## Abstract

The impact of Maxwell's theory of Saturn's rings, formulated in Aberdeen *ca* 1856, is discussed. One century later, Nielsen, Sessler and Symon formulated a similar theory to describe the coherent instabilities (in particular, the negative mass instability) exhibited by a charged particle beam in a high-energy accelerating machine. Extended to systems of particles where the mutual gravitational attraction is replaced by the electric repulsion, Maxwell's approach was the conceptual basis to formulate the kinetic theory of coherent instability (Vlasov–Maxwell system), which, in particular, predicts the stabilizing role of the Landau damping. However, Maxwell's idea was so fertile that, later on, it was extended to quantum-like models (e.g. thermal wave model), providing the quantum-like description of coherent instability (Schrödinger–Maxwell system) and its identification with the modulational instability (MI). The latter has recently been formulated for any nonlinear wave propagation governed by the nonlinear Schrödinger equation, as in the statistical approach to MI (Wigner–Maxwell system). It seems that the above recent developments may provide a possible feedback to Maxwell's original idea with the extension to quantum gravity and cosmology.

## 1. A brief historical note on Saturn's rings studies

The rings of Saturn are a series of planetary rings orbiting around the planet. The major rings are labelled. They consist largely of ice and dust. There are several gaps between the rings, all of which are caused by the gravitational pull of one or more of Saturn's moons affecting the orbits of the tiny particles that comprise the rings. Icy particles spread out into large, flat rings and make up Saturn's ring system that can be seen with even low-power telescopes on the Earth's surface.

Improved telescopes have allowed astronomers to better ‘resolve’ the images of the rings. In fact, one of the large rings was resolved into two rings (B and C), and it was found that the A ring shows a gap (the so-called Encke division). The presence of another ring (the so-called D ring) closer to the planet than the C ring was also observed.

In 1610, Galileo Galilei was the first person to observe Saturn's rings, although with his weak telescope he could barely resolve them, and thought they were two moons on either side of the planet.

After 45 years, in 1655, Christian Huygens was the first scientist to propose that there was a ring surrounding Saturn (the rings were actually discovered by him in 1659). He proposed that Saturn was surrounded by a solid ring.

In 1660, the poet Jean Chapelain, a friend of Huygens, suggested that Saturn's rings are made up of a large number of very small satellites. Chapelain's suggestion did not receive enthusiasm among the astronomers. However, 200 years later, James Clerk Maxwell arrived at similar conclusion.

A valuable discovery of Giovanni Cassini in 1675 showed the existence of a gap between the two large (A and B) rings, now called the Cassini division in honour of this astronomer.

During the period between 1856 and 1859, the theoretical studies of Maxwell showed that the rings could not be solid, but rather a swarm of particles. A solid ring would become unstable and break up. He carried out a careful theoretical treatment and concluded that the rings could not be solid or liquid, since the mechanical forces acting upon rings of such immense size would break them up. He suggested that, instead, the rings were composed of a vast number of individual solid particles rotating in separate concentric orbits at different speeds. He reported this theory in his final article on the subject entitled ‘On the stability of the motion of Saturn's rings’, published in the *Proceedings of the Royal Society of Edinburgh* in 1859 (Niven 1890).1

## 2. Some details of Maxwell's idea about Saturn's rings

Maxwell supposed that the dust particles move around the planet at the angular frequency that decreases when the energy increases. Under the action of the gravitational field of the planet, the dust particles move along closed orbits. But this is not sufficient to have a stable motion. In fact, what confines the dust is the mutual gravitational interaction between the particles: this way he proved that dust forms a stable system forced by Saturn's gravitational field to execute revolutions. In 1895, James Keeler proved that Maxwell's hypothesis was correct when he measured the Doppler shifts of different parts of the rings and found that the outer parts of the ring system orbited at a slower speed than the inner parts. The rings obeyed Kepler's third law and, therefore, must be made of millions of tiny bodies each orbiting Saturn as a tiny mini-Moon.

One century later, in the period between 1958 and 1959, an analogous mechanism was put forward by Nielsen *et al*. (1959). It was pointed out that Maxwell's mechanism would be more interesting if the particles repel each other. An example of this kind was given by the above authors in particle accelerators (for an historical review, see Lawson (1988)).

