Whether electromagnetic radiation can be accumulated and stably confined in a plasma medium at macroscopic photon number and energy density is an unresolved question, posed concretely by Tsintsadze's ball-lightning model (arXiv:0811.4640, 2008) and independently by Alanakyan's 'electromagnetic clot' mechanism (Phys. Plasmas 24, 044502, 2017). Two candidate accumulation routes exist on paper — Bose-Einstein condensation of the massive photons of a plasma, and trapping of photons with wavenumber k<kp=ωp/ck<k_p=\omega_p/c in a plasma-density well — together with order-of-magnitude formulas for the trapped-photon density and energy and a toy surface-stability criterion. Later literature has developed the equilibrium statistical mechanics of a photon gas in plasma (including the condensation condition) and has realized photon Bose-Einstein condensation in dye-filled microcavities, but has not established any dynamical accumulation mechanism in a plasma, any self-consistent derivation of achievable trapped-photon content, or any quantitative stability limit of such accumulated states, and no experimental realization or refutation exists. The problem — a quantitative, self-consistent account of whether and how radiation accumulates in a compact plasma volume, what densities and energies are reachable as functions of plasma parameters, and what stability limits the accumulated state faces — remains open.

Background

Ball lightning is a luminous, roughly spherical object, decimeters in size, occasionally reported during thunderstorms, that persists for seconds and sometimes terminates in an explosion; its energy source and confinement mechanism have no accepted explanation. Several models appeal to plasma: a plasma is a quasi-neutral ionized gas whose electrons oscillate at the plasma frequency ωp=(4πe2n0p/me)1/2\omega_p=(4\pi e^2 n_{0p}/m_e)^{1/2} (Gaussian units, electron density n0pn_{0p}), and transverse electromagnetic waves in a plasma obey the dispersion relation ω2=ωp2+c2k2\omega^2=\omega_p^2+c^2k^2. Because of this dispersion, a photon in plasma behaves as a particle with a relativistic effective mass meff=ℏωp/c2m_{\rm eff}=\hbar\omega_p/c^2 and group velocity below cc, and waves with ω<ωp\omega<\omega_p are evanescent and cannot propagate out of the plasma.

Two consequences of this 'massive photon' picture are central. First, a local depression of the plasma density (δnp<0\delta n_p<0, a density well) lowers the local effective photon mass and creates a potential well: photons with wavenumber k<kp=ωp/ck<k_p=\omega_p/c are trapped in the well. Second, if photons were effectively number-conserving (absorption weak) and could thermalize, a gas of massive bosons can in principle undergo Bose-Einstein condensation (BEC), the macroscopic occupation of the lowest available mode that occurs when the phase-space density exceeds a critical value. Photon BEC has been realized experimentally only in dye-filled optical microcavities at room temperature (Klaers, Schmitt, Vewinger and Weitz, Nature 468, 545 (2010)), where thermalization is provided by repeated absorption and re-emission of photons by dye molecules — a mechanism absent in an ordinary plasma, where photon-number-changing processes and the weak Compton-scattering thermalization channel make equilibration of a dense photon gas problematic.

