Introduction

All models are wrong,
but some are useful.

George Box

The notion of an isolated system is a theoretical idealization. In reality, everything in the Universe is connected to everything and everything interacts with everything. It is indeed difficult to imagine a free particle, perfectly isolated from the rest of the Universe. And even if such a situation were conceivable, it would be quite uninteresting. On a practical level too, it is extremely difficult to perform an experiment on a small number of particles that would be, and remain throughout the experiment, isolated from the rest of the Universe. This would indeed imply absolute control of any form of interaction or exchange between the particle or particles in question and their immediate environment; it would be a very difficult task! Thus, even if the problem involves one or a small number of particles, an ideal model would take into consideration the exchanges of energy with the rest of the Universe, which would thus play the role of heat reservoir (and/or of particles). Such a model would contain, at least in principle, the dynamics of all the particles or degrees of freedom that constitute the environment.

A lesson to be drawn from this is that a model that is used for making reliable predictions, and is easily testable and empirically verifiable, must necessarily involve (at least indirectly) a large number of degrees of freedom. This is indeed the reason why statistical methods are ubiquitous in physics.

In recent decades, prodigious technical feats have been achieved in situ, that is to say experiments carried out in the laboratory. It is indeed possible today to make measurements of extreme precision and in increasingly high energy regimes. These experiments, however, come at a cost. Moreover, it may be advantageous to turn to the world's largest laboratory, that is, look at the sky and compare our theoretical predictions with space and astrophysical observations. What do we see when we look at the sky? Essentially, stars and large expanses of partially or totally ionized gas: this is called a plasma. This is the modern interest in plasma physics (in addition to the practical interest they arouse, given their application in various technologies). Plasmas are indeed an excellent field for testing theoretical predictions and comparing them with observations.

However, it is important not to see plasma physics as a corpus of methods and tools born in plasma physics and applicable to plasmas only; the reality is more complex! Indeed, the entire kinetic approach to plasma physics arises from the kinetic theory of gases, developed essentially by Boltzmann in the 19th century (Cohen and Thirring 2012), for classic neutral gases. Furthermore, methods born within the framework of plasma physics, such as the integral transformations introduced by Landau (1946), find new life through more contemporary applications, in cosmology for example (Baym et al. 2017; Moretti et al. 2020). To illustrate this, and without going into technical details, let us take the example of quantum plasma, which can be kinetically or hydrodynamically described by the Schrödinger equation coupled with the Poisson equation for the Coulomb field. Replace the Poisson equation of the Coulomb field with the Poisson equation of the gravitational field and you have a dark matter model (Bernal and Guzmán 2006; Ourabah 2020b) (i.e. scalar field dark matter)! Thus, it is important not to see these tools as specific to plasma physics, but to see their application in plasmas as a particular case, an approach whose scope is much more general. It is precisely this aspect that this book attempts to highlight.

An ideal, though impracticable, model when dealing with a complex system composed of a large number of particles such as a plasma would be to write and solve the equations of motion of all the particles. This would be equivalent to writing N ~ 1023 partial differential equations and integrating them, which implies knowing the position and initial velocity of each particle. Such a method is clearly out of our reach. Thus, Laplace had imagined a supernatural intelligence (Laplace 1814), that is, Laplace's demon (see Figure I.1), which would know, with infinite precision, the position and speed of each particle and could thus predict the evolution of any system, even of the entire Universe: "[...] nothing would be uncertain and the future just like the past would be present before its eyes". But, we are not Laplace's demons and we have to deal with the information we have and our limitations to process such information. The best strategy, in practice, is the one that will "sort" between superfluous information and information essential to predict the evolution of the system. There are two main approaches to this end: kinetic theory and hydrodynamic models.

Figure I.1. Illustration of Laplace's demon by Ricard Solé

A first strategy is to omit all details on a microscopic scale. Thus, instead of considering the individual equations of motion of each of the particles composing the system, this strategy proposes to describe the global evolution of the density of particles in the medium. Such an approach is called hydrodynamic or fluid. Although simplistic in a sense, it makes it possible to describe, for example, the propagation of sound waves in media and explains, to a certain extent, the formation of stars from dust gases, that is, Jeans instability (see Chapter 6).

Another, perhaps more ambitious, approach consists of introducing individualism insofar as it introduces the notion of distributional function. Such an approach no longer treats the particles composing the system as being identical to each other, but makes it possible to describe the differences, in speed or energy, between them. This is called kinetic theory. It is important to understand, however, that no method is better than the other in absolute terms. While it is true that kinetic theory is more accurate than the fluid model, it is also more difficult to implement in numerical simulations. These simulations are very often essential to compare theoretical predictions with observational measurements.

This book presents the theoretical bases of these two approaches, the virtues and limits of each, as well as the bridges that exist between them. The evolution of the book is not chronological; thus, the kinetic theory will be presented before the hydrodynamic model, although historically, the fluid model was born before the kinetic theory. This choice is motivated by the fact that hydrodynamic equations can be demonstrated from kinetic theory, according to certain approximations.

The book is divided into two main parts. Part 1, aims to present the kinetic and hydrodynamic approaches in a very general theoretical framework. The models are developed so as to be applicable to different physical situations, which can be treated on an equal footing. Part 2, on the other hand, brings together applications of these methods in specific physical situations, in plasmas, gravitational systems, Bose-Einstein condensates, as well as in cosmology and the study of dark matter.

More specifically, Chapter 1 will be devoted to kinetic theory and Chapter 2 is devoted to the hydrodynamic approach, both being developed in a classical, non-quantum, and non-relativistic framework. Chapter 3 presents a generalization of these methods within the framework of quantum mechanics. It aims to present the quantum hydrodynamic model and the quantum kinetic theory of Wigner-Moyal. These two approaches are indispensable tools for modern physics and allow for describing, on an equal footing, a large number of systems (Mendonça and Terças 2013; Mendonça 2019; Ourabah 2021), ranging from quantum plasmas to Bose-Einstein condensates and assemblies of cold atoms in optical networks. Chapter 4 presents a generalization of the kinetic and hydrodynamic approaches to the theory of relativity, restricted and general. Such generalization makes it possible to adapt hydrodynamic and kinetic methods to extreme astrophysical conditions, where the speeds of the particles approach the speed of light, and to curved spaces.

Part 2 brings together applications of kinetic and hydrodynamic approaches, and their generalizations to quantum mechanics and the theory of relativity, in various physical systems. When deemed necessary, both approaches will be applied to the same problem, allowing virtues and limitations of each to be seen. In other cases, one approach will be preferred, for simplicity, but with sufficient evidence in the use of the other method.

Chapter 5 will be devoted to the study of plasmas. We will mainly deal with oscillations in plasmas, both classical and quantum or relativistic. We will also study the phenomenon of Landau damping: a purely kinetic phenomenon that cannot be studied in a hydrodynamic framework. In Chapter 6, we will use the analogy between Colombian and gravitational interactions to study gravitational systems or, more precisely, systems of self-gravitating particles, that is, systems of particles interacting with each other via gravitational interactions. We will essentially study the phenomenon analogous to oscillations in plasmas, namely Jeans gravitational instability; it is the phenomenon that explains the formation of stars from molecular clouds. We will present different variants of the problem, for example, in the presence of dark matter...