In fact, in a circular accelerating machine, a charged particle beam is forced to execute millions or billions of revolutions under the action of electromagnetic (e.m.) forces instead of the gravitational field of the planet.

## 3. The theory of Nielsen, Sessler and Symon

Due to their electric charges, the particles of the beam repel each other. Consequently, one could think that in this case, due to the space charge repulsion, the beam motion is unstable. In reality, the physical conditions in an accelerating machine are richer than the ones occurring in the mechanism proposed by Maxwell, because the angular frequency variation is related to the energy variation through a threshold called the ‘transition energy’. But, nevertheless, Maxwell's idea was very useful for Nielsen, Sessler and Symon and constituted the basic idea of their mechanism. Maxwell's idea was, in fact, extended to the collective behaviour of a charged particle system in a circular accelerating machine (e.g. a storage ring).

### (a) Kinematics of a charged particle beam in a storage ring

Let us consider a relativistic charged particle beam travelling with velocity *βc* throughout the pipe of a circular accelerating machine. In order to describe the kinematics of the beam, it is useful to introduce the concept of a synchronous particle as a particle that does not change its energy *E* and the radius *R* of its circular orbit. This particle rotates at an angular velocity *ω*=*βc*/*R* and has a linear momentum *p*=*m*_{0}*γβc*, where *m*_{0} is the particle rest mass and *γ* is the relativistic Lorentz factor.

It is useful to adopt a co-moving frame of coordinates, whose origin coincides with the position of the synchronous particle. During motion, an arbitrary particle of the beam has, in general, the physical quantities displaced with respect to the ones of the synchronous particle. Note that(3.1)and(3.2)Furthermore, it is easy to see that(3.3)and(3.4)To maintain an arbitrary particle with a given linear momentum on a circular orbit of radius *R*, an axial magnetic field with amplitude *B*=*B*(*R*) is necessary. Consequently,(3.5)Then, by combining the previous relationships, we easily get(3.6)where we have introduced the ‘momentum compaction’ as(3.7)By introducing the ‘transition energy’, i.e.(3.8)and the ‘slip factor’, i.e.(3.9)equation (3.6) can be finally cast as(3.10)Equation (3.10) relates the energy displacement to the angular frequency displacement of an arbitrary particle with respect to the synchronous one. Therefore, it describes the kinematics of the longitudinal motion of an arbitrary particle of the beam and has to be coupled with the ones describing the dynamics of the system. Furthermore, from equation (3.10) we observe that: (i) for *η*<0 (above transition energy) the angular frequency decreases as the energy increases and (ii) for *η*>0 (below transition energy) the angular frequency increases as the energy decreases.

Let us now describe the longitudinal dynamics associated with an arbitrary particle of the beam.

### (b) Dynamics of a charged particle beam in a storage ring

Owing to the interaction with the surroundings, the beam space charge (capacitive effect) is in competition with the magnetic inductive effect. In order to describe the interaction between the beam and the surroundings, let us consider a beam propagating through the accelerator pipe and denote with *λ* the longitudinal number density (number of particles per unit length along the propagation direction). In general, *λ* depends on both the transverse and longitudinal coordinates and time. For the sake of simplicity, let us consider the one-dimensional case in which the beam is considered transversally flat and therefore *λ* depends only on the longitudinal coordinate *z* and time *t*, i.e. *λ*=*λ*(*z*, *t*). Correspondingly, the current associated with the beam is *I*(*z*, *t*)=*qβcλ*(*z*, *t*), where *q* is the particle charge.

During its motion, the beam is a source of both radial electric and azimuthal magnetic fields. Furthermore, owing to its interaction with the surroundings through the image charges and image currents (mainly originated on the pipe walls), a self-interaction (collective force) is produced.

In the case of perfectly conducting walls and coasting beam, the Lorentz–Maxwell system of equations easily leads to the following expression for the total longitudinal force per unit length acting on the beam (Lawson 1988)(3.11)where *k*_{0} is a positive constant proportional to the square of *q* and *X*_{L} and *X*_{C} are the inductive reactance (magnetic effect of the beam current) and capacitive reactance (space charge effect) per unit length, respectively. Note that *F* does not depend on the sign of the particle charge.

