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The unusual properties of plasmas

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La Rivista del Nuovo Cimento Aims and scope

Abstract

Sixty years ago Enrico Persico gave two seminars, one in Messina and one in Palermo, with the title “Le strane proprietà del plasma” which we freely translated as the title of the present article. Is what was unusual 60 years ago, in a sense, still unusual now? Possibly the “non reductionist approach” that the study of plasmas requires. Again, freely translating from the seminar notes [1] “ This field of physics is characterised by the fact that, even if the laws that describe the elementary microscopic forces are known, the different phenomena that take place in plasmas are so intertwined that predicting, or even just interpreting, the plasma macroscopic behaviour is often extremely difficult”. Where does this intertwining come from? And, is it universal? Often in multibody systems the presence of thermodynamic equilibrium puts to rest most of the details in the system behaviour that arise from the peculiarities of the interaction forces. But in plasmas even local thermodynamic equilibrium is generally not present: this is a fundamental property of electromagnetic interactions in dilute systems. And plasmas constitute most of the visible matter in the Universe.

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Notes

  1. Of the order of 10 nm for water, see Kelvin’s equation for the surface tension at a curved interface.

  2. c.g.s. units are adopted. This condition goes under the name of ideal Ohm’s law.

  3. This limitation has been partially lifted e.g. in Ref. [14] where low-collisionality accretion flows, such as that near the black hole at the Galactic centre have been considered. In such regimes the combined action of kinetic, see Sect. 5, and turbulent, see Sect. 9, effects needs to be considered. However, it not yet possible to perform spatially global kinetic simulations and the analysis must be limited to a portion of the full configuration with simplified boundary conditions, the so called “shearing box”.

  4. To avoid misinterpretations we should refer to such a non-closed set of equations not as the set of fluid equations but as the set of moment equations, as will be explained in Sect. 5.

  5. Actually since the thermonuclear fusion cross-section of deuterium–tritium ions is much smaller than the Coulomb collision cross section, the plasma must remain confined and interacting for periods much longer that the collision time mentioned in Sect. 2.4.

  6. More specifically of instabilities that are magnetic variants of the well known Rayleigh–Taylor instability of a fluid in an inverted gravity field.

  7. Characteristically in a present day magnetic fusion experiment the outward electron energy transport is two orders of magnitude larger than that predicted by binary particle processes.

  8. We note however that even in far-from equilibrium plasmas T often is called kinetic temperature or simply temperature.

  9. In particular in the astrophysical community [28] the Vlasov equation is instead named the collisionless Boltzmann equation. This denomination is sometimes seen as inappropriate as the Boltzmann equation, in contrast to the Vlasov equation, does not generally include any interparticle interaction besides collisions.

  10. This assumption may no longer be valid in the high energy density interactions between relativistic plasmas and the ultra-intense laser pulses in the petawatt power range that are presently available, see Sect. 10, and will not be tenable for the laser pulses in the exawatt power range that are expected to become available in a relatively near future.

  11. Here care must be exercised as the \(N\rightarrow \infty \) limit cannot be interchanged with the \(t\rightarrow \infty \) limit that is sometimes assumed when considering the long time behaviour of a plasma described by the Vlasov equation. Furthermore in the absence of thermodynamic equilibrium the expansion in powers of g is a formal expansion since in general it is not possible to put an a priori bound from above on the magnitude of the neglected terms.

  12. A more precise analysis indicates that the truncation is at order \(\, g \, \ln {(g)}\).

  13. These waves are charge separation waves where the plasma current cancels the displacement current so that no oscillatory magnetic field is present. Their linear dispersion equation can be written as \(\omega ^2 = \omega _{pe}^2 + 3 k^2 v_{the}^2\), with \(\omega _{pe} = [ 4\pi n e^2/m_e]^{1/2}\) and \(v_{the}\) a velocity that accounts for the width of the electron distribution function and that reduces to the standard thermal velocity in case of a Maxwellian distribution. The factor 3 can be interpreted as the one-dimensional adiabatic index. It accounts for the fact that only one degree of freedom is excited by the waves since, in the absence of collisions, the compressional energy is not redistributed between the other degrees of freedom.

  14. Such resonances go under the name of Landau resonances and represent a collisionless mechanism [31, 32] of absorption or stimulated emission that is responsible for an important channel of energy transport between the electromagnetic fields and the particles in the plasma. This energy transfer may result in damping (the so-called Landau damping) or excitation of perturbations whose amplitudes would instead remain constant within a “fluid” description.

  15. Adiabatic conditions are required where the characteristic nonuniformity scales in space and time are much longer than the particle gyroradii and the cyclotron periods, respectively.

  16. The Hamiltonian referred to here is the single particle Hamiltonian in given electromagnetic fields.

  17. And in particular \(-f \ln {(f)}\) that is commonly interpreted as the plasma entropy density for each species.

  18. These properties can be introduced at different levels of mathematical sophistication. The most comprehensive formulation describes a collisionless plasma in terms of a Hamiltonian functional where the dynamical variables are the electromagnetic fields and the distribution function f (Vlasov–Maxwell system). These variables are noncanonical and the Poisson brackets which act on them are degenerate [35]. Their kernel, i.e. the set of functionals of the dynamical variables that commute with all other functionals, corresponds to the set of the Casimir invariants. An example of a Casimir invariant of the Vlasov–Maxwell system is the plasma entropy defined as \(\mathcal{S} = - \int d^3 x\, d^3 v \sum _\alpha f_\alpha \ln {(f_\alpha )}\). The reduction procedures from the Vlasov equation to the moment equations [36] and to the magnetohydrodynamic equation is most consistently performed within this Hamiltonian field formulation by constructing new reduced non canonical variables from f by integration over velocity space.

