Plasma physics is a rather young branch of modern science. The reason is that plasmas are not naturally present on Earth, except during a thunderstorm. But around 1920–1930, the pioneering works of Irving Langmuir and Lewis Tonks on electrical discharges pointed to the peculiar properties of a “new medium”: the plasma state. During this same period, some of these properties were being indirectly and independently observed in the measurements of radio-wave transmission through that part of the atmosphere which at that time was being discovered: the ionosphere, which represents the “gas” in a natural, persistent plasma state that is closest to Earth’s surface.

In a neutral gas, the main physical properties can be explained in terms of a statistical description of the mutual interactions of a large number of particles (usually neutrally charged molecules), where each particle interacts (i.e. “collides”) with just a very few other particles at a time. On the contrary, in a plasma, the presence of free charged particles (ions and electrons) points to a different and new physical behavior, since a single charge can interact with a huge number of other charged particles at the same time.

These collective properties can be explained as due to the long-range interaction of Coulomb forces, which decays as r^–2, while the number of particles at a distance r from a test particle grows as r⁺². This number can be larger than 10⁸ in astrophysical plasmas (sun, stars, interstellar medium etc.) as well as in laboratory plasmas for thermonuclear fusion experiments (by magnetic or inertial confinement). In the limit in which this number can be considered as infinite in a mathematical sense, a master equation can be derived which makes it possible to describe the collective behavior of plasmas. This is the well-known (at least by plasma physicists!) Vlasov equation, introduced by A. Vlasov in 1945 and L. D. Landau in 1946.

However, the modeling of a Vlasov plasma is not that easy, especially because of the intrinsic nonlinearities of the equations involved. As this represents an active field of research, approaching it by relying on modern scientific literature can be a quite challenging task for a newcomer, despite several excellent books that already exists about the physics of Vlasov plasmas. Most books on this subject tend to focus either on the physics of “warm/diluted plasmas”, to which Vlasov formalism applies, or to more strictly mathematical properties of the Vlasov equation, related to issues more relevant to applied mathematics (convergence, integrability, etc.). The present book aims to privilege a discussion of the general properties of the Vlasov equation and of its applications, which are presented to the reader from a historical and – hopefully – pedagogical point of view.

Although plasma physics itself is a rather young discipline, being less than 100 years old, one of us (Pierre Bertrand) has been working on this subject for more than 50 years. He has been a modest actor, but he has been involved during more than half of the “history” of plasma physics. Thus, it seemed to us that introducing some of the historical works which were carried out during all these years (a task which is mainly accomplished in the first two chapters of the present book) would help the understanding of more current works, whose discussion is developed in the following chapters: notably, the implications of the Vlasov formalism for describing the propagation of small amplitude electromagnetic waves in a plasma (Chapters 3 and 4), and the nonlinear behavior of the latter, in the electrostatic Vlasov limit (Chapter 5).

This book is addressed to students in plasma physics and to young researchers, as well as to whomever wants to get a good understanding of a Vlasov plasma, in order to be able to grasp the essence of its main properties.

Pierre BERTRAND

Daniele DEL SARTO

Alain GHIZZO

June 2019