### (c) The physical mechanism of the instability

Physically, particles with a speed very close to *c* cannot travel much faster by acceleration but can increase their momentum and consequently move with a larger radius of curvature. Above the transition energy, particles that normally repel each other seem to experience an attractive force. This is known as the negative mass effect just for its similarity with the formation of planetary rings with an otherwise attractive gravitational force outside the Roche limit. In general, to have a stable or unstable beam motion more than a combination is possible (for a detailed explanation, see table 1 and figures 1–3). This way, Maxwell's mechanism is the gravitational analogue of the case in which the system is above the transition energy and the inductive effect overcomes the space charge effect only. But, of course, when the space charge is dominating, the beam particles are paradoxically drawn in the opposite direction towards which the total force is acting, like the behaviour of particles with negative mass.

## 4. Subsequent developments: kinetic theory of coherent instabilities (Vlasov–Maxwell system)

To provide a more general description of the coherent instability, Boltzmann's kinetic theory is applied. For a collisionless charged particle beam, the Boltzmann transport equation (Lawson 1988) has no usual collision integral. However, taking into account the two-particle correlation effects due to the e.m. interactions, a macroscopic collective Lorentz force appears in the Boltzmann equation (mean field approximation). Such an equation is usually known as the Vlasov equation, widely used in plasma physics (Nicholson 1983). Thus, for a transversally flat beam, the Vlasov equation, governing its longitudinal phase–space evolution, has to be coupled with Maxwell's equations for the self-consistent e.m. fields. (The set of equations comprising the Vlasov equation and Maxwell's equations, the so-called Vlasov–Maxwell system, govern the motion of the particles under the action of the e.m. fields that they actually generate.)

By introducing the first-order perturbations in the physical quantities (e.g. distribution function and e.m. fields) the Fourier analysis of the Vlasov–Maxwell system in the case of a coasting beam leads to the following dispersion relation (Chao 1993):(4.1)where *f*=*f*_{0}(*p*) is the equilibrium momentum distribution (*p* being the longitudinal conjugate momentum coordinate); is the complex longitudinal coupling impedance of the system; and *α*_{0} is a positive real constant. To obtain equation (4.1), the first-order perturbations of the physical quantities have been assumed to be proportional to exp[i(*kz*−*ωt*)]. The real part of accounts for the resistive properties of the walls of the machine's vacuum chamber. The inclusion of the resistive effects extends the instability analysis to an arbitrary coupling impedance presented above for a purely reactive coupling impedance only. By denoting with _{R} and _{I} the real and imaginary parts of , respectively, the dispersion relation (4.1) can be cast in the following way:(4.2)where , , and the symbol stands for the Cauchy principal value. This equation determines a relationship between *V*_{R}, *V*_{I} and *β*_{ph}. In principle, *β*_{ph} is a complex quantity. Thus, we put: . Consequently, we can plot curves in the *V*_{R}−*V*_{I} plane for a given equilibrium distribution function *f*_{0}(*p*) and for different growth rates *γ*_{I}. A qualitative behaviour of these plots is given in figure 4, where is assumed.

This picture describes the coherent instability (for instance, the negative mass instability) in circular accelerating machines in which coherent instabilities compete with the stabilizing effect of Landau damping. We would like to stress that figure 4 represents a sort of universal stability chart predicted by the Vlasov–Maxwell system. Any impedance leading to a (*V*_{R}, *V*_{I}) pair belonging to the area surrounded by the curve with *γ*_{I}=0 corresponds to a stable operation.

Furthermore, in the limit of a monochromatic beam, i.e. , all the curves become parabolas and the stability region collapses into the positive part of the imaginary axis. In this limit, the case _{R}=0 recovers the instability table (table 1) of a coasting beam.

