loosely speaking disorder; and (ii) this disordered potential is taken to be weak, as will be specified in section 5.

Quantum mechanics

Schrödinger dynamical equation ∴ ψ(r, t) ∝ (complex-valued) probability amplitude in r space ∂ψ ∂t

= (ih _ ) -1 [E kin (-ih _ ∇) + U 1 (r) + U 2 (r)]ψ ↓ Wigner dynamical equation ∂f W ∂t = ( ∂f W ∂t ) kin + ( ∂f W ∂t ) U1 + U2 ∴ f W (r, p, t)/h 3 = (real-valued)
quasi-probability density in phase space

∂f W ∂t + ( ∂E kin ∂p ) . ∇f W + ∇(-U 1 ) . ( ∂f W ∂p ) = (ih _ ) -1 ∫ BZ Ω d 3 p' h 3 u W (r, p -p') f W (r, p', t) ↓ Classical (statistical) mechanics Boltzmann-Lorentz kinetic equation in phase space ∴ f 0 (r, p, t) ≡ 〈f W (r, p, t)〉 disorder ∂f 0 ∂t + ( ∂E kin ∂p ) . ∇f 0 + ∇(-U 1 ) . ( ∂f 0 ∂p ) = ∫ BZ Ω d 3 p' h 3 W p,p' [f 0 (r, p', t) -f 0 (r, p, t)]
The paper unfolds as shown in figure 1. Section 2 starts out from Schrödinger's formulation of quantum dynamics and section 3 specifies the quantum states of the spinless particle. Section 4 reformulates wave mechanics in position-momentum space where the wave function or the state operator is replaced by its Wigner transform. In section 5, the disordered environment is specified as a random energyscape. Wave mechanics in such an environment is studied in section 6. The transition to classical behaviour in Boltzmann's statistical formulation is the subject of section 7. Section 8 comes back to decoherence theory and closes the paper. Appendix A provides details about the multiplescale expansion sorting out the large-scale behaviour from the small-scale one. In appendix B, the change due to a non-static environment is sketched out, and this is tied to the issue of thermalization by the environment.

The issue: Quantum particle in a disordered environment

The present paper deals with the quantum-mechanical motion of a particle hosted in an environment which is modelled in the simplest way as a disordered static energyscape. More specifically, we are considering a spinless particle of mass m acted upon by a force derived from an external potential energy U 1 (r) where r is the position. The Hamiltonian operator is Fig. 1. Organization chart of the article, where the notations of the physical quantities involved will be introduced along the paper. The starting point is Schrödinger's differential equation governing the complex amplitude of a probability density, i.e. the matter-wave field ψ of a spinless particle. Its Wigner transform f W is h 3 times a quasi-probability density in phase space which is governed by an integro-differential equation. Both the Schrödinger and Wigner equations describe quantum dynamics endowed with phase coherence. The ending point of the article is Boltzmann's classical-statistical description which makes use of true (phaseless) probabilities.

H = (-ih _ ∇) 2 2m + U 1 (r) + U 2 (r), (1)
where h _ = h/2π is the reduced Planck constant and ∇ ≡ ∂/∂r is the gradient operator. While U 1 (r) is a given function of position known with certainty, U 2 (r) is known only statistically. Mathematically speaking U 1 (U 2 ) is a deterministic (probabilistic) function of r. For definiteness, the environment as modelled by U 2 may be thought of as a set of fixed scatterers without internal momentum space is the fundamental Brillouin zone (times h _ ), henceforth denoted by BZ, in the parlance of the physics of the crystalline solid state.

Σ r Ω N c F(r) ≈ ∫ Ω F(r) d 3 r w l (r) = 1 N c Σ p exp( ip . l h _ )b p (r), (2)
Σ p h 3 Ω G(p) ≈ ∫ BZ G(p) d 3 p
where w l (r) is a ' cellular function ' localized about position l. There are N c such positions which make up a ' ghost lattice ' in the box. In one dimension, the positions of the ghost lattice sites l are x n = na, where a is L/N c and n takes N c values. The calculation yields Kronecker symbol

↔ Dirac function δ r,r' ≈ Ω N c δ(r -r') w n (x) = 1 N c a 1/2 sin[π(x -x n )/a] sin[π(x -x n )/L] δ p,p' ≈ h 3 Ω δ(p -p') ≈ 1 a 1/2 sin[π(x -x n )/a] π(x -x n )/a if N c >> 1. ( 3
)
A warning is in order. Cyclic boundary conditions are associated with a torus topology in which Ehrenfest's This function peaks at the discrete position x n . Two cellular functions are shown in figure 3. Accordingly, the first relation does not hold for a plane wave because the expectation value of position is independent of time while that of velocity does not vanish, namely Several definitions of f W exist in the literature [7,11,13,14]; the one chosen in this paper is dimensionless. Just as there is one quantum state, there is one Wigner function. The r-to-p duality is evinced in the two equally valid formulae for f W in Table 2. The advantage of the discretized finite box of section 3 is now apparent. While the use of delta-peaked wave functions is common practice in Schrödinger's wave mechanics, it is troublesome in Wigner's formulation because then a squared delta function can arise in the Wigner transform. A similar trouble is met with unbounded plane waves such that ψ ^ is a delta function of momentum. d dt

(b p |r|b p ) = 0 while (b p | -ih _ ∇ m |b p ) = p m ≠ 0. (4)
In this paper round brackets ( | ) denote the scalar product in the Hilbert space of states.

Wigner's formulation of wave mechanics in phase space

The dynamics of the particle is governed by Schrödinger's wave equation, namely

ψ(r) ∫ Ω ψ(r + s 2 ) ψ * (r -s 2 ) exp(-ip . s h _ ) d 3 s ∂|ψ| 2 ∂t + div J = 0. ( 6
)
From the matter-wave field ψ(r) at a given time, one can build a Wigner transform f W (r, p) in phase space at the same time. The phrase ' phase space ' is used herein to mean the Cartesian product Ω×BZ whose volume is N c h 3 . Juste like r in ψ(r), r and p in f W (r, p) are not quantum operators but classical variables, namely the position of a geometric point and h _ times the reciprocal (Fourier) variable. In keeping with the cyclic boundary conditions and following the guidelines of [13], we shall consider all functions of r, such as the wave function ψ or the Wigner function f W to be introduced below, as restrictions to Ω of periodic functions on the unbounded position space, with period L in one dimension. Then, we can write ψ as a Fourier series,

ψ ^(p) ∫ BZ Ω ψ ^(p + q 2 ) ψ ^*(p -q 2 ) exp( iq . r h _ ) Ω d 3 q h 3
The function f W is real-valued. It has some, but not all, attributes of a probability distribution in phase space. The function provides the marginal distributions in position and in momentum, namely

∫ BZ f W (r, p) d 3 p h 3 = |ψ(r)| 2 , ( 10
)
∫ Ω f W (r, p) d 3 r = |Ωψ ^(p)| 2 ,
(11) whence

ψ(r) = Σ p ψ ^(p) exp( ip . r h _ ), (7) ∫∫ Ω×BZ f W (r, p) d 3 r d 3 p h 3 = 1.
(12) where p is a quantized momentum, and However, f W (r, p) d 3 r d 3 p/h 3 cannot be thought of as the probability of finding the particle in the volume d 3 r d 3 p.

For example, f W can take non-zero values at positions r (momenta p) where |ψ(r)| 2 (|ψ ^(p)| 2 ) vanishes. This happens in particular prior to the overlapping of two Gaussian wave packets which are about to meet together and interfere. The Wigner function takes significant values midway between the two packets, where ψ(r) vanishes and no particle can be detected [7,14]. Where ψ(r) vanishes, relation (10) implies that f W (r, p) takes both positive and negative values. It is said that f W (r, p)/h 3 is a quasi-probability density in phase space. Given a subvolume ϖ of Ω×BZ where f W < 0, the integral

ψ ^(p) ≡ ∫ Ω ψ(r) exp(-ip . r h _ ) d 3 r Ω (8)
has the same physical dimension as ψ. On restricting the summation over p to N c values, the equality (7) becomes approximate, but it is expected to become accurate in the limit N c → ∞. In the continuum limit (see Table 1), the equality ( 7) is replaced by

ψ(r) ≈ ∫ BZ ψ ^(p) exp( ip . r h _ ) Ω d 3 p h 3 . (9)
Likewise, we consider all functions of p, such as E kin (p) or the Wigner function f W (r, p), as restrictions to the BZ of periodic functions on the unbounded momentum space, with period h/a in one dimension. The Wigner function is given in Table 2 where the definitions only hold in the limits of large Ω and BZ. Note that, in Table 2, the integrand is periodic in s with period L in one dimension (respectively, periodic in q with period h/a) so that no boundary terms are produced upon integration by parts.

∫∫ ϖ' f W (r, p) d 3 r d 3 p h 3 (13)
over the complementary subvolume ϖ', such that ϖ ∪ ϖ' = Ω×BZ, will exceed unity. Likewise, if f W exceeds unity in one or more Planck cells h 3 , then because of (12) it will take negative values somewhere else. Because (13) may lie outside the range 0-1, it is a quasi-probability.

