2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all

Transcription

1 Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either neglected or unabiguously linked to the plasa density; then, at any given location, plasa is described just by few variables: the density, the fluid velocity, and the pressure (separately for each species). However, in reality the nuber of degrees of freedo of plasa is uch larger, because plasa consists of individual particles, and each of the is described by its independent variables such as coordinates and velocities. A coplete (Kliontovich) description would have to account for all these variables. But a significant iproveent in understanding plasa dynaics can be achieved also through a sipler, statistical theory, which is known as kinetic theory. Kinetic theory treats plasa as a fluid too, except flows are considered not in the coordinate space but rather in phase space. Below, we will discuss soe of the basics of this theory and later apply it to describing specific plasa waves. Coent: The kinetic theory that is discussed here by default assues classical plasa. Quantu effects can be captured by kinetic theory too, but then a substantial odification of the kinetic theory is needed, because quantu particles cannot be assigned specific coordinates and oenta siultaneously due to the uncertainty principle. Instead, quantu particles can be odeled as waves (then, linear plasa oscillations ust be viewed as nonlinear odulational dynaics of quantu atter) using the so-called Wigner function and the Moyal equation. A siilar foralis of wave kinetics also allows one to study nonlinear interactions of any other waves in classical plasas. These topics will be briefly discussed at the end of the course. Distribution function Suppose that the otion of a single particle is fully described by soe set of phase space variables [e.g., x, v or x, p]; then, in space, one can define the particle density, (for each given species independently). Consider two different sets of such variables apped into each other by soe function Q, Q :. First, suppose a density corresponding to a single particle with soe coordinate 0. Clearly, such density is given by 0. In ters of the particle coordinate 0 in the space, the sae function can be expressed as Q 0. Likewise, in case of ultiple particles, the corresponding density in ust be averaged over all 0, or, in other words, integrated over their density. This gives the following general rule for apping the density in to the density in : Q Q, where the latter ratio denotes the Jacobian of the corresponding variable transforation, and Q is the function inverse to Q. Aong all possible variable transforations, there exists a special class of so-called canonical transforations. They correspond to unit Jacobians, so canonical densities (which we denote with as opposed to general densities

2 lecture-06.nb ) are transfored siply as Q. In this sense, one can talk about the canonical density, whose value is invariant with respect to the choice of specific (canonical) coordinates in phase space. Liouville s theore It is well known (see any textbook on analytical echanics) that the syste evolution in tie, Q t : 0 t, can be considered as a canonical transforation too. (Here Q t can be understood as the propagator, and the index t denotes tie.) This gives an equation describing the evolution of the canonical density, t Q t More explicitly, this can also be written as t, xt, x 0, p 0, pt, x 0, p 0 0 x 0, p 0, where x, p are soe canonical coordinates, and x 0, p 0 are their initial values. (The index t has been oitted for clarity.) By differentiating this equality with respect to tie, one gets 0 t t, xt, x 0, p 0, pt, x 0, p 0 which is known as Liouville s theore. t x t x p t p, Coent: This also can be given the following interpretation. Consider particles within soe phase space eleent with volue. According to a fundaental theore of classical echanics, phase flow can distort the shape of this eleent but not the volue; i.e., const. But the nuber of particles inside this volue is conserved too, and that gives const. Hence, const along the trajectory followed by each given phase space eleent. This again leads to Liouville s theore. In nonrelativistic plasa physics, it is also custoary to work with the particle density in the x, v space, f t, x, v, instead of the canonical phase-space density t, x, p. The relation between f and is obtained using the sae forula that we derived above for general coordinate transforations, naely, f t, x, v t, x, pt, x, v x, p x, v. For nonrelativistic plasas, which we consider below, one can take pt, x, v v e At, xc, where A is the electroagnetic vector potential. The corresponding Jacobian is then siply x, p x, v 3 const. Hence, for f, we get the sae equation as for, f t v f F where F is the force. f v 0, Coent: This equation is odified in relativistic plasas and, particularly, in curved spacetie. For details, see, e.g., [Phys. Plasas 7, 8 (00)]. The entioned paper also explains how to write the kinetic