## 5. Subsequent developments: quantum-like theory of coherent instability (Schrödinger–Maxwell system)

### (a) The thermal wave model

The description of the coherent instabilities given above can be alternatively provided by the quantum formalism (quantum-like description, see Fedele & Shukla (1995)). The charged particle beam dynamics has been successfully described by a quantum formalism in a number of problems of particle accelerators by means of the so-called thermal wave model (TWM; Fedele & Miele 1991). In particular, within the TWM framework, the longitudinal dynamics of particle bunches is described in terms of a complex wave function, *Ψ*(*z*, *s*), where *s* is the distance of propagation (i.e. *s*=*ct*) and *z* is the longitudinal extension of the particle beam measured in the moving frame of reference. The particle density, *λ*(*z*, *s*), is related to the wave function according to *λ*(*z*, *s*)=|*Ψ*(*z*, *s*)|^{2} (Fedele *et al*. 1993; Anderson *et al*. 1999*a*) and the collective longitudinal evolution of the beam in a circular high-energy accelerating machine is governed by the following Schrödinger-like equation for *Ψ*, *viz*.(5.1)where *ϵ* is the longitudinal beam emittance and *U*(*z*, *s*) is a dimensionless effective potential energy. Note that here 1/*η* plays the role of an effective mass associated with the beam as a whole. In general, equation (5.1) has to be coupled with the e.m. fields (via Maxwell's equations) that allows one to determine *U* (Schrödinger–Maxwell system). In particular, *U* is related to the self-consistent e.m. fields accounting for the collective interaction between the beam and the surroundings and, consequently, to the longitudinal coupling impedance . Let us assume that no external sources of e.m. fields are present and the effects of charged particle radiation damping are negligible. Then, the self-interaction between the beam and the surroundings, due to the image charges and the image currents originated on the walls of the vacuum chamber, makes *U* a functional of the beam density. It can proven that, for a coasting beam of an unperturbed density *λ*_{0} travelling in a high-energy circular accelerating machine, the governing equation (5.1) can be cast as the following integro-differential equation (Anderson *et al*. 1999*a*):(5.2)where , , *α*=*ϵη* and *Ψ*_{0} is a complex constant such that *λ*_{0}=|*Ψ*_{0}|^{2}. Equation (5.2) belongs to the family of nonlinear Schrödinger (NLS) equations governing the propagation and dynamics of wave packets in the presence of non-local effects. The modulational instability (MI) of such an integro-differential equation has been investigated for the first time in literature by Anderson *et al*. (1999*a*). Some non-local effects associated with the collective particle beam dynamics have been recently described with this equation (Johannisson *et al*. 2004). Note that, by comparing equation (5.1) with equation (5.2), *U* is now expressed by the functional(5.3)

### (b) MI of a monochromatic coasting beam and its identification with coherent instability

In order to provide a suitable quantum-like description of the coherent instability, we cast equation (5.2) in the pair of Madelung fluid equations (Madelung 1926). To this end, let us assume(5.4)then substitute (5.4) in (5.2). After separating the real from the imaginary parts, we get the following representation of (5.2) in terms of a pair of coupled fluid equations (continuity and motion, respectively):(5.5)(5.6)where the current velocity *V* is given by(5.7)Under the conditions assumed above, let us consider a monochromatic coasting beam travelling in a circular high-energy machine with an unperturbed velocity *V*_{0} and an unperturbed density *λ*_{0}=|*Ψ*_{0}|^{2} (equilibrium state). In these conditions, all the particles of the beam have the same velocity and their collective interaction with the surroundings is absent. Within the Madelung's fluid representation, the beam can be thought as a fluid with both current velocity and density which are uniform and constant (monochromatic beam). In this state, the Madelung fluid equations (5.5) and (5.6) vanish identically. Let us now introduce small perturbations in *V*(*z*, *s*) and *λ*(*z*, *s*), i.e.(5.8)(5.9)By introducing (5.8) and (5.9) in the pair of equations (5.5) and (5.6), after linearizing, and assuming that *V*_{1}, , we finally get the following dispersion relation:(5.10)where we have introduced the complex quantity , proportional to the longitudinal coupling impedance per unit length of the beam. In general, in equation (5.10) *ω* is a complex quantity, i.e. . If *ω*_{I}≠0, an instability takes place in the system. It has been recently proven that such an instability is a sort of MI predicted by the integro-differential NLS equation (5.2) (Fedele *et al*. 1993; Anderson *et al*. 1999*a*). If now we substitute the complex form of *ω* in equation (5.10), separating the real from the imaginary parts, the dispersion relation can be cast as(5.11)This equation fixes, for any values of the wavenumber *k* and any values of the growth rate *ω*_{I}, a relationship between real and imaginary parts of the longitudinal coupling impedance. For each *ω*_{I}≠0, running the values of the slip factor *η*, it describes two families of parabolas in the complex plane *Z*_{R}−*Z*_{I}. Each pair (*Z*_{R}, *Z*_{I}) in this plane represents a working point of the accelerating machine. Consequently, each parabola is the locus of the working points associated with a fixed growth rate of the MI. According to figures 5 and 6, below the transition energy (*γ*<*γ*_{T}), *η* is positive and therefore the instability parabolas have a negative concavity, while above the transition energy (*γ*>*γ*_{T}), since *η* is negative, the instability parabolas have a positive concavity (negative mass instability). Let us suppose that *η*>0. From equation (5.11) one can easily see that, approaching *ω*_{I}=0, the parabolas reduce asymptotically to a straight line lower unlimited located on the imaginary axis, as shown in figure 5. If *η*<0, in the same limit, parabolas reduce to a straight line upper unlimited located on the imaginary axis, as shown in figure 6. The straight line represents the only possible region below (above) the transition energy where the system is modulationally stable against small perturbations in both density and velocity of the beam, with respect to their unperturbed values *λ*_{0} and *V*_{0}, respectively. (Note that density and velocity are directly connected with amplitude and phase, respectively, of the wave function *Ψ*.) Any other point of the complex plane belongs to an instability parabola (*ω*_{I}≠0).