Thus, the function f W should be understood as a mathematical tool which fully specifies the state of the particle, just as the state operator |ψ)(ψ| also known as the probability density operator in quantum statistical mechanics. In particular, the Wigner function enables one to calculate the expectation value, denoted by an overbar, of observables such as E kin (p) ≡ p 2 /2m or U(r), in the manner of classical statistical mechanics, namely in the force ∇(-U 1 ) (this occurs in particular with a quadratic potential) and f W obeys Liouville's equation of classical physics, ∂f W ∂t

+ ( ∂E kin ∂p ) . ∇f W + ∇(-U 1 ) . ( ∂f W ∂p ) = 0. ( 23
)
But it should be kept in mind that f W /h 3 is only a quasiprobability density. If its integral over a subvolume of phase space is negative or lies above unity at t = 0, then at a later time this non-classical feature will show up in another subvolume. This comes about because the range of f W is preserved during the evolution, as can be seen by rewriting [START_REF] Verdet | Leçons d'Optique physique I[END_REF] as

E kin __ = ∫∫ Ω×BZ E kin (p) f W (r, p) d 3 r d 3 p h 3 , ( 14
) U _ = ∫∫ Ω×BZ U(r) f W (r, p) d 3 r d 3 p h 3 . ( 15
) f W (r, p, t + dt) = f W (r -( ∂E kin ∂p ) dt, p -∇(-U 1 ) dt, t).
The Wigner function also provides the probability current density according to the familiar formula, ( 24)

J = ∫ BZ ( ∂E kin ∂p ) f W (r, p) d 3 p h 3 , (16)
In many problems, the knowledge of the system cannot be specified as a pure state (reine Fall). Instead of one wave function ψ, one has to consider several wave functions ψ i with weights w i ≥ 0 such that Σ i w i = 1. The weights have to be specified at a given instant such as t = 0. The state of the system is then called a statistical mixture and it is embodied in the state operator Σ i w i |ψ i )(ψ i |. This straightforwardly carries over to the Wigner-function formalism, where the pure-state f W is replaced by a weighted sum Σ i w i f W,i . Because the weights do not depend on time, the Wigner function of the mixture obeys the same dynamical equation as f W .

where ∂E kin /∂p is the group velocity of the matter wave.

How does the Wigner function f W (r, p, t) change in time? Just as ∂ψ/∂t is a sum of kinetic-and potentialenergy contributions in Schrödinger's formulation of wave mechanics, here one can write ∂f W ∂t

= ( ∂f W ∂t ) kin + ( ∂f W ∂t ) pot . ( 17
)
The kinetic-energy contribution is calculated as

( ∂f W ∂t ) kin = -( ∂E kin ∂p ) . ∇f W . (18)
The next section is devoted to the mathematical specification of the environment, and how it affects the particle's behaviour according to classical physics. Its effect in quantum physics is dealt with in section 6.

The pattern is simple because third-and higher-order derivatives of E kin (p) vanish; otherwise, see [START_REF] Woodward | Electron nonlocality in semiconductors[END_REF]16]. The potential-energy contribution is calculated as

( ∂f W ∂t ) pot = 1 ih _ ∫ BZ Ω d 3 p' h 3 u W (r, p -p') f W (r, p', t), ( 19
)
5 The disordered environment as a static energyscape

where

u W (r, q) ≡ ∫ Ω [U(r + s 2 ) -U(r -s 2 )] exp(-iq . s h _ ) d 3 s Ω (20)
In this work the real-valued function U 2 is a random scalar field specified at every location by its mean, 〈U 2 (r)〉, and an unbiased fluctuation, u(r), that is to say and U ≡ U 1 + U 2 . The dynamical equation obeyed by the Wigner function is thus

U 2 (r) ≡ 〈U 2 (r)〉 + u(r). ( 25
) ∂f W ∂t + ( ∂E kin ∂p ) . ∇f W =
The brackets 〈..〉 denote the averaging over all instantiations (' realisations ') of the energyscape.

Averaging is an operation which associates with a random variable a non-random or ' sure ' number. A possible U 2 (r) in two dimensions is pictured in figure 2; the static function U 2 (r) may be thought of as a spacewise noise. The random field has a covariance C U ≡ 〈u(r) u(r + s)〉. The field is taken to be homogeneous (translationally invariant), i.e. 〈U 2 〉 and C U do not depend on r. We further assume isotropy (rotational invariance), i.e. C U is a function of the modulus |s| only. The function C U is characterized by the variance of the fluctuation, namely u 0 2 ≡ C U (s = 0), and its typical decay length, denoted by l c and called the correlation length. The Zitterkraft (stochastic force) exerted by the field is Z(r) ≡ ∇(-U 2 ). Homogeneity entails a vanishing 〈Z(r)〉, i.e. the Zitterkraft is unbiased. It is characterized by a covariance matrix which is derived from the covariance of the random potential energy,

1 ih _ ∫ BZ Ω d 3 p' h 3 u W (r, p -p') f W (r, p', t). ( 21
)
Alternatively, the right-hand side may be written in terms of the Fourier transform U ^(q) of U(r),

( ∂f W ∂t ) pot = i h _ ∫ BZ Ω d 3 q h 3 U(q) ^ [f W (r, p + q 2 ) - f W (r, p -q 2 )] exp( iq . r h _ ). ( 22
)
By integration upon momentum, the right-hand side of [START_REF] Henkel | Transfert radiatif et transport d'atomes[END_REF] does not contribute and we arrive at (6) given (10) and (16).

If U 2 is absent and U 1 is smooth enough to allow neglect of its third-and higher-order derivatives, the associated u W is L (∂U 1 /∂x) δ'(q) in one dimension. Then, the potential-energy contribution reduces to a differential term, namely ∇U 1 . (∂f W /∂p) in three dimensions. The rate of change of f W due to the potential energy becomes local

〈Z i (r) Z j (r + s)〉 = -( ∂ 2 C U
∂s i ∂s j ), i, j = x, y, z. [START_REF] Zhang | Fermi's golden rule: its derivation and breakdown by an ideal model[END_REF] first Born approximation of quantum scattering theory [START_REF] Landau | Quantum Mechanics: Non-Relativistic Theory 2nd edn[END_REF], it is calculated that

σ v (p) = 2σ 0 ( p 0 2p ) 4 {ln[1 + ( 2p p 0 ) 2 ] -[1 + ( p 0 2p ) 2 ] -1 }, (34)
As an example, consider that U 2 (r) is due to N s identical scatterers, of individual potential energy v(ρ), haphazardly interspersed in the volume Ω. In this example, where

σ 0 ≡ 4πρ 0 2 ( 2mv 0 p 0 2 ) 2 (35) U 2 (r) = Σ j = 1 ... N s v(r -R j ).
(27) This is called a Poisson random field in the mathematical literature. Its randomness lies in the position R j of each scatterer j = 1 ... N s . The probability of finding R j in a volume d 3 R j is d 3 R j /Ω, it being assumed that the number density of scatterers n s ≡ N s /Ω is low. An analytically workable example is the Yukawa potential, is its value as p → 0. The function σ v is plotted against momentum p in figure 4 along with c(p). It is seen that σ v (p) takes its highest value σ 0 as p vanishes. The momentum-relaxation rate 1/τ v is plotted against the momentum in figure 5. The rate takes its highest value ≈ n s σ 0 p 0 /2m as p ≈ p 0 /2; that is to say the de Broglie wavelength of the matter wave, h _ /p, is on a par with the range ρ 0 of the Yukawa potential.

v(ρ) = v 0 ( ρ 0 ρ ) exp(-ρ ρ 0 ), (28)
where |v 0 | and ρ 0 respectively assess the strength and range of the potential. In figure 2, N s = 3 and the coordinates have been scaled by ρ 0 . In this example, 〈U 2 〉 = 2πn s ρ 0 3 v 0 where n s ρ 0 3 is the number of scatterers in a volume ρ 0 3 . Taking this number to be much less than unity makes 〈U 2 〉 negligible. The covariance of this energyscape is calculated to be exponential [START_REF] Bar | Encyclopedic Dictionary of Condensed Matter Physics[END_REF]

σ v /σ 0 c = 1 -σ v /σ C U (s) = u 0 2 exp(- s l c ), (29)
where

l c = ρ 0 , ( 30
) u 0 = 2πn s ρ 0 3 |v 0 |. (31)
A weak disorder is obtained either in the weak-coupling (v 0 → 0) or the low-density (n s → 0) limit [START_REF] Erdös | Scaling limits of Schrödinger quantum mechanics[END_REF]. The covariance matrix of the force is

〈Z i (r) Z j (r + s)〉 = ( u 0 l c ) 2 exp(-s/l c ) s/l c [δ ij -( s l c + 1) s i s j s 2 ], i, j = x, y, z. ( 32
)
Fig. 4. The total scattering cross section (σ) and velocityrelaxation scattering cross section (σ v ) of the Yukawa potential v(ρ) are plotted as functions of the momentum p. The momentum scale p 0 is h _ /ρ 0 , where ρ 0 is the range of v(ρ). The cross sections have been scaled by the value σ 0 at a vanishing energy. Also shown is the ratio of velocity persistence c(p), or average direction-cosine of the scattering angle (p, p'), as a function of p = p'.

