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Thusv ¼ 2π 1=2 vth ¼ 2ð2KT=πmÞ1=2ð7:11Þ2147 Kinetic TheoryThe velocity component in a single direction, say vx, has a different average.Of course, vx vanishes for an isotropic distribution; but jvx j does not:ðð7:12Þjvx j ¼ jvx j ^f m ðvÞd 3 v!ð 2 m 3=2 ð 1v2y 1vzdv y exp¼dvexpz22πKTv2thv11th 2ð1v2vx exp 2 x dvxvth0ð7:13ÞFrom Eq. (7.6), each of the first two integrals has the value πl/2vth. The last integralis simple and has the value v2th . Thus we have3=2 4πvth ¼ π 1=2 vth ¼ ð2KT=πmÞ1=2jvx j ¼ πv2thð7:14Þ11Γrandom ¼ njvx j ¼ nv24ð7:15ÞThe random flux crossing an imaginary plane from one side to the other is given byHere we have used Eq.
(7.11) and the fact that only half the particles cross the planein either direction. To summarize: For a Maxwellian,vrms ¼ ð3KT=mÞ1=2jvj ¼ 2ð2KT=πmÞ1=2jvx j ¼ ð2KT=πmÞ1=2vx ¼ 0ð7:7Þð7:11Þð7:14Þð7:16ÞFor an isotropic distribution like a Maxwellian, we can define another function g(v)which is a function of the scalar magnitude of v such thatð10gðvÞdv ¼ð11f ðvÞd3 vð7:17ÞFor a Maxwellian, we see from Eq. (7.9) thatgðvÞ ¼ 4πnðm=2πKT Þ3=2 v2 exp v2 =v2thð7:18ÞFigure 7.3 shows the difference between g(v) and a one-dimensional Maxwelliandistribution f(vx). Although f(vx) is maximum for vx ¼ 0, g(v) is zero for v ¼ 0.7.1 The Meaning of f(v)215Fig. 7.3 One- and three-dimensional Maxwellian velocity distributionsFig.
7.4 A spatiallyvarying one-dimensionaldistribution f(x, vx)This is just a consequence of the vanishing of the volume in phase space (Fig. 7.2)for v ¼ 0. Sometimes g(v) is carelessly denoted by f(v), as distinct from f(v); but g(v)is a different function of its argument than f(v) is of its argument. From Eq. (7.18), itis clear that g(v) has dimensions s/m4.It is impossible to draw a picture of f(r, v) at a given time t unless we reduce thenumber of dimensions. In a one-dimensional system, f(x, vx) can be depicted as asurface (Fig. 7.4). Intersections of that surface with planes x ¼ constant are thevelocity distributions f(vx). Intersections with planes vx ¼ constant give densityprofiles for particles with a given vx.
If all the curves f(vx) happen to havethe same shape, a curve through the peaks would represent the density profile.The dashed curves in Fig. 7.4 are intersections with planes f ¼ constant; these arelevel curves, or curves of constant f. A projection of these curves onto the x vx2167 Kinetic TheoryFig. 7.5 Contours ofconstant f for atwo-dimensional,anisotropic distributionFig.
7.6 Contours of constant f for a drifting Maxwellian distribution and a “beam” in twodimensionsplane will give a topographical map of f. Such maps are very useful for getting apreliminary idea of how the plasma behaves; an example will be given in thenext section.Another type of contour map can be made for f if we consider f(v) at a givenpoint in space. For instance, if the motion is two dimensional, the contours off(vx, vy) will be circles if f is isotropic in vx, vy. An anisotropic distribution wouldhave elliptical contours (Fig.
7.5). A drifting Maxwellian would have circularcontours displaced from the origin, and a beam of particles traveling in thex direction would show up as a separate spike (Fig. 7.6).A loss cone distribution of a mirror-confined plasma can be represented bycontours of f in v⊥, v|| space. Figure 7.7 shows how these would look.7.2 Equations of Kinetic Theory217Fig. 7.7 Contours ofconstant f for a loss-conedistribution.
