A Possible Quantum-like Approach to
Non-conventional Plasmas

Renato Fedele
Dipartimento di Scienze Fisiche, Università "Federico II" and INFN, Napoli
Complesso Universitario di Monte S. Angelo, Via Cintia I-80126 Napoli, Italy
fedele@osfna1.na.infn.it

I.  Introduction
Plasmoids involved in Hessdalen phenomena [1], luminous effects produced during discharges due to piezoelectricity, thermoluminescence, and triboluminescence in connection with earthquakes [2], luminous phenomena that seem to be produced by the interaction of Extraterrestrial antimatter with the upper atmosphere [3], some kind of dusty plasmas [4], ball lightning (BL) [5], etc.,constitute examples of non-conventional plasmas. For the most part, they involve generation and/or propagation of relatively intense electromagnetic (e.m.) waves. In some cases, static or quasi-static electric and magnetic fields seem to be involved, as well. Unfortunately, their behaviour is still not sufficiently modelled. Among them, BL are surely the most surprising manifestations for which special attention has been devoted. This interest is testified by several very important investigations (for a review, see papers in [5, 6]). Nevertheless, BL are still difficult to explain. On the other hand, the development registered in conventional plasma physics (see, for instance, papers in [7]), especially, for parametric processes and parametric instabilities, laser-plasma interaction, generation of large amplitude fields for many scientific and technological applications, nonlinear wave propagation, shows clearly a wealth of phenomena, whose description is quantum-like instead of the proper quantum one [8]. Furthermore, quantum-like approaches for describing charged-particle bunch dynamics [9] and particle traps [10] have been also proposed. In this paper, I try to show the suitability to apply the above quantum-like description to the physics of non-conventional plasmas. This is done by taking into account some important nonlinear collective effects, connected both to e.m. radiation and to particles that may take place in a charged-particle systems. In the next Section, a brief description of the quantum-like approaches is given for radiation as well as for particles, while in Section III some nonlinear collective effects are briefly described by means of suitable nonlinear Schrˆdinger-like equations, giving some examples. In Section IV, on the basis of the material presented in the preceding sections, a possible quantum-like description for BL is proposed as a possible novel approach. Finally, Section V is devoted to perspectives concerning investigations to be carried out in future within the above quantum-like framework.

II.  Quamtum-like approaches for radiation and particle bunches

A. Radiation bunches. In 1946, Fock and Leontovich [11], showed that the equation which governs the propagation of an e.m. bunch in an arbitrary medium is a sort of Schrˆdinger equation, where and the time are replaced by the inverse of the wave number and the propagation coordinate, respectively. This quantum-like equation can be essentially obtained in the so-called slowly-varying amplitude approximation, widely used in nonlinear optics and in plasma physics [12]. The 1-D form of the Fock-Leontovich equation is in fact:

(1)

where the time-like variable s is the propagation coordinate, is the inverse of the wave number (, being the wavelength) associated with the e.m. wave, and the effective potential U (refractive index), in general, can depend on which, in this case, accounts for an e.m. beam propagation through a nonlinear medium. In 1969, Gloge and Marcuse [13] performed a transition from geometrical optics to wave optics, developing a formal quantization ala Bohr of geometrical optics (in which is replaced by ). In (1) F is a complex e.m. wave amplitude. Provided that

gives the normalized e.m. power density (in the quantum-like language: the probability density for the system of photons).

B. Charged-Particle Bunches: The Thermal Wave Model. In the recent years a procedure ala Gloge and Marcuse has been proposed to describe the collective behavior of charged-particle bunches [9a, 14, 15]. The corresponding quantum-like model developed is theso-called thermal wave model (TWM), which assumes that the collective behaviour of a charged-particle bunch is governed by the following evolution equation for a complex function , (1-D case):

(2)

where is the timelike variable (c and t being the speed of light and time, respectively), x is the space coordinate, U is an effective potential made dimensionless dividing by the quantity mc²(m being the particle mass), and a is a dispersion parameter proportional or eventually equal to the bunch emittance e, which accounts for the bunch thermal spreading [16]. Provided that the normalization condition

is satisfied, the number density of the bunch particles is given by
,
where N is the total number of particles. Eq.n (2) is formally identical to Eq.n (1). In this analogy the inverse of the wave number is replaced by the bunch emittance, and the inhomogeneous refractive index is replaced with the effective potential.