Since (u 0 /l c ) 2 = 2πn s ρ 0 3 (v 0 /ρ 0 ) 2 , the typical Zitterkraft u 0 /l c is smaller than the typical force |v 0 |/ρ 0 of a single Yukawa scatterer by a factor (2πn s ρ 0 3 ) 1/2 . Classically speaking the Zitterkraft scatters particles in a chance-like manner. The average momentum 〈p〉 of an ensemble of particles which start out with the same initial momentum is damped. The modulus p, however, is unchanged as scattering is elastic. In classical kinetic theory, the rate of momentum -or velocity-relaxation is

1 τ v = p m n s σ v , (33)
where σ v , called the velocity-relaxation scattering cross section off the potential v(ρ), includes velocity persistence in a scattering event. The velocity-persistence ratio c is defined as the average direction-cosine of the velocities before and after the event. For scattering by a Yukawa potential, at vanishing kinetic energies p 2 /2m the differential cross section becomes independent of the scattering angle so that c ≈ 0. At increasing energies, the differential cross section falls off increasingly rapidly as a function of the scattering angle. Once p >> p 0 ≡ h _ /ρ 0 , only a forward peak of angular width ≈ h _ /pρ 0 is left so that the collision becomes ineffective and c → 1. In the Often the quantum scattering cross section is quite different from the classical one. This is why frequently classical-physics models are built which make use of the quantum cross section or a phenomenological one. In those models, it is considered that in multiple scattering the probability current densities of matter waves add up incoherently, just like in optics the intensities of light beams reflected by several obstacles add up without interfering. Whereas single scattering is handled quantum-mechanically i.e. coherently, coherence effects are ignored in multiple scattering. The next section investigates the motion of a particle through a disordered set of obstacles within the Wigner formalism where coherence is fully accounted for.

If the weakness of the fluctuation can be assessed by a dimensionless parameter ε, we may attempt to express f W -〈f W 〉 as an expansion in powers of ε or equivalently

ε 1/2 , namely f W = f 0 + ε 1/2 f 1 + ε 1 f 2 + O(ε 3/2 ). ( 36
)
In this expression, f 0 ≡ 〈f W 〉 is independent of the realisation of the fluctuation u(r) while f 1 , f 2 ... do depend on the realisation. By construction of the expansion, 〈f 1 〉 , 〈f 2 〉 ... vanish 1 . To find out a small dimensionless parameter ε, we know of two small physical parameters, namely the strength u 0 of the fluctuation and its characteristic length l c . The former is taken to be small against the typical energy scale of f W ≈ 〈f W 〉, hereafter denoted by E typ and estimated as E kin __ such as given by ( 14). The length l c is taken to be much shorter than the scale over which 〈f W 〉 undergoes a significant variation. That scale, denoted by L (usually not identical with the L of section 3), depends on the applied force ∇(-U 1 ) or, in the absence thereof , the box side Ω -1/3 . Here L will be defined in connection with u 0 [16,[START_REF] Henkel | Transfert radiatif et transport d'atomes[END_REF]. Over a length l c , energy U 2 varies by about +u 0 or -u 0 with equal probabilities. Over a length L >> l c consisting of L/l c segments l c , the root-mean-square variation of U 2 is u 0 (L/l c ) 1/2 . A significant variation will ensue in the matter-wave field or its average Wigner transform, if u 0 (L/l c ) 1/2 is on a par with the typical energy scale E typ of f W . Thereby we let

Wave mechanics in a disordered environment

We have to do with a matter wave propagating through an energyscape which, as a stochastic field, can have many realisations. Given an initial condition ψ(r, t = 0), a matter wave has a unique history determined by the peculiar realisation of U 2 (r). Each one gives rise to a matter-wave field ψ(r, t > 0) or Wigner transform thereof. Thereby ψ and f W themselves are stochastic quantities. We have to consider a statistical ensemble of particles, and we are primarily interested in the average 〈f W 〉 over all realisations of U 2 . The fluctuating U 2 will entail a fluctuating Wigner function f W . If the fluctuation u(r) is weak, then a realisation of f W will be close to the average 〈f W 〉. This is sketched in figure 6. The fact that covariance is short-ranged implies that two contiguous volumes l c 3 are statistically independent. Thereby each one is the seat of a realisation of the random variable; and the spatial average over a large volume is identical with the average over a large number of realisations, that is to say an ensemble average [16].

ε 1/2 ≡ ( l c L ) 1/2 = u 0 E typ . ( 37
)
This yields L = l c (E typ /u 0 ) 2 . For u 0 much smaller than E typ , a length L much larger than l c is needed for the average Wigner function of the matter wave to be significantly affected by the small but numerous kicks.

The main correction f 1 to f 0 is expected to scale as the relative fluctuation in U 2 , namely ε 1/2 ; and ( 36) is also an expansion in powers of the perturbation strength u 0 . As a function of position, f W will vary over both the long scale L and the short scale l c . This state of affairs is reminiscent of the dynamics of a damped harmonic oscillator, the position function f of which varies over both the scale of the reduced period 1/ω (owing to the elastic force) and the scale of the decay time τ (owing to the damping force). Provided that 1/ω << τ, it is mathematically convenient to consider that the instantaneous oscillation f formally depends on two time variables, a ' fast ' one t' and a ' slow ' one t = εt', with ε standing here for the small dimensionless parameter (ωτ) -1 [START_REF] Nayfeh | Perturbation Methods[END_REF]. Accordingly, the time derivative of f(t, t') is ∂f/∂t + ε -1 (∂f/∂t'), and the second contribution dominates. In a similar way, in the present issue f W (r, r') will be 1 Actually it is not possible to make 〈f 1 〉 , 〈f 2 〉 ... simultaneously vanish if the expansion is limited to a finite order. Here we take 〈f 1 〉 = 0, and dropping the second-order term 〈f 2 〉 serves as a closure relation of the expansion [START_REF] Nayfeh | Perturbation Methods[END_REF]. Fig. 6. Typical position dependence of the Wigner function f W (x, p, t) in one space dimension x at given p and t. The smallscale fluctuation of the Wigner function arises from the envisaged as a function of two formally independent position variables, r' and r = εr' where ε is the ratio l c /L of the characteristic variation lengths, besides the variables p and t. The gradient of f W is ∇f W + ε -1 ∇'f W where ∇' ≡ ∂/∂r'. The deterministic potential energy is taken as a function U 1 (r) of the slow variable while the random fluctuation u(r') is a function of the fast variable. Omitting the p and t dependencies for a simpler notation, the expansion (36) of f W is recast as 2Lorentz-type scattering term on the right-hand side, in which W p,p' d 3 p'/h 3 is the transition probability per unit time that a particle of momentum p gets scattered to momentum p'. Equations (40)(41) mean that, if the scatterers are arranged independently, the total scattering is proportional to the number N s of scatterers.

We can see that W p,p' is given by first-order perturbation theory as originally worked out by Dirac [START_REF] Dirac | The quantum theory of emission and absorption of radiation[END_REF] and called the second golden rule by Fermi [START_REF] Fermi | Nuclear Physics[END_REF]. That golden rule plays a major role also in decoherence theory [1]; and it continues to arouse a number of comments [START_REF] Zhang | Fermi's golden rule: its derivation and breakdown by an ideal model[END_REF][START_REF] Merlin | Rabi oscillations, Floquet states, Fermi's golden rule, and all that: Insight from an exactly solvable model[END_REF]. We remark that strict energy conservation, as expressed by the delta function of energy in (40), is obtained here without assuming longlived plane waves. The time-energy uncertainty relation invoked in textbooks [10] is unnecessary. Even for a high scattering rate causing short-lived and energy-broadened states, expression (40)(41) of W p,p' is valid if used in (39) 3 . A similar conclusion had been reached in a different way in the case of a particle colliding inelastically [START_REF] Richards | Evolution of energy distribution in a model system without conventional Lorentzian lifetime broadening[END_REF].

f W (r, r') = f 0 (r, r') + ε 1/2 f 1 (r, r') + ε 1 f 2 (r, r') + O(ε 1/2 ). [START_REF] Balian | From Microphysics to Macrophysics[END_REF] It will be shown that the dominant term f 0 = 〈f W 〉 only depends on the slow variable r, in keeping with the fact that f 0 does not depend on the peculiar realisation of the fluctuation u(r'). Contrariwise the other terms f 1 , f 2 ... do vary over the small scale of the fluctuation. For the expansion to be a solution of the Wigner dynamical equation of section 4, the zero-order term will have to satisfy a certain equation. A closed-form equation not involving the higher-order terms is obtained in appendix A and it is discussed in the next section.

The present paper deals with a weak disorder; the opposite case of a strong disorder is shown in Table 3. The golden rule does not hold for a strong U 2 which cannot be handled perturbatively owing to the nonexistence of delocalized energy eigenstates of H (plane waves); and then localization in Anderson's sense arises [29]. In a weak disorder, a particle may be seen to propagate as a plane wave, the lifetime of which is finite owing to scattering events.

