Contents20 Kinetic Theory of Warm Plasmas 120.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.2 Basic Concepts of Kinetic Theory and its Relationship to Two-Fluid Theory 220.2.1 Distribution Function and Vlasov Equation . . . . . . . . . . . . . . 220.2.2 Relation of Kinetic Theory to Two-Fluid Theory . . . . . . . . . . . 420.2.3 Jeans’ Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620.3 Electrostatic Waves in an Unmagnetized Plasma: Landau Damping . . . . . 720.3.1 Formal Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . 720.3.2 Two-Stream Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 920.3.3 The Landau Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020.3.4 Dispersion Relation For Weakly Damped or Growing Waves . . . . . 1220.3.5 Langmuir Waves and their Landau Damping . . . . . . . . . . . . . . 1420.3.6 Ion Acoustic Waves and Conditions fo r their Landau Damping to beWeak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1620.4 Stability of Electrostatic Waves in Unmagnetized Plasmas . . . . . . . . . . 1920.5 Particle Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2120.6 N Particle Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . 240Chapter 20Kinetic Theory of Warm PlasmasVersion 0820.1.K.pdf, 8 April 2009.Please send comments, sugge stions, and errata via email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 91125Box 20.1Reader’s Guide• This chapter relies significantly on:– Portions of Chap. 2 on kinetic theory: Secs. 2.2.1 and 2.2.2 on the distributionfunction, and Sec. 2.6 on Liouville’s theorem and the collisionless Boltzmannequation.– Section 19.3 on Debye shielding, collective behavior of plasmas and plasmaoscillations.– Portions of Chap. 20: Wave equation and dispersion relation for dielectrics(Sec. 19.2), and two- fluid formalism a nd its application to Langmuir and ionacoustic waves (Secs. 1 9.3 and 19 .7 )• Chapter 22 on nonlinear dynamics of plasmas relies heavily on this chapter.20.1 OverviewAt the end of Chap. 20, we showed how to generalize cold-plasma two -fluid theory so as toaccommodate several distinct plasma beams, and thereby we discovered an instability. If thebeams are not individually monoenergetic (i.e. cold), as we assumed they were in Chap. 2 0,but instead have broad velocity dispersions that overlap in velocity space (i.e. if the beamsare warm), t hen the approach of Chap. 20 cannot be used, and a more powerful description ofthe plasma is required. Chapter 20’s approach entailed specifying the positions and velocities12of specific groups o f particles (the “fluids”); this is an example of a La g rangian description.It turns out that the most robust and powerful method for developing the kinetic theory ofwarm plasmas is an Eulerian one in which we specify how many particles are to be found ina fixed volume of one-particle phase space.In this chapter, using this Eulerian approach, we develop the kinetic theory of plasmas.We begin in Sec. 20.2 by introducing kinetic theory’s one-particle distribution function,f(x, v, t) and recovering its evolution equation (the collisionless Boltzmann equation, alsocalled Vlasov equation), which we have met previously in Chap. 2. We then use this Vlasovequation to derive the two-fluid formalism used in Chap. 2 0 and to deduce some physicalapproximations that underlie the two-fluid description of plasmas.In Sec. 20.3 we explore the application of the Vlasov equation to Langmuir waves—theone-dimensional electrostatic modes in an unmagnetized plasma that we met in Chap. 20.Using kinetic theory, we rederive the Bohm-Gross dispersion relation for Langmuir waves,and as a bonus we uncover a physical damping mechanism, called Landau damp i ng, that didnot and cannot emerge from the two-fluid analysis of Chap. 20. This subtle process leadsto the transfer of energy from a wave to those particles that can “surf” or “phase-ride” thewave (i.e. those whose velocity resolved parallel to t he wave vector is just slightly less thanthe wave’s phase speed). We show that Landau damping works because there a re usuallyfewer particles traveling faster than the wave and augmenting its energy density tha n thosetraveling slower and extracting energy from it. However, in a collisionless plasma, the particledistributions need not be Maxwellian. In particular, it is possible for a plasma to possessan “inverted” par t icle distribution with more fast ones than slow ones; then there is a netinjection of particle energy into the waves, which creates an instability. In Sec. 20.3 we usekinetic theory to derive a necessary and sufficient criterion for instability due to t his cause.In Sec. 20.4 we examine in greater detail the physics of Landau damping and show thatit is an intrinsically nonlinear phenomenon; and we give a semi-quantitative discussion o fNonlinear Landau damping , prefatory to a more detailed treatment of some ot her nonlinearplasma effects in the following chapter.Although the kinetic-theory, Vlasov description of a plasma that is developed and usedin this chapter is a great improvement on the two-fluid description of Chap. 20 , it is still anapproximation; and some situations require more accurate descriptions. We conclude thischapter by introducing greater accuracy via N-particle distribution functions, and as applica-tions we use them (i) to explore the approximations underlying the Vlasov description, and(ii) to explore two- par t icle correlations that are induced in a plasma by Coulomb interactionsand the influence of those correlations on a plasma’s equation of state.20.2 Basic Concepts of Kinetic Theory and its Rela-tionship to Two-Fluid Theory20.2.1 Distribution Function and Vlasov EquationIn Chap. 2 we introduced the number density of particles in phase space, called the dis-tribution function N(P, ~p). We showed that, this quantity is Lorentz invariant and that3it satisfies the collisionless Boltzmann equation (2.61) and (2.62); and we interpreted thisequation as N being constant along the phase-space trajectory of any freely moving particle.In order to comply with the conventions of the plasma-physics community, we shall usethe name Vlasov equation in place of collisionless Boltzmann equation,1and we shall changenotation in a manner described in Sec. 2.2.2: We
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