Quantum-like Approach to
II. Quamtum-like approaches for radiation and particle bunches
A. Radiation bunches. In 1946, Fock and Leontovich , showed that the equation which governs the propagation of an e.m. bunch in an arbitrary medium is a sort of Schrdinger 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 . The 1-D form of the Fock-Leontovich equation is in fact:
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
U (refractive index), in general, can depend
which, in this case, accounts for an e.m. beam
propagation through a nonlinear medium. In 1969, Gloge and
Marcuse  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
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):
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
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 .
Provided that the normalization condition
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
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:
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 . 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  as well as of the particle bunch . In both cases, the probability density given by (3) has the form:
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 . 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 . 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 .
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  as well as for charged-particle bunches (in this second case it describes the so-called Bennett profile).
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 . 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 , 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.
where ee is the
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
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
is replace by ee).
where ei 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 ni (r,s) which, inturn, is proportional to , namely . Consequently, (6) is in principle a sort of nonlinear Schrdinger 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.
1. Zou Y.-S, Phys. Scripta
52, 726 (1995) and references therein; Strandt E. and
Teodorani M., Experimental Method for Studying the
Hessdalen-Phenomenon in the Light of the Proposed Theories:
a Comparative Overview, Hfgskolen
Rapport 1998:5 (1998)
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