Classical statistical mechanics recovered

With the local force term included and letting v(p) ≡ ∂E kin /∂p, the function f 0 is governed by a linear Boltzmann kinetic equation of the Lorentz type [START_REF] Lorentz | Le mouvement des électrons dans les métaux[END_REF] (see appendix A), namely ∂f 0 ∂t

+ v(p) . ∇f 0 + ∇(-U 1 ) . ( ∂f 0 ∂p ) = Table 3. Weak vs strong disordered energyscape. ∫ BZ W p,p' d 3 p' h 3 [f 0 (p') -f 0 (p)] , (39)
Weak disorder We can see that 〈f W 〉 = f 0 obeys a closed-form equation involving the covariance of the random potential-energy field instead of the full probability law governing the random field itself. If u in C U (s) is written as a Poisson sum [START_REF] Merlin | Rabi oscillations, Floquet states, Fermi's golden rule, and all that: Insight from an exactly solvable model[END_REF] of N s identical potentials v(ρ) shifted at random with respect to one another, then in (40) we have to do with a random sum of N s independent phasors. The Verdet-Rayleigh theorem gives [START_REF] Verdet | Leçons d'Optique physique I[END_REF] With [START_REF] Bringuier | [END_REF](40) we have recovered classical nonequilibrium statistical physics, save for two points:

u 0 → 0 Strong disorder u 0 → ∞ where W p,p' ≡ 2π h _ ∫ Ω C U (s) exp( i(p -p') . s h _ ) d 3 s Ω δ(E(p') -E(p)). ( 40
∫ Ω 〈u(R) u(R + s)〉 exp( i(p -p') . s h _ ) d 3 s Ω = (
i) the classical scattering cross section is replaced by the quantum one in the first Born approximation;

(ii) the unknown f 0 is the disorder average of a quasiprobability density.

N s | ∫ Ω v(ρ) exp( i(p -p') . ρ h _ ) d 3 ρ Ω | 2 . ( 41
)
The first point should not come as a surprise: the size of atoms (bound states of electrons, as opposed to The right-hand side involves a matrix element of the particle-scatterer interaction potential reminiscent of the first Born approximation of scattering theory. Equation [START_REF] Bringuier | [END_REF] is the Liouville equation ( 23) supplemented with a 3 In textbook derivations of the golden rule, a pseudo-problem arises from arbitrarily picking out an initial time t 0 in addition to the current time t. In a rate-equation description, no duration ∆t = tt 0 and no energy uncertainty h _ /∆t arise. Accordingly, W p,p' reflects a tendency or propensity in line with a probabilistic description of the phenomenon. continuum states) is given by Bohr's radius a 0 which involves the non-classical constant h _ . For h _ → 0, a 0 vanishes and the discontinuous structure of matter does not arise. The emergence of classicality has to do with the second point. The function f W enfolds the information allowing to calculate the expectation values of most observables as if f W were a probability density in phase space. Its classicality is conditional upon or so. A sufficient condition to achieve this is a smooth variation of the potential energy, besides the initial condition 0 ≤ f W ≤ 1 at t = 0. A contrario, a fast spatial variation of the deterministic potential energy U 1 usually entails non-classical values of f W , as actually occurs in a resonant tunnelling diode where the particle is an electron and U 1 is the fast-wiggling applied electrostatic energy [START_REF] Mizuta | The Physics and Applications of Resonant Tunnelling Diodes[END_REF].

0 ≤ ∫∫ ϖ f W (r, p) d 3 r d 3 p h 3 ≤ 1, (42)
A non-classical behaviour also occurs in a disordered potential energy U 2 , but then it is liable to zoom out to scales exceeding the correlation length of the disorder. Averaging f W over a correlation volume l c 3 shrinks the range of 〈f W 〉 which may then fall within the 0-1 interval. To the lowest order in the disorder strength, it is found that 〈f W 〉 satisfies a closed-form linear Boltzmann equation of the Lorentz type. That kinetic equation contains a scattering term which further shrinks the range of values of 〈f W 〉. One classical feature is out of reach, however, namely an arbitrarily sharp function of (r, p) forbidden by Heisenberg's indeterminacy principle. whatever the subvolume ϖ of Ω×BZ, that is to say quasiprobabilities reduce to probabilities. On replacing f W by an approximation f 0 which ignores fine-grained details on a scale l c , the range of values spanned by the function is reduced and condition (42) is easier to satisfy. Then, f 0 admits of a classical interpretation. If (42) is not satisfied at t = 0, the dynamics will be non-classical at short times. But the Boltzmann-Lorentz evolution of f 0 combines advection in phase space, on the left-hand side of [START_REF] Bringuier | [END_REF] which preserves the range of values of f 0 as implied by [START_REF] Dirac | The quantum theory of emission and absorption of radiation[END_REF], with scattering, on the right-hand side of (39) which shrinks that range to one single value at long times as isoenergetic scatterings tend to distribute the values of p equally over the energy shell E kin (p) = E [16]. Thereby the approximation f 0 to f W will eventually satisfy condition (42) and it will become interpretable as the occupancy of a classical state defined as a pair (r, p) i.e. a point in phase space.

The Wigner representation treats position and momentum on the same foot; this is at odds with decoherence theory where position is often the preferred observable which becomes effectively superselected. The dynamical theory of decoherence involves a time scale τ dec much shorter than the characteristic damping time τ v of classical physics. In the present study no time scale is involved because the environment is a non-dynamical entity. Coherent i.e. non-classical effects are present in the correction f 1 to the classical behaviour embodied in f 0 . The coherence correction f 1 , so to speak a quantum noise, is given by (A.7) in appendix A. It depends on the peculiar realisation of the disorder, and it varies over the scale l c . This length is akin to the spatial coherence length in optics. Let us compare the matter-wave coherence length with the momentum-damping length of the random energyscape. The latter length is the mean free path λ(p) = v(p)τ v (p). An upper bound of the ratio l c /λ(p) is 2πn s ρ 0 3 (2mv 0 /p 0 2 ) 2 or, in terms of the stochastic energyscape, An important limitation is in order, however. Classical physics allows for occupancies f 0 such that both r and p are defined with arbitrarily sharp accuracy. This cannot arise in the present description because f W is bounded by a Cauchy-Schwarz inequality |f W | ≤ 8; see also [7]. This feature outlives the ε → 0 limit of f W in [START_REF] Balian | From Microphysics to Macrophysics[END_REF]. Therefore, not all features of classical physics are recovered in the ε → 0 limit. In this study the particle is elastically bounced off a static energyscape, which embodies a set of infinitely heavy scatterers. The perfect Lorentz limit of kinetic theory ensues. Appendix B discusses the modification brought about when scatterers are not infinitely heavy; and this is tied to the distinct issue of thermalization by the environment.

l c λ(p) ≤ ( u 0 E 0 ) 2 , (43)
where E 0 stands for (h _ /l c ) 2 /2m. The waning of the phase coherence takes precedence over damping if u 0 < E 0 . This inequality may be viewed as a condition on the rootmean-square amplitude of disorder u 0 being less than E 0 at a given l c , or as a condition on the correlation length of disorder l c being smaller than h _ /(2mu 0 ) 1/2 at a given u 0 . In the present approach, a classical behaviour is not obtained dynamically after a time exceeding a decoherence time scale. Instead, this behaviour comes out by zooming out to spatial scales exceeding a coherence length equal to the correlation length of the static disordered environment. Classical physics emerges (subject to some limitations) as a coarse-grained description of the quantum reality.

Closing summary

This paper has made use of the Wigner function, instead of the state vector or von Neumann's state operator, to describe the quantum state of a simple system. The Wigner function is a tool which allows one to calculate the expectation values of most major observables as did the phase-space occupation function in classical statistical physics. Actually, however, f W /h 3 is a density of quasi-probability in phase space, whose integral over a subvolume of phase space may lie outside the range 0-1. A strongly non-classical dynamics is typified by a large positive quasi-probability being almost cancelled by a negative quasi-probability to give the normalization (12) to unity. On the contrary, in nearly classical dynamics, the integral of f W /h 3 over any subvolume lies within 0-1 I am indebted to Jean-Claude Serge Lévy (Université de Paris-Denis Diderot) for inviting me to participate in the Eighth Complexity-Disorder Meeting.

Appendix A

Now, if a motionless scatterer has a mass m e largely exceeding m, an m-m e scattering event causes a small recoil of m e whereby the guest particle imparts kinetic energy to the environment hosting it. In this imperfect Lorentz model (1 << m e /m < ∞), scattering events are slightly inelastic; and the energy-relaxation time τ E (p) ∝ m e /m is not infinite although it largely exceeds the momentum-relaxation time τ v (p). The latter assesses the friction undergone by the particle, i.e. the damping of the ensemble-averaged velocity. If the heavy scatterers are not motionless but thermally agitated at a well-defined and stable temperature T e , they tend to bring the guest particle's momentum distribution to a thermalequilibrium one at the temperature T e . In other words, the scatterers act, with respect to the guest particle, as a thermostat 4 . It is possible to extend the present treatment to a random time-varying environment, as was done by Henkel [START_REF] Henkel | Coherent transport[END_REF], and then the particle can undergo inelastic scattering events. To the lowest order in u, in the golden rule the delta function δ(E -E') becomes δ(E ± h _ ω(q) -E') where h _ ω(q) is the energy quantum of a vibrational mode of the environment. Thermalisation, however, does not arise unless the time-varying field is quantized [16].