III.  Some nonlinear and collective effects described in the quantum-like framework

Eq.s (1) and (2) and their two- and three-dimensional extension have been used in a number of theoretical studies. Some of them are briefly resumed here for the relevance that they may have in BL investigations. To this end, it is important to point out that when a relatively intense e.m. bunch propagates through a nonlinear medium, such as optical fibres or plasma, the refractive index in (1) becomes a function of the field amplitude F. In many cases it becomes

n being a constant which accounts for the nonlinear frequency shift [12]. This effect is produced by the ponderomotive force associated with the e.m. bunch (radiation pressure mechanism). In a fully similar way, when a charged-particle bunch is travelling in an accelerating machine or in a plasma, the effective potential in (2) becomes a function of Y, as well. In several cases already successfully investigated with TWM it becomes
     [14, 15].
This dependence is due to the wake-field interaction between the bunch and the surroundings, and the constant n¥ plays the role of coupling coefficient which, for instance, accounts for the reactive coupling impedance). In the above physical situations, both radiation and particle bunches suffer some electro-mechanical actions in such a way that they modify their shape, and, in turn, the properties of their propagation. It is useful to distinguish the case in which this action is longitudinal (parallel to the propagation direction) from the transverse one (orthogonal to the propagation direction).

A. Self-modulation. Let x represents in (1) and in (2) the longitudinal space-coordinate, and let us put these equations in the unified form:

(3)

Thus, under the condition PQ > 0 (Lighthill condition), the (3) describes the (longitudinal) instability of an inhomogeneous amplitude modulation which may evolve in such a way to produce the self-bunching of the system. In case of e.m. radiation, this effect is referred to as modulational instability. In the opposite regime (PQ < 0), we get stability. Physically, it come from combination of nonlinearity (third term of Eq. (3)), due to the ponderomotive force, with dispersion (second term of Eq. (3)), due to the diffraction. On the other hand, recently it has been pointed out that, in the TMW framework, the above longitudinal self-modulation represents the so-called coherent instability that takes place when a density perturbation is produced in a charged-particle bunch travelling in an accelerating machine as well as in plasmas [15]. Physically, it comes from the combination of the nonlinearity due to the wake-field interaction and the dispersion due to the thermal spreading.

B. Solitary waves. The competition between dispersion and nonlinearity in Eq.n (3) can produce wave envelope with stationary soliton-like profile. In particular, in case of self-modulation (PQ > 0), the soliton envelope is the natural asymptotic evolution of the self-bunching of radiation [12] as well as of the particle bunch [15]. In both cases, the probability density given by (3) has the form:

(4)

where V₀ is the soliton velocity; its maximum amplitude r₀ and its width D are related to P and Q as: r₀ D² = 2P/Q = constant. Remarkably, the shape of the density given by (4) remembers the one of the solitons described by the Korteweg-de Vries equation [12]. The main property of the solitons is that they are very stable structures against external perturbations.

C. Self-focusing. When a 2-D transverse dynamics is taken into account (the second-order derivative in (3) is replaced with ), another important effect can be considered. In fact, the self-interaction, which is still in competition with the dispersion, acts now transversally to compress the bunch. When the two effects are balanced, a self-equilibrium of the bunch is reached, but when nonlinearity is dominating, the bunch self-focusing starts. This condition represents a sort of instability of the system leading to the bunch collapse. However, as the bunch compression becomes very intense, higher-order nonlinearities provide for saturating the process. Self-focusing is also the basic mechanism for the filamentation (filamentary instability). Self-focusing of an e.m. bunch in plasmas has been extensively described in terms of 2-D versions of Eq.n (3) and experimentally investigated [12]. Furthermore, self-focusing and self-pinching of charged-particle bunches in plasmas have been successfully described by 2-D versions of Eq.n (3) in the framework of TWM [14].

D. Self-channelling. In the limiting case of 1-D transverse dynamics, in suitable conditions, the transverse profile of the bunch will assume a soliton-like shape. However, when the bunch width associated with an arbitrary initial profile largely exceeds the one of the soliton having approximately the same amplitude, the bunch decays transversally, during its (longitudinal) propagation, into several plane-parallel channels, similar in form to solitons. This effects, called self-channelling, is analogous to the above self-modulation. Given (1) and (2) in the case of cubic nonlinearity as in (3), self-channelling obviously happens in plasmas for e.m. bunches [12] as well as for charged-particle bunches (in this second case it describes the so-called Bennett profile[16]).

E. Self-trapping. Let us observe that one- and two-dimensional self-focusing are sort of transverse self-trapping. However, when longitudinal and transverse self-compression are working simultaneously, a 3-D self-trapping of the bunch should be expected. In particular, 3-D solitons have been predicted, by using a relativistic extension of (3), for a steady propagation of an e.m. pulse in plasmas [17]. In this physical situation, the initial spherical symmetry of the system can be preserved. Because of the great difference between electron and ion masses, the ponderomotive force essentially acts on the plasma electrons. So, they are pushed radially outward producing a sphere whose internal region has a refractive index larger than the external one. This way the radiation is trapped and continuously internally reflected at the sphere surface where the most of the electrons are pushed.

