Collective Phenomena in Plasmas and Elsewhere E-Book

Kamel Ourabah

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 206

Veröffentlichungsjahr: 2023

Beschreibung

The Universe is made up of systems consisting of a very large number of particles interacting in a very complex way. When studying these systems, a precise microscopic approach is unattainable. In practice, the best strategy is one that is able to "distinguish" between superfluous information and the information needed to make predictions about the evolution of the system. There are two main competing approaches: kinetic and hydrodynamic. Collective Phenomena in Plasmas and Elsewhere presents an overview of the theoretical bases of these kinetic and hydrodynamic approaches, but also discusses their limitations, the links between them and their extension to quantum mechanics and relativity. These methods were born in part out of the study of plasmas, but they also have more universal applications. A general framework for these approaches is outlined, followed by some applications in plasmas, gravitation, Bose-Einstein condensates and dark matter. Particular emphasis is placed on the analogies that can be made between all these different systems.

Leseprobe

Contents

Cover

Table of Contents

Title Page

Copyright Page

Preface

Introduction

Part 1: Theory

1 Kinetic Theory

1.1. The impossibility of an accurate approach

1.2. Boltzmann equation

1.3. Collision term

1.4. Steady state: Boltzmann distribution

1.5. Maxwell–Boltzmann distribution: properties

1.6. H-theorem

1.7. Paradoxes related to H-theorem

2 Hydrodynamic Approach

2.1. Fluid model: a heuristic approach

2.2. Macroscopic transport equation

2.3. Fluid model

2.4. Pressure tensor

2.5. A deadlock in the fluid model: closure relationships

2.6. The collision effect

3 Quantum Models

3.1. Physical interest

3.2. Quantum hydrodynamic model

3.3. Quantum kinetic approach: Wigner–Moyal theory

4 General Relativity

4.1. Relativistic hydrodynamics

4.2. Relativistic kinetic theory

Part 2: Applications

5 Plasmas

5.1. Electronic oscillations in classical plasmas: a hydrodynamic approach

5.2. Ion waves: hydrodynamic approach

5.3. Classic plasmas: kinetic approach

5.4. Quantification of electronic oscillations: hydrodynamic and kinetic approaches

5.5. In a relativistic plasma: kinetic approach

5.6. Beyond the linear approach: the Korteweg–De Vries equation

6 Gravitational Systems

6.1. Jeans instability: hydrodynamic approach

6.2. Kinetic approach and collision effect

6.3. Jeans instability in the presence of dark matter

6.4. Jeans instability: alternative theories

7 Bose-Einstein Condensates

7.1. Quantum hydrodynamic representation

7.2. Quantum kinetic representation

7.3. Kinetic approach to Bogoliubov oscillations

8 Cosmology and Dark Matter Models

8.1. Hydrodynamics of the Universe

8.2. Dark matter models

Appendix. Language of Relativity

References

Index

Other titles from iSTE in Fluid Mechanics

End User License Agreement

List of Tables

Chapter 1

Table 1.1. Table summarizing the mean values of the Maxwell–Boltzmann distribu...

Chapter 3

Table 3.1. Density and temperature parameter (electronics) for a variety of na...

Chapter 6

Table 6.1. Values of for different values of ratio σ

d

/σ

b

, for

Guide

Cover

Table of Contents

Title Page

Copyright Page

Preface

Introduction

Begin Reading

Appendix

References

Index

Other titles from iSTE in Fluid Mechanics

End User License Agreement

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Collective Phenomena in Plasmas and Elsewhere

Kinetic and Hydrodynamic Approaches

Kamel Ourabah

First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2023The rights of Kamel Ourabah to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2023938460

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-769-9

Preface

I am astounded by the number of peoplewho want to “know” the universewhen it’s hard enough to findyour way around Chintatown.

Woody Allen

During my physics studies, plasma physics classes always seemed boring to me. I could not say exactly why, but I aspired to fundamental things and struggled to see an essential aspect in plasma courses. It is therefore quite natural that I have turned to more fundamental branches of physics. It was only when I found myself joining a plasma physics laboratory, by a combination of circumstances, that I questioned my beliefs. I discovered that the tools and methods I associated with plasma physics were not confined to the study of plasmas but had a much more general scope.

Gradually, I discovered the exciting theories behind the study of plasmas in a clearer way, namely statistical physics and the kinetic theory of gases. Moreover, I also discovered, much to my surprise, the usefulness of methods that I thought were specific to plasmas in fields as diverse and varied as astrophysics, cosmology, dark matter models, the study of Bose–Einstein condensates, particle physics, etc. I reconciled myself, so to speak, with plasma physics to such an extent that I decided to write this book.

Beyond certain phenomena specific to plasmas, there are two major approaches in the study of plasmas and, more generally, in the study of complex systems, composed of a large number of degrees of freedom, namely the hydrodynamic approach and the kinetic approach. Each approach has its virtues and limitations. This book summarizes the theoretical bases of these two approaches and presents the qualities and limitations of each, as well as the bridges that exist between them. These two approaches have innumerable applications. Some of them are presented in this book; a drop of water in an ocean!

