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showing that the Vlasov equation is satisfied for arbitrary g, since theζi are constants and their total derivative is by definition zero. Thus, (3.36) is a solution of Vlasov’s equation regardless of the nature of the ∂g/∂ζi terms. In other words, any function of the constants of motion of a particle is a solution of the Vlasov equa- tion, a result which is known as Jeans’s theorem. As an example, consider a plasma in the absence of an electric field, i.e., E= 0. In this case, there can be no acceleration of any particle and its energy remains constant, so 1₂mv2 is a constant of motion. Thus, any function of 1₂mv2 is a solution of the Vlasov equation, e.g., the Maxwell–Boltzmann distribution, f(v) = e−mv2/(2kBT).

Expanding on this concept, we can find the equilibrium distribu- tion function for a gas in the presence of an external conservative force. A conservative force is one that can be written as a gradient of a scalar potential. This is another way of saying that the action of such a force is reversible and leads directly to a change in potential energy. Gravity and the electric and magnetic forces of the Lorentz force are conservative forces. Frictional and drag forces, on the other hand, are not conservative since they lead to a loss of kinetic energy that cannot be retrieved. Under the action of a conservative force, the sum of the potential and kinetic energies of particles should therefore be a constant of motion. If the force is specified in terms of a potential energy U(r) by

F(r) = −∇U(r)

then the equilibrium distribution function of a gas in the presence of this force is given by

f(v) = N0  m 2πkBT 3 2 e  −1 2mv2−U  /(kBT)_, (3.38) which is the unperturbed Maxwell–Boltzmann distribution of Equation (3.20), multiplied by a correction term e−U(r)/kBT known as the Boltzmann factor. We can now consider the problem of a plasma in the presence of an electrostatic field. If there is an electro- static field E(r) present it can be specified by an electrostatic poten- tial(r). The Boltzmann factor in this case is simply e−q(r)/kBT, making the distribution function

f(v) = N0  m 2πkBT 3 2 e  −1 2mv2−q  /(kBT)_.

The number density for this situation, obtained by integrating over velocity space, is

N(r) = N₀e(−q(r))/(kBT), (3.39) which is the expression that was used in Chapter 1 (Equation (1.6)) to derive the Debye length.

3.8 Summary

In this chapter we introduced kinetic theory, the description of a plasma using a probability distribution function f(r, v, t) describing the likelihood of finding particles at a given position, at a given velocity, and at a given time. Employing a probability distribution function is a logical way to proceed from discussing single-particle motions, as was done in Chapter 2, to the description of the col- lective behavior of the particles that make up a plasma. In kinetic theory a plasma is described in a six-dimensional phase-space coor- dinate system with three spatial dimensions (x, y, z) and three velocity dimensions (vx, vy, vz). The way a particle distribution changes under the influence of external forces is described by the Boltzmann equation. Derivation of the Boltzmann equation follows from the fact that velocity changes will originate from the Lorentz force and position changes can be derived from velocities. If we neglect collisions, the Boltzmann equation reduces to the Vlasov equation. In a limiting case, the Vlasov equation can be interpreted as stating that, without external forces or collisions, the distribution function (as measured by a total derivative) in fact does not change, which is an intuitively pleasing result.

In the second half of the chapter we discussed the most probable distribution function that a plasma will assume under equilibrium. This is the Maxwell–Boltzmann distribution, which all plasmas and gases will approach, given enough time, if there is no net energy flow into or out of the system. The Maxwell–Boltzmann distribu- tion is prevalent in many applications of plasma physics and has remarkable properties. At the same time there are many applica- tions where a Maxwell–Boltzmann distribution cannot be assumed nor be observed. It is important to emphasize that kinetic theory, including the Boltzmann and Vlasov equations, is general and can be applied to all plasma distributions. Plasma fluid theories, dis- cussed in the following chapters, follow directly from kinetic theory but are not as general and are applicable only to situations where equilibrium or quasi-equilibrium conditions can be assumed.

3.9 Problems 81

3.9 Problems

3-1. Write down the distribution function for the following cases: (a) two infinite particle beams each with density N0 moving

in opposite directions along the x axis at a speed of v; (b) an infinite particle population with all speeds less than the maximum speedvmaxbeing equally probable.