  19. This condition ensures that the magnetic helicity is a gauge invariant quantity. Less restrictive conditions have been discussed in the literature.

  20. In particular in Reversed Field Pinch configurations, see Ref. [40].

  21. The time it takes a low frequency magnetic field to penetrate a conductor due to the effect of resistivity.

  22. The Alfvèn velocity, also called the magnetic sound velocity, is the characteristic propagation velocity of magnetohydrodynamic waves. It arises from the restoring forces due to the different components of the magnetic field contribution to the Maxwell stress tensor and the inertia of the plasma elements that drag the magnetic field lines according to the “frozen-in” condition.

  23. This separation of scales allows for the use of elegant asymptotic matching techniques that make it possible to explore analytically the dependence on the plasma properties of the onset and development of the reconnection process (see e.g. [45,46,47,48,49]).

  24. The formation of such highly inhomogeneous structures is a consequence of the nonlinear plasma dynamics constrained by the frozen-in condition and is another example how nonlinear Hamiltonian systems naturally develop small spatial scales.

  25. Mass ejection can cause geomagnetic storms that can perturb the Earth magnetosphere, temporarily weakening its protective shielding. The study of the effects of solar disturbances on Earth has recently developed into a fast growing research field on Space weather, [50].

  26. In such regimes the word magnetic reconnection is likely to be a misnomer and the term magnetic field annihilation would be more proper as no magnetic topology constraints apply in such short time-scale regimes.

  27. A similar sentence has also been attributed to Werner Heisenberg.

  28. The coronal activity belt is constituted of the coronal streamers and of the active regions localised in a ribbon shaped domain around the Sun’s equatorial plane.

  29. Note that in the other sections of this paper, following a different convention, we have called \(\omega = 2\pi f\) frequency while it should have been more precisely called angular frequency.

  30. Lévy functions are obtained from the central limit theorem by relaxing the hypothesis that the variance of the variables is finite.

  31. Self-affine time series are time series where the power spectral density scales as a power of the frequency.

  32. The Lundquist number corresponds to the ratio between the resistive time \(\tau _R\) and the Alfvèn time \(\tau _A\) defined in Sec. 7 where for the sake of simplicity only the resistive term had been included in the non-ideal Ohm’s law.

  33. These terms can be derived by properly combining the electron and the proton moment equations described in Sec. 5. In particular the Hall term accounts for the decoupling between the electron and the proton fluid velocities.

  34. In Hybrid simulations the Vlasov equation for protons is solved explicitly while electrons are described as an isothermal fluid which contributes to determine the plasma dynamics through the generalised Ohm’s law (15).

  35. On the other hand the fact that the speed of a particle is limited by c has been used in Sec.7 to explain the relativistic magnetic reconnection model based on the current starvation.

  36. We recall that such charge separation electric fields are at the root of the laser plasma acceleration of charged particle[126]. This acceleration scheme can achieve electric fields that exceed hundreds of GV/m. These fields are generated by large amplitude Langmuir waves propagating in the plasma in the wake of an ultraintense laser pulse. This important and fast developing line of research will not be described here and we refer the reader to the review article [127]. Similarly we do not address the laser ion acceleration mechanism where a charge separation field is instrumental in transferring the momentum from the accelerated electrons to the ions. For the different mechanisms at work in the ion acceleration mechanism we refer the reader to Ref. [128].

  37. For example for laser pulses with power approaching \(10^{23} \, W/cm^2\) in the 1mm wavelength range of near future interest.

  38. This is obtained by inserting the unperturbed Lorentz acceleration term in the Lorentz-Abraham-Dirac equation [149]

  39. See Sec. VI B of Ref.[146] for the condition discriminating the classical and the quantum regimes in the presence of intense electromagnetic fields.

  40. The nonlinear dynamics of the electromagnetic field in vacuum is described in the long wavelength limit by the Euler-Heisenberg Lagrangian [56] that depends on the electromagnetic Lorentz invariants \(|\mathbf{E}|^2 -|\mathbf{B}| ^2\) and \(\mathbf{E} \cdot \mathbf{B}|\) which vanish for a plane wave in vacuum.

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Acknowledgements

P. V. acknowledges useful discussions with F. Malara, S. Perri and S. Servidio, F. P. a longstanding collaboration with S. V. Bulanov and Ph. J. Morrison.

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Authors and Affiliations

  1. Enrico Fermi Department of Physics, University of Pisa, Pisa, Italy

    Francesco Pegoraro

  2. National Research Council, National Institute of Optics, Pisa, Italy

    Francesco Pegoraro

  3. Dipartimento di Fisica, Università della Calabria, Arcavacata di Rende, CS, Italy

    Pierluigi Veltri

  4. Istituto Nazionale di Astrofisica (INAF), Rome, Italy

    Pierluigi Veltri

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Pegoraro, F., Veltri, P. The unusual properties of plasmas. Riv. Nuovo Cim. 43, 229–279 (2020). https://doi.org/10.1007/s40766-020-00005-4

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