Complesso Universitario di Monte S. Angelo, Via Cintia I-80126 Napoli, Italy fedele@osfna1.na.infn.it
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
where the time-like variable
where
is the timelike variable ( 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 conditionis 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 frameworkEq.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
Thus, under the condition
where _{0} and its width
D are related to
P and
Q as: r_{0}
D^{2} =
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.
## IV. A possible connection with ball lightningOn 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 perspectivesIn 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. 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
ii. F
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 _{p} r is the
radial coordinate with respect to the centre of the ion
core, e is the
absolute value of electron charge,
R and _{i}
R 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 _{e} U =
Cf , where ^{2} /2
C is the capacity of the equivalent
capacitor. 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. iii. 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. iv. 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: v.
where e
q depends on the intensity of the
e.m. radiation (by means of _{0}
F). 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_{p}_{e}). 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: vi.
where e
1. Zou Y.-S,
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