Multiple-scale expansion of the Wigner dynamical equation

Plugging the expansion (38) into the dynamical equation [START_REF] Henkel | Transfert radiatif et transport d'atomes[END_REF][START_REF] Lorentz | Le mouvement des électrons dans les métaux[END_REF] results in

1 ε v . ∇'f 0 + 1 ε v . ∇'f 1 + ε 0 ( ∂f 0 ∂t + v . ∇f 0 + v . ∇'f 2 ) + O( ε) = ( ∂f 0 ∂t ) u + ε( ∂f 1 ∂t ) u , (A.1)
where the ∇U 1 contribution is omitted. The right-hand side of (A.1),

( ∂f ∂t ) u ≡ i h _ ∫ BZ Ω d 3 q h 3 u ^(q) [f(r, r', p + q 2 ) - f(r, r', p - q 2 )] exp( iq . r' h _ ), (A.2)
is the rate of change of f ≈ f 0 + ε f 1 due to the Zitterkraft. We note the scaling

( ∂ ∂t ) u ∝ Z ∝ 1 ε . (A.3)
Equating terms of like powers on the two sides of (A.1) yields a recursive chain of equations,

The opposite limit of a heavy guest particle knocked about in a host medium of light scatterers (m e << m) is called a Brownian particle; one also speaks of a Rayleigh particle [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF][START_REF] Lifshitz | Physical Kinetics[END_REF][START_REF] Bringuier | [END_REF]. This is shown in Table 4. The particle interchanges energy with the environment when bouncing off the light scatterers. This allows for the thermalisation of the guest particle at the temperature T e of the environment. The mathematical proof of this statement is simple because the scattering operator of Brownian motion is differential instead of integral as occurs in the Lorentz case, and it is given here.

O(ε -1 ) v . ∇'f 0 = 0, (A.4) O(ε -1/2 ) v . ∇'f 1 = ( ∂f 0 ∂t ) u , (A.5) O(ε 0 ) ∂f 0 ∂t + v . ∇f 0 + v . ∇'f 2 = ( ∂f 1 ∂t ) u . (A.6)
At the leading order (ε -1 ), we find that f 0 (r, r') does not depend on the fast variable, so that it will be denoted by f 0 (r). In other words, in f 0 short-scale details of f W are dismissed upon zooming out to scales exceeding l c . The next-order (ε -1/2 ) equation may be solved in the distributional sense upon Fourier transformation of f 0 and f 1 in the variable r', Table 4. Light particle (m << m e ) governed by the Lorentz integro-differential equation contrasted with heavy (Brownian) particle (m >> m e ) governed by the Klein-Kramers differential equation.

f 1 (r, r') = ( ∫ BZ Ω d 3 q h 3 u ^(q) f 0 (r, p + q 2 ) -f 0 (r, p -q 2 ) -iη + q . v(p) × exp( iq . r' h _ )) η → 0 . (A.7)

Lorentz particle m << mass of environmental scatterers

Brown particle m >> mass of environmental scatterers Finally, at order ε 0 , disorder averaging yields 〈f

0 〉 = f 0 and 〈∇'f 2 〉 = ∇'〈f 2 〉 is neglected, whence ∂f 0 ∂t + v . ∇f 0 = 〈( ∂f 1 ∂t ) u 〉. (A.8)
Large change of p per scattering event Small change of p per scattering event Now f 1 is known from f 0 according to (A.7). Thereby a closed-form equation on the zero-order function f 0 is obtained. The calculation of the right-hand side of (A.8), which involves the Fourier transform C ^U(q) of C U (s), is given elsewhere [16,[START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF]. Equation [START_REF] Bringuier | [END_REF](40) The scattering term on the right-hand side is differential, instead of integral, in p because momentum is infinitesimally changed in a scattering event with an environmental particle of mass m e << m. In the scattering term, T e denotes the temperature of the environmental particles, assumed to be in thermal equilibrium (this demands that T e be homogeneous), and γ is the rate of momentum relaxation in force-free motion (U 1 = 0). The rate γ is expressible in terms of the m-m e differential scattering cross section [START_REF] Balian | From Microphysics to Macrophysics[END_REF]. Unlike the momentumdependent rate 1/τ v (p) of section 5, here γ is an average over the thermal distribution of momenta of the environmental particles. The mean kinetic energy u(r, t) This is a local energy-balance equation in which the source term, on the right-hand side, is the net rate of energy change per unit volume. Now u is 3kT/2 for a thermal-equilibrium momentum distribution of the guest particle at temperature T. Thus, the net rate of energy change is positive if the environmental bath has a temperature T e higher than the guest particle's temperature T so that the bath heats up that particle. On the contrary, a particle which is ' hotter ' than the environment, i.e. such that u > 3kT e /2, will be cooled down until it eventually gets thermalized at the bath's temperature. To finish with, we note from (B.6) that the rate of energy variation of the Brown particle is twice the rate γ of momentum variation, in stark contrast with the Lorentz particle for which the rate 1/τ E is much less than 1/τ v [START_REF] Bringuier | [END_REF].

of

Fig. 2 .

 2 Fig. 2. Random potential-energy field U 2 (x, y) in two dimensions x and y. The field shown is a sum of three Yukawa potentials of identical strength and range. The potentials only differ in their central positions. Two instantiations (' realisations ') of the random field are shown in the figure.Fig. 3. Cellular functions w n,m complementary to the discrete plane waves, in two dimensions x and y. Two functions are shown in the figure, namely w 5,2 and w -3,-6 , centred at {x n = 5a, y m = 2a} and {x n = -3a, y m = -6a} respectively. The cellular functions make up an orthonormal basis set of the Hilbert space. In the continuum limit a → 0, the cellular function w n,m tends to δ(xx n )δ(yy m ).

Fig. 3 .

 3 Fig. 2. Random potential-energy field U 2 (x, y) in two dimensions x and y. The field shown is a sum of three Yukawa potentials of identical strength and range. The potentials only differ in their central positions. Two instantiations (' realisations ') of the random field are shown in the figure.Fig. 3. Cellular functions w n,m complementary to the discrete plane waves, in two dimensions x and y. Two functions are shown in the figure, namely w 5,2 and w -3,-6 , centred at {x n = 5a, y m = 2a} and {x n = -3a, y m = -6a} respectively. The cellular functions make up an orthonormal basis set of the Hilbert space. In the continuum limit a → 0, the cellular function w n,m tends to δ(xx n )δ(yy m ).

3

  The quantum states of a particle in a boxThe mathematical set-up of the quantum states is simpler in a finite volume such as a cubic box of side L = Ω -1/3 [10]. With cyclic (Born-von Kármán) boundary conditions, the momentum p of a plane wave b p (r) = exp(ip . r/h _ )/Ω 1/2 takes discrete values. In one dimension they are p n = n(h/L) with n = 0, ±1, ±2 ... Cyclic boundary conditions allow for a steady particle flow through the volume with a probability-current density (to be defined below) J = Ω -1 p/m. As two neighbouring values of p n differ by h/L, the momentum-space volume per state is (h/L) 3 = h 3 /Ω in three dimensions. It is mathematically easier to restrict n to N c values, i.e. -N c /2 < n ≤ +N c /2, and eventually let N c → ∞. The quantum states make up a Hilbert space of finite dimension N c . The momentum space has a volume N c (h 3 /Ω) in three dimensions. The case of an unbounded continuum is recovered by letting Ω → ∞ and N c → ∞ [10, 12].

  Velocity persistence ratio c(p)Scaled total and velocity-relaxation cross sections Scaled momentum p/p 0 σ/σ 0

  p/p 0 Momentum-relaxation rate 1/τ v ∝ pσ v Momentum-relaxation cross section σ v /σ 0

Fig. 5 .

 5 Fig. 5. The rate of momentum relaxation (1/τ v ) is plotted as a function of the scaled momentum together with the momentum-relaxation scattering cross section (σ v ) of the

P

  osition x (arb. units)Wigner function f W (x, p, t) due to U 1 + U 2 Disorder-averaged 〈f W 〉

γ

  div p [pf + mkT e ( ∂f ∂p )]. (B.1)

j 2

 2 U (r, t) ≡ ∫ E kin (p) v(p) f(r, p, t) d 3 p h 3 . (B.4) 10. A. Messiah, Mécanique quantique (Dunod, Paris, 1959) sections V.11, VIII.13 and XVII.4 From the Klein-Kramers equation (B.1), we can derive a Maxwell equation of change of the energy [40, 41]. Multiplying the kinetic equation by E kin (p) and integrating over momentum yield 11. E. C. McIrvine and A. W. Overhauser, Phys. Rev. 115 (1959) 1531-1536 12. J.-L. Calais and W. Weyrich, Finite and infinite Born-von Kármán regions, Int. J. Qu. Chem. 63 (1997) 223-227 ∂(nu) ∂t+ div j U = -γ ∫ v(p) . [pf + mkT e ( ∂f ∂p )] d 3 p h 3 , (B.5) 13. P. Degond and P. A. Markowich, A quantumtransport model for semiconductors: the Wigner-Poisson problem on a bounded Brillouin zone, M2AN. Mathematical modelling and numerical analysis 24 (6) (1990) 697-709 where the right-hand side has been integrated by parts. Given that p . (∂f/∂p) = div p (pf) -3f, we finally obtain ∂(nu) ∂t + div j U = -2γn(u -3 kT e ). (B.6) 14. L. E. Ballentine, Quantum Mechanics: A Modern Development (World Scientific, Singapore, 1998) chap. 15

Table 1

 1 shows that, in this double limit, a Darboux sum is replaced by a Leibniz integral and a Kronecker symbol goes over to a Dirac delta function. As noted by Messiah [10] and expressed by the notation ≈ in Table1, this procedure is not mathematically rigorous, but it provides the physically pertinent results.