Here v|| and v⊥stand for the componentsof v along andperpendicular to themagnetic field, respectively7.2Equations of Kinetic TheoryThe fundamental equation which f(r, v, t) has to satisfy is the Boltzmann equation:∂fF ∂fþv∇f þ ¼∂tm ∂v∂f∂tcð7:19ÞHere F is the force acting on the particles, and (∂f/∂t)c is the time rate of change off due to collisions. The symbol ∇ stands, as usual, for the gradient in (x, y, z) space.The symbol ∂/∂v or ∇v stands for the gradient in velocity space:∂∂∂∂¼ x^þ y^þ z^∂v∂vx∂v y∂vzð7:20ÞThe meaning of the Boltzmann equation becomes clear if one remembers that f is afunction of seven independent variables.
The total derivative of f with time is,therefored f ∂ f ∂ f dx ∂ f dy ∂ f dz ∂ f dvx ∂ f dv y ∂ f dvz¼þþþþþþdt∂t ∂x dt ∂y dt ∂z dt ∂vx dt ∂v y dt∂vz dtð7:21ÞHere, ∂f/∂t is the explicit dependence on time. The next three terms are just v · ∇f.With the help of Newton’s third law,mdv¼Fdtð7:22Þthe last three terms are recognized as (F/m) · (∂f/∂v). As discussed previously inSect.
3.3, the total derivative df/dt can be interpreted as the rate of change as seen in2187 Kinetic TheoryFig. 7.8 A group of pointsin phase space, representingthe position and velocitycoordinates of a group ofparticles, retains the samephase-space density as itmoves with time.a frame moving with the particles. The difference is that now we must consider theparticles to be moving in six-dimensional (r, v) space; df/dt is the convectivederivative in phase space. The Boltzmann equation (7.19) simply says that df/dtis zero unless there are collisions. That this should be true can be seen from theone-dimensional example shown in Fig.
7.8.The group of particles in an infinitesimal element dx dvx at A all have velocity vxand position x. The density of particles in this phase space is just f(x, vx). As timepasses, these particles will move to a different x as a result of their velocity vx andwill change their velocity as a result of the forces acting on them.
Since the forcesdepend on x and vx only, all the particles at A will be accelerated the same amount.After a time t, all the particles will arrive at B in phase space. Since all the particlesmoved together, the density at B will be the same as at A. If there are collisions,however, the particles can be scattered; and f can be changed by the term (∂f/∂t)c.In a sufficiently hot plasma, collisions can be neglected. If, furthermore, theforce F is entirely electromagnetic, Eq. (7.19) takes the special form∂fq∂fþ v ∇ f þ ðEþv BÞ ¼0∂tm∂vð7:23ÞThis is called the Vlasov equation.
Because of its comparative simplicity, this is theequation most commonly studied in kinetic theory. When there are collisions withneutral atoms, the collision term in Eq. (7.19) can be approximated by ∂ff f¼ n∂t cτð7:24Þwhere fn is the distribution function of the neutral atoms, and τ is a constantcollision time. This is called a Krook collision term. It is the kinetic generalization7.2 Equations of Kinetic Theory219Fig.
7.9 Representation in one-dimensional phase space of a beam of electrons all with the samevelocity v0. The distribution function f(x, vx) is infinite along the line and zero elsewhere. The lineis also the trajectory of individual electrons, which move in the direction of the arrow.of the collision term in Eq. (5.5). When there are Coulomb collisions, Eq.
(7.19) canbe approximated by2df∂1 ∂¼ ð f hΔviÞ: ð f hΔv ΔviÞdt∂v2 ∂v∂vð7:25ÞThis is called the Fokker–Planck equation; it takes into account binary Coulombcollisions only. Here, Δv is the change of velocity in a collision, and Eq. (7.25) is ashorthand way of writing a rather complicated expression.The fact that df/dt is constant in the absence of collisions means that particlesfollow the contours of constant f as they move around in phase space. As anexample of how these contours can be used, consider the beam-plasma instabilityof Sect.
