# Basic microscopic plasma physics from *N*-body mechanics

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## Abstract

Computing is not understanding. This is exemplified by the multiple and discordant interpretations of Landau damping still present after 70 years. For long deemed impossible, the mechanical *N*-body description of this damping, not only enables its rigorous and simple calculation, but makes unequivocal and intuitive its interpretation as the synchronization of almost resonant passing particles. This synchronization justifies mechanically why a single formula applies to both Landau growth and damping. As to the electrostatic potential, the phase mixing of many beam modes produces Landau damping, but it is unexpectedly essential for Landau growth too. Moreover, collisions play an essential role in collisionless plasmas. In particular, Debye shielding results from a cooperative dynamical self-organization process, where “collisional” deflections due to a given electron diminish the apparent number of charges about it. The finite value of exponentiation rates due to collisions is crucial for the equivalent of the van Kampen phase mixing to occur in the *N*-body system. The *N*-body approach incorporates spontaneous emission naturally, whose compound effect with Landau damping drives a thermalization of Langmuir waves. O’Neil’s damping with trapping typical of initially large enough Langmuir waves results from a phase transition. As to Coulomb scattering, there is a smooth connection between impact parameters where the two-body Rutherford picture is correct, and those where a collective description is mandatory. The *N*-body approach reveals two important features of the Vlasovian limit: it is singular and it corresponds to a renormalized description of the actual *N*-body dynamics.

## Keywords

N-body dynamics Debye shielding Landau damping Wave–particle interaction Spontaneous emission Coulomb scattering## Notes

### Acknowledgements

D. F. E. is grateful to the members of Equipe Turbulence Plasma in Marseilles, since the theory reviewed in this paper is the result of three decades of collaboration with them. He thanks Professor M. Kikuchi for suggesting him to write this review. He also thanks Professor A. Sen for many useful suggestions, and Professor P. Huneman for pointing out to him the book “Reductionism, emergence and levels of reality” by Chibbaro et al. He thanks Drs F. Bonneau, M.-C. Firpo, and F. Sattin for helpful comments on the manuscript. Also D. F. G. Minenna who brought the precious views of a newcomer in the field. One of the authors (D. Z.) has been supported by the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR).

### Compliance with ethical standards

### Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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