Table 1 .

 1 Discrete vs continuum description.

	Darboux sum	↔ Leibniz integral

Table 2 .

 2 Definition of the Wigner function f W from the wave function ψ in either the position or the momentum representation.

	ih _ ∂ψ ∂t	= H ψ,	(5)

where ψ(r, t) is a complex-valued function. Its squared modulus is the particle's probability of presence per unit volume at position r and time t. Its argument ϕ(r, t) provides the probability-current density J(r, t) = |ψ| 2 h _ ∇ϕ/m appearing in the local conservation equation,

Wave function

Wigner function f W (r, p)

  the guest particle is given by 4. D. Giulini, J. Kupsch, C. Kiefer, E. Joos, I.-O. Stamatescu and H.-D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, Berlin, 2003 5. K. Camilleri, A history of entanglement: Decoherence and the interpretation problem, Studies in History and Philosophy of Modern Physics 40 (2009) 290-302 M. Schlosshauer, Quantum decoherence, Phys. Reports 831 (2019) 1-57 6. C. Cohen-Tannoudji, Théorie quantique du cycle de pompage optique, Annales de Physique 7 (1962) 423-461 7. C. Cohen-Tannoudji, B. Diu and F. Laloë, Mécanique quantique tome III (CNRS Editions, Paris, 2017) pp. 636-637 and 735-769 nu = ∫ E kin (p) f(r, p, t) d 3 p Einstein and L. Infeld, The Evolution of Physics (Cambridge University Press, Cambridge, 1938) p. 313 Likewise, the energy-current density (reckoned in watt per m 2 in SI unit) is given, in the absence of U 1 (r), by 9. E. Schrödinger, Are there quantum jumps? Part II, British J. Phil. Sci. 3 (1952) 233-242

	h 3	(B.2)
	h 3 .	(B.3)	8. A.

where n is the particle density, C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics vol. 3 (Wiley, New York, 2020) pp. 2199-2120 and 2297-2323 (English translation) n(r, t) ≡ ∫ f(r, p, t) d 3 p

The general scheme of a multiple-scale expansion involves an infinite ladder of scales[START_REF] Nayfeh | Perturbation Methods[END_REF], while this expansion of f W is limited to two length scales.

Appendix B Non-static energyscape and thermalization

In the perfect Lorentz limit considered in section 7, energy is exactly conserved in a scattering event, which is to say that the energy-relaxation time τ E is infinite. 4 There is no interaction between guest particles which would mediate energy, which is the usual thermalisation mechanism in a molecular gas.

In classical statistical physics, the Brownian particle's phase-space occupation f(r, p, t) obeys the Klein-Kramers kinetic equation [START_REF] Klein | Zur statistischen Theorie der Suspensionen und Lösungen[END_REF][START_REF] Kramers | Brownian motion in a field of force and the diffusion model of chemical reactions[END_REF][START_REF] Balian | From Microphysics to Macrophysics[END_REF][START_REF] Bringuier | [END_REF],


Into the 21 st century : The integration of principles of global governance in space law

This analysis aims at proposing a fresh legal view of the Outer Space Treaty (OST) that would conciliate the presence of both nations and private entities in the industry. The latter, who has been revealed to be a major player of the space industry in the last two decades, is today slowed down by the absence of clear international legislation in several domains pertaining to the exploitation of outer space. In order to allow for the growth of the space sector, it then seems essential for the international community to act accordingly. On the 50 th anniversary of the Outer Space Treaty, it is fitting to envisage a substitute that would need to take into account the commercial aspirations of States, but also protect the ability of firms to generate profit in order to foster investments. This could be accomplished by integrating principles of global governance to international space law in order to dynamize the sector, all the while setting up a regime that affirms the status of States as patrons of the protection of the general principles of the original OST. This way a balance could be found between the ambitions of the private sector and the sovereignty of nations.

Introduction 1 While we are already well into the 21 st century, it appears clearer now that new actors are progressively becoming the leaders of the space industry to the detriment of States. Nations have been the primary subjects of international space law ever since the Outer Space Treaty of 1967, and are still for a large part responsible for the production of norms pertaining to space activities, but the private sector is becoming increasingly important to the survival and development of the industry.

Private actors are indeed not only a major source of financing, but also the instigators of space projects on a grand scale, and their expansion in the space sector is steadily ousting nations as the number one space actors. This tendency is not a particularity of the space industry, but only a mere facet of a global phenomenon of denationalization of international law 2 . It seems it should be time for the international community to react accordingly and to adapt space law to the challenges it faces today.

The main issue is that while the OST implemented a number of general principles with the intent of regulating a space-race between opposing superpowers, it left out many important details on the account of there not being sufficient technology to fully exploit outer space at the time. As a result, vague definitions and divergence of interpretations 3 have led most private actors to hold back on their projects, fearful that there would be unforeseen consequences to their space activities. These divergences, ranging from the definitions of 'space object' or 'mankind', to the many interpretations of the non-appropriation principle or the delimitation of outer space, have made investors wary of the potential fallout of their businesses. For example, the actual effervescence about the exploitation of natural resources of outer space, and the relative failure of the Moon Agreement of 1979, have proven the necessity to calm the fears of the private sector concerning the "equitable sharing by all 1 Valentin DEGRANGE, valentin.degrange@hotmail.fr, Université Jean Moulin Lyon III. 2 Christian CHAVAGNEUX, "La montée en puissance des acteurs non étatiques", in JACQUET Pierre, PISANI-FERRY Jean, TUBIANA Laurence (eds), Gouvernance mondiale, (Rapport du conseil d'Analyse économique n°37, La documentation française, 2010) 233 ; Lider BAL, Le mythe de la souveraineté en Droit international, La souveraineté des États à l'épreuve des mutations de l'ordre juridique international (Thesis, Centre d'études internationales et européennes, Université de Strabourg, 2012). 3 Brendan COHEN & Elena CARPNELLI, "Interpretating 'Damage Caused by Space Objects' under the Liability Convention" (2013), 56, 56 th IISL colloquium on the law of Outer space, Proceedings of the International Institute of Space Law; E. Fasan, "The meaning of the term mankind in space legal language" (1974), Journal of space law, 125; Fabio TRONCHETTI, "The non-appropriation principle under attack: using article II of the Outer Space Treaty in its defense" (2007) State parties in the benefits derived from those resources"4 .

This has unfortunately slowed down the development of space activities since the beginning of the 21 st century and condemned private entities to a standstill while nations are still arguing over the interpretations of the articles of a treaty concluded 50 years ago. A possible solution to this legal deadlock between states would be to try and conciliate the presence of both nations and private actors in the industry. This would mean taking into account the commercial aspirations of States on the one hand, and protecting the ability of firms to generate profit in order to foster investments on the other hand. Such a goal could be accomplished by integrating principles of global governance5

to international space law and allowing bottom-up international lawmaking6 to become an effective way of producing space law, all the while setting up a regime designed to temper the influence of private actors in order to keep the general principles of the original OST from being violated.

I. Governance in Outer Space, an instrument of development

In order to make international space law a more comprehensive set of rules, lawmakers would have to take into account the needs of the industry and its actors in a dynamic sector, and not only national interests of states that invariably end up slowing down or even blocking its development. To do so, this analysis proposes to integrate principles of governance to space law as well as to establish an institutional framework adapted to the specificities of space activities.

Governance is a controversial notion because it is defined in a variety of ways, sometimes contradictory, but also used in a variety of fields (public, private, environmental, corporate, global, etc.). For the purpose of this discussion, it will be defined as the establishment of policies, and continuous monitoring of their proper implementation, by the members of the governing body of an organization. It includes the mechanisms required to balance the powers of these members, with the associated accountability, and to enhance the prosperity and viability of the organization through efficiency. According to Stoker 7 , governance implies five essential elements: the intervention of many actors, which do not all belong to the governmental sphere; an erasure of frontiers between public and private sectors; an interdependency between the institutions associated with collective action; there are networks of autonomous actors; and finally, a possibility of acting without relying on state power or authority. In the present case, we will focus on global governance, which can be defined as "the complex of formal and informal institutions, mechanisms, relationships, and processes between and among states, markets, citizens and organizations, both inter-and nongovernmental, through which collective interests on the global plane are articulated, right and obligations are established, and differences are mediated" 8 . Rosenau has used the term "global governance" to denote the regulation of interdependent relations in the absence of an overarching political authority [START_REF] Roseneau | Toward an Ontology for Global Governance[END_REF] .

In the field of space activities, that last statement is particularly accurate considering that space law is a part of international law in which the primary subjects are states. In regard to the principle of sovereignty, proclaimed in the UN Charter [START_REF]Charter of the United Nations[END_REF] , and to the Article I of the OST [START_REF]Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, Including the Moon and Other Celestial Bodies[END_REF] which states that the freedom of exploration and uses of outer space only benefits nations, the absence of an overarching political authority competent with all space-related matters is especially noteworthy.