In the limit of small dispersion, i.e. *ϵk*≪1, the second term of the r.h.s. of equation (5.10) can be neglected and equation (5.11) reduces to(5.12)Furthermore, for purely reactive impedances (*Z*_{R}≡0), equation (5.11) reduces to the cubic NLS equation and the corresponding dispersion relation gives (note that in this case *ω*_{R}=*V*_{0}*k*)(5.13)from which it is easily seen that the system is modulationally unstable () under the following conditions:(5.14)(5.15)Condition (5.15) implies that the instability threshold is given by the non-zero minimum intensity .

Remarkably, we observe that condition (5.14) is the well-known Lighthill criterion associated with the standard cubic NLS equation (Lighthill 1965, 1967). On the other hand, according to what has been described in §3, it reproduces the coherent instability condition (see table 1) of a monochromatic coasting beam in the presence of a purely reactive impedance.

Furthermore, equation (5.12) predicts the MI associated with the cubic integro-differential NLS equation (5.2). In particular, the MI is predicted in terms of parabolas in the plane *Z*_{R}–*Z*_{I}, but, remarkably, it coincides with the results of coherent instability of a monochromatic coasting beam in the presence of an arbitrary longitudinal coupling impedance presented in §4.

To complete the above correspondence, one has to extend the MI analysis to the case of a non-monochromatic beam. This is done in §5*c*.

### (c) Statistical approach to MI analysis of a non-monochromatic coasting beam

The dispersion relation (5.10) allows one to write an expression for the ‘longitudinal coupling admittance’ of the coasting beam, *Y*≡1/*Z*:(5.16)Let us now consider a non-monochromatic coasting beam. Such a system may be thought as an ensemble of incoherent coasting beams with different unperturbed velocities (white beam). Let us call *f*_{0}(*V*) the distribution function of the velocity at the equilibrium. The subsystem corresponding to a coasting beam collecting the particles having velocities between *V* and *V*+d*V* has an elementary admittance d*Y*. Replacing *λ*_{0} with *f*_{0}(*V*)d*V* in equation (5.16), the expression for the elementary admittance is easily given as follows:(5.17)All the elementary coasting beams in which we have divided the system suffer the same electric voltage per unit length along the longitudinal direction. This means that the total admittance of the system is the sum of all the elementary admittances, as happens for a system of electric wires all connected in parallel. Therefore,(5.18)Of course, this dispersion relation can also be cast in the following way:(5.19)where *Z*=1/*Y* stands for the total impedance of the system which is the inverse of the total admittance.

An interesting equivalent form of equation (5.19) can be obtained (Fedele *et al*. 2006). To this end, we first observe that the following identity holds:Then, using this identity, equation (5.19) can be easily cast in the following form:(5.20)and finally in the following form:(5.21)We soon observe that, assuming that *f*_{0}(*V*) is proportional to *δ*(*V*−*V*_{0}), from equation (5.21) we easily recover the dispersion relation for the case of a monochromatic coasting beam (see equation (5.10)). In general, equation (5.21) takes into account the non-monochromatic character of the velocity distribution.