There are excellent books covering plasma physics in a much more comprehensive way. This book is perhaps less conventional in that it attempts to emphasize the more general character of the methods usually attributed (rightly or wrongly) to plasma physics. The methods are developed in a very general theoretical framework, followed by applications in plasmas, as well as in gravitational systems, the study of Bose–Einstein condensates, cosmology and dark matter models. The student I once was, who had no particular interest in plasma physics, would certainly have been delighted to open his eyes to the richness of these methods and the many fields in which they prove indispensable.

This book will hopefully be useful to master’s and doctoral students, as well as young researchers, working in fields involving the study of large numbers of particles and collective phenomena, such as plasma theory, condensed matter theory, astrophysics and cosmology. As scientific research becomes more and more interdisciplinary, this book may be of increasing interest in the future. Indeed, it is essential today for a physicist to have an overview of the different branches of physics, and to have even somewhat superficial knowledge. It is also essential to know the bridges that exist between the different branches and to aspire to build new ones.

As the process of scientific research is itself a collective phenomenon, I have benefited from a lot of exchanges with colleagues. These exchanges contributed greatly to guiding my thinking and allowed me to crystallize these ideas in this book. In particular, exchanges with Hugo Terças and Tito Mendonça opened my eyes to the richness and flexibility of the quantum kinetic approach. Many colleagues and friends also encouraged me (sometimes almost to the point of threatening!) to write this book. I will not be able to name them all here, but they will know who they are.

June 2023

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 and in the framework of alternative theories of gravity.

Chapter 7 will be devoted to the use of hydrodynamic and kinetic approaches to the study of Bose–Einstein condensates (a particular state of a quantum gas at very low temperature). Since Bose–Einstein condensates are quantum, we will present the use of quantum generalizations, namely the hydrodynamic-quantum model and the Wigner–Moyal kinetic theory. We shall see that, again, a phenomenon similar to the oscillations in plasmas exists in the Bose–Einstein condensates; this is what is called the Bogoliubov excitations. Finally, in Chapter 8, we will present applications of hydrodynamic and kinetic methods in the context of cosmology and the study of dark matter. We will mainly insist on the formulation of hydrodynamic equations in an expanding universe. We will also study a dark matter model, composed of scalar fields.

The list of applications presented here is, of course, far from exhaustive. The hydrodynamic and kinetic approaches are, in fact, highly flexible and find applications in an increasing number of fields. The applications discussed in this book can, however, be seen as a starting point for studying other systems that can be formally described by similar equations. Also, we will direct the reader, through a rich bibliography covering the state of the art of research in this field, to other applications in very active fields of scientific research.

PART 1Theory

1Kinetic Theory

The general struggle for existence of animate beings is nota struggle for raw materials [...], all abundantly available –nor for energy, which exists in plenty in any body in theform of heat Q, but of a struggle for entropy S, whichbecomes available through the transition ofenergy from the hot sun to the cold earth.

Ludwig Boltzmann

A complex system, such as a plasma or a system of self-gravitating particles, shows collective behavior. Thus, the evolution of one particle strongly depends on the evolution of the other particles in the medium. This collective behavior is actually the consequence of multiple interactions between a large number of particles and the fields they generate (electric and magnetic fields for a plasma or the gravitational field for systems of self-gravitating particles). Thus, the movement of each particle has the consequence of modifying fields, which in turn influences the trajectory of all particles, thus generating a modification of fields, and so on.

Given the large number of particles making up the system, a dynamic description (i.e. based on the individual equations of motion of each particle) is clearly out of reach. Fortunately, such an impractical approach is not necessarily desirable. Indeed, it is often useless to know the individual behavior of each particle. It is more useful to know the statistical behavior of the system. The most precise statistical approach is theoretical kinetics, to which this chapter is devoted.

1.1. The impossibility of an accurate approach

As explained earlier, an accurate approach, based on the dynamics of each particle, is an insoluble problem. However, let us try to develop such a model. This will make it possible, firstly, to better see why such an approach is feasible in practice and, secondly, it will constitute a good starting point for building the foundations of kinetic theory.

A dynamic approach first requires defining the trajectories of each particle that make up the system. Thus, it is interesting to consider the position and velocity of each particle as independent variables a six-dimensional space. This is called the phase space. Note the position of the particle and its velocity vector. Thus, a particle is characterized, at a given instant t, by a point located in a volume d3xd3v in the phase space. Thus, the evolution of the particle (whose position and velocity vary over time) corresponds to a trajectory in this six-dimensional space.

To simplify, let us first consider the simple case of a single point particle whose position in phase space can be described by a Dirac1. In real space (that of positions), this particle has a trajectory , representing the positions occupied by the particle over time. Similarly, the particle has a trajectory in velocity space. By combining the coordinates of space and velocity , in a six-dimensional space, we can thus write the density of a particle as follows (Savoini 2010):

[1.5]