International space law, up until now, has been laboriously produced through lengthy negotiations between sovereign states defending national interests. Thus, it only encompasses general principles and grand declarations of intention which, while setting up major tenets of this particular field of activity, only grazed the surface and never offered an in-depth international regime addressing the specifics of space activities. Even worse, the lack of a centralized means of lawmaking or even of a way of coordinating national legal efforts to regulate all aspects of space activities, or at least of harmonizing existing laws, seems to have provoked a sort of regulatory competition between states [START_REF] Dimitri | The impact of national space legislation on private space undertakings: a regulatory competition between States?[END_REF] . In order to attract investments and to encourage private firms to base themselves within their borders, nations now adapt their legislation regarding for example the authorization of space activities as well as liability and insurance requirements. The additional observation made concerning the increasing implication of private entities, and the subsequent multiplication of actors involved in space activities, proves the necessity of the establishment of an effective way of regulation.

there are several possibilities that need to be addressed here. Possibilities that can be sorted out in two different kinds of models: classical models and emerging models. These models are directly inspired from those of global governance 13 , considering the nature of space activities. In order to offer an exhaustive analysis of these institutional shapes, it is imperative to start with the two classical forms of global governance even though they don't offer an operational model in today's economy. The first of these two classical models is that of the world government 14 , which is basically the transposition on a global scale of a federal government. It would then use a single unified set of rules under the supervision of a global parliament. There are many advantages such a model could procure to the world and to the space industry in particular -a unique set of rules for all private actors of the world, space programs run by the global government, etc. -and it has sparked many brilliant ideas from researchers all over the world (e.g. the Tobin Tax 15 , the cosmopolitan democracy of Held 16 , or the global federalism of Dani Rodrik 17 ). It is however very unlikely that we will ever see that kind of model in action. The mere example of the European Union shows the difficulty to unify different countries under one federal government. Furthermore, it seems unnecessary to go to such length only to make the space industry more efficient.

The second classical model of governance that needs to be presented here is that of institutionalized cooperation of nations 18 . Contrary to the world government which is largely of an hypothetical nature, this one is on the contrary quite operational as it is the one that has been used ever since the end of the second world war. According to this neo-realistic approach, states are the sole source of legitimacy and organize themselves to cooperate according to their needs. This cooperation can be somewhat limited (e.g. League of Nations) or more demanding (e.g. United Nations), but is of an inter-governmental nature in any case. In effect, it is more or less the model that has been used in space-related matters since the first Space Race. Nations are asked to cooperate, either through the system of the UN or all five space treaties, but the level of cooperation between states is still relatively low. Space powers generally prefer to conduct their own activities and confine situations of cooperation to specific operations, as it is the case with the International Space Station for example 19 . While it seemed like the best model to use for a long time, the institutionalized cooperation of nations does have limits that prevents it from being an effective way to govern space activities in today's world. The strength of this model lies in the fact that it should combine the effectiveness of proven solutions with the legitimacy enjoyed by democratic governments. However, it is unsatisfactory on both accounts because when national interests differ, intergovernmentalism transforms every issue into a bargaining object, resulting in the formation of coalitions and the elaboration of compromises whose efficiency is questionable at best. In conclusion, this model of governance, while respecting principles of sovereignty and making nations the sole originators of laws, also slow down the development of the space industry in that every step forward becomes incredibly difficult especially since it is already worth over US$330 billion 20 and generates hundreds of thousands of jobs.

The classical models of global governance seem to be rather ineffective at being applied to space activities and already don't correspond to the reality of contemporary economy, but it is however conceivable that the emerging models of governance that recently sprung up in international law could. The classical models are based on the assumption that legitimacy is derived from elections, while in fact most specialized institutions (both national and international) do not humor democratic requirements. Experts are appointed, not elected, in order to bring a certain competence to the table and to make decisions based on efficiency. The models of governance that have been emerging in the international society for the past few years clearly stand on that side of the line, and will be presented here as potential solutions for a governance of space activities.

Emerging models of global governance

The first of these models is that of the network of independent authorities 21 , which is the most representative of these emerging models. The literal or figurative "shareholders" of these authorities are the States. Therefore, their legitimacy is ultimately based on the democratic process that has established their missions and the method of appointing their officials, specified their obligations of transparency and the conditions under which they are to report on the execution of their mandate, but their distance from politics is an asset rather than a handicap. Tirole 22 reckons that delegation to an independent body is preferable to political responsibility, for example when decisions are too technical for voters to exercise direct control over the elected officials, when consequences are only known with great delay, or when the preferences of the majority can be severely harmful to a minority. This model has a certain relevance in terms of global governance, and even more so concerning space activities where institutions of experts could be given authority to legislate on space matters.

The second model that is used in international law and which could be used to set up a space governance is the model of Law without States 23 . It is based on the action of supranational judicial bodies and its central argument is that on the basis of a legal corpus (which can be very limited), the dynamics of jurisprudence are likely to produce an efficient legal system24 . This model builds on both the movement of private international law production at the initiative of enterprises, and the intensification of production of public international law following the establishment of the World Trade Organisation's (WTO) Dispute Settlement Body (DSB). The first movement, through the combination of private and public initiatives, led to the emergence of a "law of globalization" with an economic vocation. The second is sufficiently striking for having given rise to the disputes that we know. In both cases it is no longer possible to ignore the fact that new sources of law have emerged, beyond the usual procedure of international negotiation between governments. In the specific context of space law, it can be noted that no international jurisdiction except the International Court of Justice (ICJ) is competent to sit in judgment for space-related litigation. The creation of a specialized supranational judicial body however, could be a sufficiently flexible method of creating standards to accommodate the evolutions of space technologies and activities. It might not only be the occasion to put specialized judges up to the task but also to allow non-states parties, such as private firms, to make their cases before the court. In any case, the creation of such a judicial body might become necessary considering that with the increase in space activities, the number of disputes will almost certainly escalate as well and the ICJ is not equipped to deal with a potentially huge number of cases.

Finally, the last emerging model of global governance that could be used to foster the development of the space industry is that of private auto-regulation 25 . It would indeed be foolish to limit the scope of alternatives to public regulations. In their absence, private regulations soon take root, occupy space, create norms and establish jurisprudence. This is eloquently illustrated by the case of the Internet, where, despite its weaknesses (weak legitimacy, uncertain authority, institutional complexity, jurisdictional conflicts, legal uncertainty, lack of sanctioning power), the Internet Corporation for Assigned Names and Numbers (ICANN) and other private regulators have, thanks to their speed and flexibility, decisive advantages over public regulation, and essentially determine the rules of the game. Furthermore, the modalities of private regulation can be sources of inspiration for public regulation. To overcome the shortcomings of traditional public regulation, Rischard has recently proposed the creation of Global Issues Networks26 , which involve governments, civil society, companies and international organizations, on a number of international issues. The objective would be to produce recommendations or non-coercive codes of conduct.

Rather than waiting for the implementation of general obligations and associated compliance mechanisms, Rischard proposes to rely on soft law tools, and on reputation effects to ensure their effectiveness. Regarding the space industry, this model could present the undeniable advantage of putting the responsibility of regulating space activities in the hands of those directly involved: professionals and industrials. This form of bottom-up lawmaking27 would not only allow for the growth of the space sector but would also be able to cope with its rapid evolution, contrary to public instances.

In conclusion, it seems that there are a number of options at the disposal of the international society in order to not only regulate space activities more efficiently but also to enable the law to keep up with their constant development. Any of these models could be implemented by a treaty and then used to apply principles of good governance to the space sector. However, it should be kept in mind that the OST has proclaimed general principles of great moral value and that those principles should be protected by a regime that takes into account not only the needs of the industry, but those of all nations as well. Thus, it would seem imperative for the international community to take both into account if it were to ever amend the Outer space Treaty, or to conclude a new one.

II. The necessity to mitigate the influence of the private sector

Indeed, though it may seem desirable to include private actors in the lawmaking process, (either through specialized institutions, judicial precedent or private regulation) in order to make the industry more competitive and foster its development, precautions must however be taken in order to avoid a tyranny of the market. Unbridled private space activities could go against all the principles that were proclaimed in the OST to protect outer space as a res communis as well as the rights of all nations of the world. The idea is then for the international community to build a regime that stimulate the growth of the space industry by making laws that fill in the blanks left by the OST and by offering satisfying interpretations of obscure points of regulations, all the while conciliating with the intention of the general principles proclaimed by the treaty.