In this context Tsintsadze ('A new Concept of Ball Lightning', arXiv:0811.4640 (2008)) proposed that ball lightning is a weakly ionized gas — one in which electron/ion collisions with neutral molecules dominate over Coulomb collisions — in which electromagnetic radiation accumulates through (i) BEC of the plasma photons, an idea he had developed earlier (Phys. Plasmas 11, 855 (2004)), and/or (ii) photon trapping in the plasma density well, with trapped-photon number density nγtrap=kp33π2∣δnpn0p∣3/2n_\gamma^{\rm trap}=\frac{k_p^3}{3\pi^2}\left|\frac{\delta n_p}{n_{0p}}\right|^{3/2}, trapping energy Utrap=ℏωp10π2kp3V∣δnpn0p∣5/2U_{\rm trap}=\frac{\hbar\omega_p}{10\pi^2}k_p^3 V\left|\frac{\delta n_p}{n_{0p}}\right|^{5/2} (presented in the letter as the energy of a trapped photon), and total accumulated energy E=NtrapUtrap=ℏωp30π4(kp3V)2∣δnpn0p∣4E=N_{\rm trap}U_{\rm trap}=\frac{\hbar\omega_p}{30\pi^4}(k_p^3 V)^2\left|\frac{\delta n_p}{n_{0p}}\right|^4, where NtrapN_{\rm trap} is the number of trapped photons and VV the ball volume, building on his earlier thermodynamics of a dense photon gas in plasma (J. Plasma Phys. 55, 407 (1996); Phys. Rev. E 76, 016406 (2007)). Treating the ball as an incompressible one-fluid plasma (electrons, ions, and neutral molecules treated as a single fluid) with surface tension α\alpha and surface charge density σ\sigma, he linearized the surface dynamics and obtained the capillary-gravity-electrostatic dispersion relation ω2=1R0{αρMR02(l−1)(l+2)+g−4πσ2(l+1)ρMR0}\omega^2=\frac{1}{R_0}\left\{\frac{\alpha}{\rho_M R_0^2}(l-1)(l+2)+g-\frac{4\pi\sigma^2(l+1)}{\rho_M R_0}\right\} for spherical-harmonic index ll, mass density ρM\rho_M, ball radius R0R_0, and gravitational acceleration gg: an uncharged ball is stable, while a charged surface becomes unstable above a threshold (for l=1l=1, σ>ρMR0g/8π\sigma>\sqrt{\rho_M R_0 g/8\pi}), leading to explosive release of the stored radiation. Independently, Alanakyan (Phys. Plasmas 24, 044502 (2017)) proposed a related route to the same underlying question: an oscillating electromagnetic field localized in a self-sustaining plasma resonator is compressed by its own Maxwell stress (the tension of field lines), shrinking the resonator volume and raising the confined field energy density — an 'electromagnetic clot'.

What is missing, and what both sources leave open, is a quantitative and self-consistent physical account of radiation accumulation in plasma: the microscopic accumulation mechanism (how a dense photon gas is built up and thermalized or retained against escape and absorption in a plasma), the achievable trapped-photon number and energy as functions of the plasma parameters, and the stability limits of the accumulated state. The equilibrium statistical mechanics of a photon gas in plasma is now partly developed — Mati (J. Stat. Mech. 2020, 023102) derived the BEC condition for plasma photons and the modified Planck spectrum with a condensation peak below ωp\omega_p, in line with the microcavity photon-BEC program — but the dynamical accumulation and stability question for a compact plasma volume remains untouched by quantitative theory and by experiment.

Problem Statement

Can electromagnetic radiation be accumulated and stably confined in a plasma at macroscopic photon number and energy density — and if so, by what quantitative mechanism, to what achievable densities and energies as functions of the plasma parameters, and within what stability limits? Concretely, for a weakly ionized plasma (or a self-sustaining plasma resonator) with equilibrium density n0pn_{0p}, plasma frequency ωp\omega_p, and density perturbation δnp\delta n_p: (1) establish a microscopic, quantitative accumulation mechanism — candidates being Bose-Einstein condensation of the plasma's massive photons (meff=ℏωp/c2m_{\rm eff}=\hbar\omega_p/c^2) and trapping of photons with k<kp=ωp/ck<k_p=\omega_p/c in a plasma-density well, as in Tsintsadze's ball-lightning model, or Maxwell-stress self-compression of a plasma resonator as in Alanakyan's electromagnetic clot — including the rates of the processes that build up, thermalize, and retain the photon population against escape and absorption; (2) derive the trapped-photon number and energy content self-consistently as functions of the plasma parameters, accounting for the back-reaction of the accumulated radiation on the confining density profile (the literature estimates nγtrap=kp33π2∣δnp/n0p∣3/2n_\gamma^{\rm trap}=\frac{k_p^3}{3\pi^2}|\delta n_p/n_{0p}|^{3/2} and E=ℏωp30π4(kp3V)2∣δnp/n0p∣4E=\frac{\hbar\omega_p}{30\pi^4}(k_p^3 V)^2|\delta n_p/n_{0p}|^4 are fixed-profile results); and (3) determine the stability limits of the accumulated state — the conditions under which it remains confined and the thresholds (for example, in surface charge density, radiation pressure, or perturbation wavenumber) beyond which it becomes unstable and releases its energy. These three components jointly serve one overarching objective: a self-consistent quantitative theory (and, ultimately, experimental demonstration or exclusion) of radiation accumulation and stable confinement in plasma, the open question being whether such an account can be established or whether instead no stable high-density accumulated photon state can form in a plasma.

References

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