This dispersion relation extends the standard MI analysis to a non-monochromatic beam thought of as a statistical ensemble of monochromatic coasting beams. It predicts a Landau-type damping, a phenomenon very similar to the well-known Landau damping, predicted by L. D. Landau in 1946 for plasma waves (Landau 1946). The latter is recovered in the limit of small dispersion, i.e. *ϵk*≪1. In fact, the dispersion relation (5.21) reduces to a form very similar to dispersion relation (4.1) obtained in classical kinetic theory (Vlasov–Maxwell system).

### (d) Quantum-like kinetic description of coherent instability (Wigner–Maxwell system)

Equation (5.21) can be also obtained within the kinetic description provided by the quasidistribution (Wigner 1932; Ville 1948; Moyal 1949), as it has been done for the first time in the context of the TWM (Anderson *et al*. 1999*b*; Fedele *et al*. 2000) soon extended to nonlinear optics (Fedele & Anderson 2000; Hall *et al*. 2002; Helczynski *et al*. 2002), plasma physics (Fedele *et al*. 2002; Marklund 2005), surface gravity waves (Onorato *et al*. 2003) and lattice vibration physics (molecular crystals, Visinescu & Grecu 2003; Grecu & Visinescu 2004, 2005).

According to the former quantum kinetic approaches (Klimontovich & Silin 1960; Alber 1978), the basic idea is to transit from the configuration space description, where the NLS equation governs the particular wave-envelope propagation, to the phase space, where an appropriate kinetic equation is able to show a random version of the MI. This has been accomplished by using the mathematical tool provided by the ‘quasidistribution’ (the Fourier transform of the density matrix) that is widely used for quantum systems. In fact, for any nonlinear system, whose dynamics is governed by the NLS equation, one can introduce a two-point correlation function that plays a role similar to the one played by the density matrix of a quantum system (Landau 1927; Weyl 1931). Consequently, in the TWM framework, the governing kinetic equation is nothing but a sort of nonlinear von Neumann–Weyl (or Wigner–Moyal) equation (Wigner 1932; Moyal 1949) coupled with the effective potential (via Maxwell's equations) accounting for the interaction with the beam and the surroundings.

A linear stability analysis of the von Neumann–Weyl equation for the quasidistribution leads to a dispersion relation that coincides with equation (5.12).

## 6. The impact of Maxwell's theory during the twentieth century and concluding remarks

In this section, we describe the impact that Maxwell's theory of Saturn's rings has produced on the physics of the twentieth century. To this end, we present a summary of the above results in the form of a historical discussion.

In this paper, we have outlined that, 100 years later, by replacing the gravitational interaction with the e.m. one, Maxwell's theory of Saturn's rings was extended to the more rich context of charge particle beam dynamics in high-energy circular accelerators. With this extension, an important transfer of knowledge from planetary physics to accelerator physics allows one to conceive the physical mechanism of coherent instability (including the negative mass instability). During the sixth and seventh decades, this mechanism soon stimulated the development of the general theory of coherent instability based on the kinetic theory (Vlasov–Maxwell system). Another important process meanwhile already known in physics was the MI. Since the 6th decade, the MI (also known as the Benjamin–Feir instability, Benjamin & Feir 1967; Zakharov 1968; Yuen 1982) was already referred to as ‘a general phenomenon encountered when a quasi-monochromatic wave is propagating in a weak nonlinear medium, whose dynamics is governed essentially by a NLS equation coupled with a set of equations accounting for the interaction between the wave and the medium’. It was discussed and even observed experimentally in many fields of physics, such as deep waters physics (ocean gravity waves), plasma physics (electrostatic and e.m. plasma waves) and nonlinear optics (Kerr media, optical fibres). More recently, it has also been studied in electrical transmission lines, matter wave physics (Bose–Einstein condensates), lattice vibrations physics (molecular crystals) and in the physics of anti-ferromagnetism (dynamics of the spin waves).

For a review of the above developments of the MI studies, see Karpman (1975), Sulem & Sulem (1999) and Abdulaev *et al*. (2002).