The most relevant example would be that of the non-appropriation principle 28 , especially considering the interest recently expressed by numerous private firms to exploit the natural resources of outer space [START_REF]Several examples can be given here, such as Planetary Resources[END_REF] . If the 1967 Treaty prohibits the national appropriation of outer space, the Moon and other celestial bodies, it does not explicitly forbid other forms of appropriation. Over the years, several interpretations of Article II of the OST have been made. Some people think that individual ownership by a private company or an international organization is therefore possible, since it is only a question of national appropriation [START_REF] Gorove | [END_REF] . Others believe, on the contrary, that the principle of non-appropriation is absolute and prohibits not only the creation of public rights but also private rights, first because Articles VI and VII cover both governmental and nongovernmental activities and second because the preparatory work of the 1967 treaty shows that the drafters' will was "to totally prohibit national appropriation" [START_REF] Williams | [END_REF] . Finally, a third, more nuanced interpretation, indicates that there is a flagrant but not intractable contradiction between the idea of freedom of use and that of non-appropriation. The installation of bases, factories, airstrips laboratories, etc. on a celestial body will create de facto exclusive rights of use on the corresponding plots of land. The possibility for entities to lease or sell the facilities that it has built will give rise to the legal title "real estate" and not "land" and therefore respecting the principle of nonappropriation. We can therefore envisage private rights relating to the use of body and of this space [START_REF] Mireille | Droit Spatial[END_REF] . The fact is that the lack of action from the international community concerning the interpretation of Article I has led private actors to hold back any project of exploitation of natural resources, which only changed recently with the recent adoption of the "Space Act" in the USA [START_REF]114th Congress[END_REF] .

results of any excavation, transformation, or trade, of the non-renewable natural resources of the celestial bodies, must be "de lege lata" for the benefit of all States, irrespective of their stage of development " 35 . We can assume that the same could be said about any form of commercial space activity. The main argument here is that redistribution of profits is perfectly consistent with and follows logically Article I of the 1967 Treaty. Nothing, therefore, prohibits the lucrative private commercial activity as long as the benefits derived therefrom are universalized either by being redistributed or by serving the common interest of all countries. For this reason, this interpretation is the most popular among developing countries. This theory, however, is questionable for several reasons. First of all, it can be considered "unnatural" in regard to commercial motivation. Second, it is contrary to the general principle of equity of international law, since only a few countries would then contribute to the development of space activities while the benefits are redistributed to all. And finally, the question of losses is totally ignored, to speak only of the benefits, whereas it would be logical in a scheme of universal collectivization to share the two. One could, however, imagine an equitable and proportional distribution of the efforts made for the development of these activities in order to reconcile principles of common interest and private interest.

Restrictive Interpretation

The second interpretation is the restrictive interpretation, according to which the article I of the 1967 Treaty is interpreted as a mere declaration of principle expressing a wish 36 . This principle would therefore not be self-executing. Only outer space and celestial bodies, or scientific information derived from their exploration, should be used for the benefit of all, the economic results of their use normally belonging to the State / group of states / private firm that acquires them. That is the most reasonable interpretation, if only because it preserves the interest of the private sector, driven by the search for profit, to actually invest in the space industry. More subtle solutions must then be found to promote the equal exploitation of space, depending on the situation and the type of exploitation envisaged: institutionalized sharing of technologies, establishment of concessions for mining activities, etc. At first glance, the idea of making private companies an official actor in space activities and especially in normative production, through the implementation of an adapted model of governance, does not necessarily give advantages to a particular State or group of States. Yet, the fact is that the majority of these firms are of the nationality of the main space powers and this would inevitably affect the market and the place of developing countries in space activities. However, it can be imagined that this situation could be tempered by market forces, as it has been the case with the International Telecommunications Union (ITU) and the practice of "paper satellites" 37 . The idea of an international regime akin to the OST of 1967 but that would include principles of governance to stimulate space activities is promising, as long as the ambitions of private actors are tempered with skill. Which begs the question of the finality of governance.

If the international community ever decides to integrate principles of global governance to the space industry, the former should first of all decide towards which goal the latter must tend. To that regard, two conceptions of global governance coexist. For the first of these conceptions 38 , it must essentially be based on stable rules of the game, the function of which is to coordinate the actions of States and those of economic agents. Its ideal is to be sufficiently clear and universal so that it doesn't need an interpretation or to be supplemented with an executive body capable of discretionary actions (e.g. commercial treaties). In the same way, it is possible to design rules favouring automatic adjustments of flux (of persons, funds, etc.). In this perspective, which can be compared to the German conception of Ordnungspolitik 39 , governance is essentially based on a predominantly economic legal order. States agree to establish a few rules of good conduct, without sharing other values or sharing other purposes. The fact that a partner would disrespect moral principles (e.g. oppressing minorities, destroying the environment, etc.) is ultimately indifferent, as it does not remove or add anything to the advantage of the partnership on a strictly economic point of view. It is only in the case of tortious behaviour (e.g. hindering competition, financial instability, identified market failure, depletion of natural resources) that it is necessary to intervene. It shares many similarities with the Law without States model presented above. While this approach could be applicable to the space sector despite its specific aspects, it should however be noted again that space law contains many general principles of a moral nature of which the violation would certainly be detrimental to developing countries and generally all nations that are not space powers, as noted above.

The second conception, global executive body/bodies 40 , presupposes the existence of common goals, whether economic (e.g. growth, full employment, expansion of trade, monetary stability) or not (e.g. peacekeeping, protection of the environment). The finality of governance is therefore defined on the basis of goals that nations set themselves to achieve together, which may 37 which constitute a form of speculation on spectral resources, since they are not associated with real projects but correspond to the acquisition of frequencies for their market value. See Laurence RAVILLON, Droit des activités spatiales -Adaptation aux phénomènes de commercialisation et de privatisation (Travaux du Credimi, vol. 22, Paris, Litec, 2004), 223. 38 JACQUET Pierre, PISANI-FERRY Jean, TUBIANA Laurence (eds), (n 5), 64-65. 39 Ordnungpolitik/Ordoliberal theory holds that the state must create a proper legal environment for the economy and maintain a healthy level of competition (rather than just "exchange") through measures that adhere to market principles. Patricia COMMUN, L'ordolibéralisme allemand: Aux sources de l'économie sociale de marché (CIRAC, 2003, 272 p.). 40 JACQUET Pierre, PISANI-FERRY Jean, TUBIANA Laurence (eds), (n 5), 65-66.

have been previously fixed or derive from the consequences induced by their interdependence. The method used to achieve these objectives can be based on the same techniques as in the previous case, but the purpose and basis of governance are not the same. In particular, the idea of the role multilateral organizations should fill differs greatly from that of Ordnungspolitik. They are rather the components of a joint executive body dedicated to the predefined common goals, giving them an irreducible political nature which was absent in the first conception. This approach obviously has a basis in the existence of global issues calling for collective action, but it can also respond to a political logic. This conception of governance would seem to be easily adapted to space activities, considering the highly political nature of the OST, and could be based on common goals of space exploration, utilization and exploitation. An executive body (or multiple specialized executive bodies) whose staff would be composed of experts and private actors appointed with the benediction of States, would allow for both the production of adapted laws and the protection of all nations' interests. Furthermore, even if Ordnungspolitik has the advantage of giving a lot of freedom to economic actors of the space sector, it is based on a defined set of rules. While those rules could be drafted with the help of private actors, they could however follow to the development of the industry and technologies only with difficulty and constant amendment of the original treaty. On the contrary, an executive body would prove much more adaptable to the evolution of space activities.

Conclusion

In conclusion, even though the idea of a comprehensive set of economic rules governing space activities would certainly allow them to foster their development, the creation of rules of both an economic and general nature in conjunction with that of specialized bodies -whether they be of an executive or judicial nature -would be much more efficient at maintaining a balance between the needs of private actors and Nations. A new take on the OST should therefore not abandon any of the rules it has set up 50 years ago, but on the contrary add new ones taking into account the new commercial uses of outer space and implement structures capable of enforcing them. Obviously, none of the models of governance presented above provides the basis for a comprehensive scheme around which to organize the governance of space activities. This calls for the construction of an original and sustainable model made by borrowing from these different categories, in order to set up a hybrid governance 41 of the space sector. Such a model could be made up of a network of independent authorities, each specific to a particular branch of space activities and with their own set of prerogatives. Meanwhile, private actors could produce recommendations or non-coercive codes of conduct rather than wait for the implementation of general obligations, and in turn inspire public regulators when their intervention is needed. Finally, all these actors would be subjected to the authority of a specialized international judicial body charged with enforcing space law and settling disputes. However, in order to guide the elaboration of such a model of governance, a number of principles that respond to general objectives of efficiency, legitimacy and transparency should be identified. The French Council of Economic Analysis proposes to select six principles: the specialization of institutions, so that citizens of the world could, through national and international associations and non-governmental organizations, exercise a critical role on a clear basis by overseeing the way in which their mandates are fulfilled; political accountability, which refers to discussions on the mode of global governance and the nature of institutions; the balance between areas of expertise and associated institutions; the transparency and democratization of procedures which, associated with the specialization of institutions, should guarantee the legitimacy of decisions; subsidiarity, that is to say that decisions must be taken at the most decentralized level if the transition to a higher level is not required in order to assure a certain level of efficiency; and finally solidarity, in order to mitigate market failures that prevent or delay development, compensate the losers of globalization and / or ensure against the economic hazards it entails and redistribute wealth to the poorest 42 . The alliance of both the general principles of the OST on the one hand, and of principles of governance of the other hand, tempered by the use of those six principles should be enough to meet the objectives set out at the beginning of this analysis. To that end, the 50 th anniversary of the Outer Space Treaty is the perfect occasion to reflect on the evolutions the space sector has undergone during the last fifty years and to take the necessary steps to adapting its rules to the 21 st century.

42 JACQUET Pierre, PISANI-FERRY Jean, TUBIANA Laurence (eds), (n 5), 74 to 92.

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Biography

Valentin Degrange is a PhD Law student since October 2015, specialized in Public International Law and Space Law. He studied at the Université Jean Moulin Lyon 3, where he earned a licence degree in Public Law as well as a master's degree in Public International Law, and is a member of the European Center of Space Law. Valentin wrote his master's thesis on the responsibility and liability of the EU for the Galileo program and now teaches to younger students at the Université Jean Moulin Lyon 3 and the Université Lumière Lyon 2. Today, he lives in the city of Lyon, France, and is working on a thesis related to the question of international cooperation regarding space activities.

E-mail address: valentin.degrange@hotmail.fr