3-2. Calculate the average number density N(x, t) and the average kinetic energy)1₂mv2*for the following distribution functions: (a)f(r, v, t) = K0δ(vx)δ(vy− v0)δ(vz); (b)f (r, v, t) =



A2₀− v2_x



A2₀− v2_yA₀2− v2_zfor|vi| < A0 (i = x, y, z).

3-3. The electrons inside a system of two coaxial magnetic mir- rors can be described by the so-called loss-cone distribution function, f(v) =  4 π 3 2 ₁ α2 ⊥α||  v2 ⊥ α2 ⊥  exp  −v⊥2 α2 ⊥ − v 2 || α2 ||  ,

where v⊥ and v|| denote the electron velocities in the direc- tions perpendicular and parallel to the magnetic bottle axis, respectively, and whereα_⊥2 = 2kBT_⊥/meandα_||2= 2kBT||/me. (a) Determine the number density of electrons N0 in the

magnetic bottle. (b) Determine the average perpendicular and parallel energies.

3-4. Given a particle distribution function of the form

f(r, v) = ⎧ ⎪ ⎨ ⎪ ⎩ Ae−x/a  1+ cos  πv3 v3 0  v ≤ v0 0 v ≥ v0 ,

where v0 is a constant and v = |v|: (a) Determine the con-

stant A, given that the particle number density at x= 0 is N0.

(b) If an application requires that the spatial dependence of f remain constant, how could the spatial distribution be maintained? Give a quantitative description of the proposed scheme.

3-5. Consider an equilibrium (Maxwellian) plasma with electron temperature Te and electron density Ne immersed in a con- stant uniform magnetic field B. (a) Find an expression for μ, the average value of the magnetic moment of the elec- tron due to its gyration around B, where μ = mev2_⊥

/

(b) For B= 5 × 10−5T and Te = 300 K, compare the magni- tude ofμ to the spin magnetic moment μB of the electron, which is called the Bohr magneton and is given by

μB = |qe|h 4πme,

where h = 6.626 × 10−34J s is Planck’s constant.

3-6. In some plasmas confined in tokamak machines, distinct pop- ulations of energetic ions are created which do not have a Maxwellian distribution function. An example is the so-called “slowing-down” distribution, given by

f_{i on} = ⎧ ⎨ ⎩ A v3_{+ a}3 v ≤ v0 0 v > v0 ,

wherev0is the velocity at which the energetic ions are created

(by fusion reactions) and a is a constant determined by the rate of collisions. (a) Determine the constant A if the average ion density is Ni. (b) Determinevmax, the most probable value ofv for this distribution.

3-7. Use direct substitution to show that a distribution function of the form

f = f



mv2/2 + qφ



is a steady-state solution of the Boltzmann equation, where

φ is an electric potential. Assume only one dimension and

neglect collisions.

3-8. Use the Saha equation to show that the solar wind, which is an electron and proton plasma, must be almost com- pletely ionized. The electron density of the solar wind is 3.5 × 106m−3 and the temperature is 105K. The ionization potential of hydrogen is 13.6 eV.

3-9. A candle flame has a maximum temperature of ∼1600 K.

Find the degree of ionization of the flame and discuss whether any plasma properties can be observed. Also find the Debye length.

3-10. A cylindrical hot-water heater, initally OFF, has a vertical temperature gradient of 50◦C m−1increasing from bottom to top. The heat source surrounding the cylinder is turned ON, increasing the temperature at a rate of 1◦C s−1. What will be the temperature change experienced by a rock, dropped into

References 83

the water at the top of the cylinder, that falls to the bottom at a speed of 0.5 m s−1? The rock is dropped in at the same time the heater is turned ON.

References

[1] E. H. Holt and R. E. Haskel, Foundations of Plasma Dynamics. (New York: Macmillan, 1965), Section 5.4.

[2] J. A. Bittencourt, Fundamentals of Plasma Physics, 3rd edn (New York: Springer-Verlag, 2004), 589–607.

[3] S. Chandrasekhar, Principles of Stellar Dynamics (Chicago: University of Chicago Press, 1942).

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Moments of the Boltzmann