IV.  A possible connection with ball lightning

On the basis of the peculiarity of the unusual phenomena, and according to what has been presented in the preceding sections (in particular, Section III), it seems suitable to transfer both the knowledge already reached and the methodologies already used for describing nonlinear collective effects in ordinary plasmas to the non-conventional ones. The idea is to construct, in the context of the non-conventional plasmas, a suitable quantum-like description, with the inclusion of nonlinear collective effects. The aim is to try to understand better the behaviour of the above unusual phenomena, especially BL. Of course, this attempt does not pretend to be exhaustive. On the contrary, it could be a possible small contribution to give within the great effort that very qualified scientists are doing since some decades in the BL investigations [5, 6]. I think that it could produce more insights in the scenario of the possible explanations proposed for these phenomena. instance, the inclusion in the description of the self-trapping allows to deal with (and probably to justify) the spherical symmetry usually (but not always) exhibited by BL. Additionally, the inclusion in the description of mechanisms to produce solitons would be helpful, because it provides to deal with structure that, due to their strong stability, can survive for relatively long time, like BL. According to both theoretical and experimental investigations [12], a typical behaviour of soliton-like structures is to reconstruct exactly their initial profile after scattering through potential wells or potential barriers (i.e. obstacles). It is clear, from the literature, that BL exhibit a similar feature in several cases reported. A third but non-minor point is the great variety of parametric processes associated with the self interaction of both e.m. and particle bunches mentioned in Section III. In particular, nonlinear coupling of e.m. and space-charge waves can lead to an instability which results in a simultaneous growing of all the wave amplitudes. But it is not excluded that in case of initial apparently steady state, breakdown or decay of the system happens. These physical circumstances could be put in correspondence with the typical explosions or decays of BL.

V.  Preliminary perspectives

In order to carry out in future the above quantum-like approach for BL, it is worth to mention the main steps in which the study should be organised.
i. The starting point is to assume the existence of relatively intense e.m. radiation sources, with spherical symmetry, in a ionised region of the atmosphere. Thus, the self-trapping will be the basic ingredient of this approach. According to the self-trapping mechanism described above, the plasma electrons are mostly pushed radially outward to form a spherical layer. The expansion of these electrons is against both the kinetic pressure of the outer gas and the electric restoring force produced by the ions.
ii. If the electrons are pushed outward very fast, the ions may be inertially and radially compressed, reducing sensitively their occupation volume in the sphere created by the self-trapping. Consequently, the most part of ions and the most part of electrons constitute a positively charged spherical core and a negatively charged layer, respectively. This physical situation is fully similar to the one of a charged spherical capacitor. Denoting by F_p the (radially-oriented) ponderomotive force, we can say that radiation provides to charge the equivalent capacitor associated with the system with an electromotive force , where r is the radial coordinate with respect to the centre of the ion core, e is the absolute value of electron charge, R_i and R_e are the effective radius of ion core and electron layers, respectively. So that, at the expense of the radiation field, an electric energy is stored in the system which is U = Cf ² /2, where C is the capacity of the equivalent capacitor.
iii. In principle, the above system is not stable. In fact, after a certain time, the blow-up of the core, caused by the ions space charge, should be expected. It enhances the core radius more and more, up to reaching the electron layer. This way, the system gives back the stored electric energy, and this physical situation corresponds to a sort of small explosion. However, how long is the effective time of the system to survive before the electron-ion recombination takes place (decay-time) will depend on the competition between inertial compression and blow-up.
iv. Additionally, in principle resistive effects should be taken into account, because they also contribute to the decay-time estimate. However, in this preliminary analysis, they will be neglected for the sake of simplicity.
v. From the quantitative point of view, the collective behaviour of the electron layer in the TWM framework is described by the following Schrˆdinger-like equation:

(5)

where e_e is the electron emittance, q₀ = Cf is the effective charge of the ion core. Note that q₀ depends on the intensity of the e.m. radiation (by means of F_p). Furthermore, is proportional to the electron layer density. Consequently, the electron equilibrium distribution should be found by solving (5) for stationary states. These solutions are very well known, because they are formally identical to the ones given by the radially-symmetric eigenstates of the hydrogen atom (here is replace by e_e).
vi. On the other hand, also for the collective behaviour of the ion core the following Schrˆdinger-like equation should be written in the TWM framework:

(6)

where e_i is the ion emittance, and V(r,s) is the space-charge potential produced by the core itself. It is worth to note that V is a function of the ion core density n_i (r,s) which, inturn, is proportional to , namely . Consequently, (6) is in principle a sort of nonlinear Schrˆdinger equation. After solving (6), the decay-time can be easily calculated. In conclusion, by taking into account some collective nonlinear effects that both electromagnetic radiation bunches and particle bunches may produce in a charged-particle system, a quantum-like approach for describing the ball lightning dynamics has been put forward. In a future work, by solving (5) and (6) with suitable boundary conditions, the above model will be developed quantitatively.

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