For a long time, the theories of coherent instability and MI were neither compared, one to the other, nor connected to each other. Around the beginning of the 9th decade, a quantum-like theory of charged particle beam dynamics was proposed in the literature. On the basis of valuable physical analogies, the TWM was formulated (Fedele & Miele 1991). Later on, by means of the TWM, a connection with the conventional description of a particle accelerator in the presence of collective effects was given for the first time in terms of a standard cubic NLS equation (Fedele *et al*. 1993) and an integro-differential cubic NLS equation (Anderson *et al*. 1999*b*). In this way, the coherent instability was easily re-described as a sort of MI process. It is worth pointing out that this discovery has shown, in particular, that Maxwell's original idea on Saturn's ring stability was also imported and extended to a new context where the ‘elements’ involved in the dynamics of the system were not the particles but the waves.

Classical statistical approaches or Vlasov-type kinetic equations to describe systems of waves were used since the 1960s to describe wave interactions and/or the phenomenon of the Landau damping (see Dawson 1960; Hasselmann 1962; Vedenov & Rudakov 1965; Bingam *et al*. 1997; Zakharov 1999). More recently, within the context of the quantum-like kinetic description based on the Wigner quasidistribution, a statistical approach to MI has been formulated (Anderson *et al*. 1999*a*; Fedele & Anderson 2000; Fedele *et al*. 2000, 2006). Remarkably, this approach, which has been applied to a number of physical problems, unifies the classical descriptions of coherent instability and MI in one general approach to the instability of any weakly nonlinear system in which the propagation of partially incoherent waves is involved (Fedele *et al*. 2002; Hall *et al*. 2002; Helczynski *et al*. 2002; Onorato *et al*. 2003; Santos *et al*. 2007). Consequently, it can be thought as the generalization of Maxwell's original idea to a random wave system.

The studies of the extreme physics exhibited by astrophysical objects, such as the superfluid two-stream instability neutrons stars (Andersson *et al*. 2003), Bose–Einstein condensation properties in collapsing and exploding stars (Ball 2001), density waves in spiral structures (Lin & Bertin 1995) and the quasi-linear kinetic approach to the planetary rings (Griv *et al*. 2003), have already imported, directly or indirectly, the basic idea of Maxwell's theory.

The very rapid developments in quantum gravity have recently shown valuable applications of the quantum formalism in astrophysics and cosmology (Penrose 1996, 1997, 1998; Moroz *et al*. 1998; Tod & Moroz 1999). It is reasonable to assume that suitable Schrödinger-like equations, coupled with a set of equations describing the gravitational interaction, govern the collective dynamics of the astrophysical systems in terms of a complex wave function as it has been assumed for the entire Universe (Hartle & Hawking 1983; Hawking 1984).

As a natural consequence, the quantum-like kinetic description of the coherent/MI and the related statistical approach may provide a random version of Maxwell's theory whether applied to planetary problems or to different astrophysical environments. In particular, a kinetic theory, based on the Schrödinger–Newton (Schrödinger–Einstein) system or Wigner–Newton (Wigner–Einstein) system, is underway and it would provide a more general description of the instability of such systems.

It should be noted that, when the knowledge of the quantum-like formulation of the coherent/MI is transferred back to astrophysics and cosmology, it is natural that one would need to determine, in the gravitational context, a possible counterpart of the e.m. repulsion. The latter has been detected through the accelerated expansion of the Universe and might be interpreted as an effective ‘repulsive gravitation’. A suitable theory of MI including this effect is also underway.

## Footnotes

One contribution of 20 to a Theme Issue ‘James Clerk Maxwell 150 years on’.

↵Most of the historical information given in this article are available at the following web sites:

↵http://en.wikipedia.org/wiki/Rings_of_Saturn;

↵www.enchantedlearning.com/subjects/astronomy/planets/saturn/saturnrings.shtml.

↵Craig Howie, http://heritage.scotsman.com/profiles.cfm?cid=1&id=39592005 (article). Nick Strobel,

↵www.astronomynotes.com/solarsys/s16.htm#A9.5.1 (article). Calvin J.

↵Hamilton, www.solarviews.com/eng/saturnbg.htm (article).

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