6.6. In the unperturbed plasma, the electrons all have velocity v0, and thecontour of constant f is a straight line (Fig. 7.9). The function f(x, vx) is a wall risingout of the plane of the paper at vx ¼ v0. The electrons move along the trajectoryshown. When a wave develops, the electric field E1 causes electrons to sufferchanges in vx as they stream along. The trajectory then develops a sinusoidal ripple(Fig. 7.10). This ripple travels at the phase velocity, not the particle velocity.Particles stay on the curve as they move relative to the wave. If E1 becomes verylarge as the wave grows, and if there are a few collisions, some electrons will betrapped in the electrostatic potential of the wave.
In coordinate space, the wavepotential appears as in Fig. 7.11. In phase space, f(x, vx) will have peaks whereverthere is a potential trough (Fig. 7.12). Since the contours of f are also electrontrajectories, one sees that some electrons move in closed orbits in phase space; theseare just the trapped electrons.Electron trapping is a nonlinear phenomenon which cannot be treated bystraightforward solution of the Vlasov equation. However, electron trajectoriescan be followed on a computer, and the results are often presented in the formof a plot like Fig.
7.12. An example of a numerical result is shown in Fig. 7.13.Fig. 7.10 Appearance of the graph of Fig. 7.9 when a plasma wave exists in the electron beam.The entire pattern moves to the right with the phase velocity of the wave. If the observer goes to theframe of the wave, the pattern would stand still, and electrons would be seen to trace the curve withthe velocity v0 vϕ.Fig. 7.11 The potential of a plasma wave, as seen by an electron. The pattern moves with thevelocity vϕ. An electron with small velocity relative to the wave would be trapped in a potentialtrough and be carried along with the wave.Fig.
7.12 Electron trajectories, or contours of constant f, as seen in the wave frame, in which thepattern is stationary. This type of diagram, appropriate for finite distributions f(v), is easier tounderstand than the δ-function distribution of Fig. 7.10.7.2 Equations of Kinetic Theory221Fig. 7.13 Phase-space contours for electrons in a two-stream instability. The shadedregion, initially representing low velocities in the lab frame, is devoid of electrons. As theinstability develops past the linear stage, these empty regions in phase space twist into shapesresembling “water bags” [From H. L.
Berk, C. E. Nielson, and K. V. Roberts, Phys. Fluids 13,986 (1970)].This is for a two-stream instability in which initially the contours of f have a gapnear vx ¼ 0 which separates electrons moving in opposite directions. The development of this uninhabited gap with time is shown by the shaded regions in Fig. 7.13.This figure shows that the instability progressively distorts f(v) in a way whichwould be hard to describe analytically.2227.37 Kinetic TheoryDerivation of the Fluid EquationsThe fluid equations we have been using are simply moments of the Boltzmannequation. The lowest moment is obtained by integrating Eq.
(7.19) withF specialized to the Lorentz force:ðððð ∂fq∂f∂fðE þ v B Þ dv þ v ∇ f dv þdv ¼dv∂tm∂v∂t cð7:26ÞThe first term givesð∂f∂dv ¼∂t∂tðf dv ¼∂n∂tð7:27ÞSince v is an independent variable and therefore is not affected by the operator ∇,the second term givesððv ∇ f dv¼∇ v f dv ¼ ∇ ðnvÞ ∇ ðnuÞð7:28Þwhere the average velocity u is the fluid velocity by definition. The E term vanishesfor the following reason:ððð∂f∂dv ¼ ð f EÞdv ¼Ef E dS ¼ 0∂v∂vS1ð7:29ÞThe perfect divergence is integrated to give the value of f E on the surface at v ¼ 1.This vanishes if f ! 0 faster than v2 as v ! 1, as is necessary for any distributionwith finite energy. The v B term can be written as follows:ððv BÞ ðð∂f∂∂dv¼ ð f v BÞdv f ðv BÞdv ¼ 0∂v∂v∂vð7:30ÞThe first integral can again be converted to a surface integral.
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