3 lecture-06.nb 3 equation in curvilinear coordinates in flat spacetie. Vlasov equation In general, both f and F can be split into acroscopic quantities, which we denote with index (not to be confused with ass), and fluctuating quantities, which we denote with prefix. Then, after averaging over fluctuations, one obtains f t v f F f v F f v. where the right hand side is known as the collision ter and is coonly denoted as C f. In neutral gases, longrange interactions are weak, so the collision ter doinates; then, f v f C f. t In plasas, the situation is the opposite. If the nuber of particles within the Debye sphere is large, each particle feels any neighbors siultaneously, so the corresponding fluctuations utually cancel out, and their cuulative effect is weak. Below, we consider the liit of ideal plasa, where this effect is sall enough to be negligible. Then the collision ter can be oitted, and one gets f t v f F f v 0, or, with F explicitly written as the Lorentz force, f v f e t E c vb f v 0. This is known as the (collisionless) Vlasov equation. Below, we adopt it as our priary odel of plasa dynaics. The index will be oitted for brevity. Linearized Vlasov equation As usual, for studying waves we will assue weak oscillating fields, so f n 0 f 0 f, where f is linear in the wave field, and f 0 is the unperturbed distribution. The coefficient n 0 is added to ensure the following noralization: f 0 v 3 Υ. Then, f satisfies f t v f c vb 0 f v e n 0 E c vb f 0 v 0, where ters quadratic in the wave field have been dropped. (Note that, in contrast to the fluid description that we studied earlier, the velocity here is an independent variable; thus, v f is a linear ter and thus ust be retained.) Assuing that the background plasa is hoogeneous and stationary, we also have f 0 f 0 v. Hence the equation can be Fourier-transfored in space, f t kv f c vb 0 f v e n 0 E c vb f 0 v 0 (here, we dropped the index k for clarity), and solved as an initial value proble ore or less straightforwardly. Below, we do so for various cases of interest.

4 4 lecture-06.nb Electrostatic oscillations in nonagnetized plasa Longitudinal conductivity We start with the case of nonagnetized plasa (B 0 0) and consider electrostatic waves first (B 0). In this case, the electric field is parallel to k, E φ k φ. Let us choose axes such that x points along k. Hence, the linearized Vlasov equation becoes f t k Υ x f e n 0 E f 0 0, Υ x or, equivalently, f t k Υ x f e n 0 E f 0. Υ x This is a driven Schrödinger equation, which we already discussed at earlier lectures. Hence we can iediately write its general solution, f t, v f 0t, v e n 0 f 0 v Υ x 0 t k Υ xtt' E t ' t ', where f 0 is deterined by initial conditions but not by the field. In particularly, for a delta-shaped field, E t t 0, the induced perturbation is f t, v f t, v f 0t, v e n 0 f 0 v k Υ x t. Υ x The corresponding induced current, e Υx f t, v 3 Υ, as usual, gives the conductivity, Σ k t e n 0 Υ x f 0 v Υ x k Υ x t 3 Υ. We now change the notation as follows: f 0 v Υ y Υ z f 0 Υ x, so f 0 Υ is understood as the distribution of velocity coponents parallel to k, and f0 Υ Υ [as opposed to f0 v 3 Υ that we adopted so far]. Then, Σ k t p Υ f 0 ' Υ k Υ t Υ. Phase ixing It is also instructive to express Σ k t as follows: Σ k t p p t t k k f 0 ' Υ k Υ t Υ f 0 Υ Υ k Υ t Υ

5 lecture-06.nb 5 p k k f 0 Υ k Υ t Υ p k k 0 k t, where 0 is the Fourier iage of f 0. In case when the background plasa is cold and stationary, i.e., f 0 Υ Υ, we get 0, so Σ k t p. This is in agreeent with our earlier discussion of cold-plasa dispersion. But the cold-plasa approxiation is only a crude odel. In reality, f 0 Υ is soe sooth function of finite width. In this case, the plasa response exhibit a qualitatively different behavior, which we will now discuss. It is well known that the behavior of the Fourier iage depends on the soothness of the original. (For exaple, if the original is sooth, the Fourier iage decays in tie exponentially; see the basic properties of the Fourier transfor.) In any case, unless f 0 Υ is delta-shaped or k is zero, then 0 k t 0 at t. More specifically, if f 0 has soe characteristic scale Υ T, then 0 has a scale Υ T ; then Σ k t decreases to zero on the tie scale k Υ T. This effect is known as phase ixing, because it is caused by the destructive interference of the contributions to Σ k produced by f at different velocities. Coent: To better understand the phase ixing effect, it is instructive to nuerically explore the behavior of sus like n g n n t with sufficiently any different n (e.g., distributed randoly within a large enough interval) and ore or less arbitrary g n. Note also that the decay of the plasa cuulative response, which is described by Σ k t, does not ean that oscillations of f decays. Once perturbed, the distribution function continues to oscillate forever (ore precisely, until collisions coe into play). In other words, a plasa can store inforation even after acroscopic currents have dissipated. (For a echanical analogy, see One interesting effect that results fro this is the so-called plasa wave echo [Phys. Rev. Lett. 9, 9 (967)]. Exaple: Maxwellian plasa To illustrate our result, let us calculate the response function Σ k t for Maxwellian plasa, i.e., f 0 Υ Υ exp Π Υ T Υ. T Assuing the notation Α k Υ T t, we then obtain Σ k t p k k f 0 Υ k Υ t Υ p p p k k Α Α Α IΑ. Α Υ exp k Υ t Υ Π Υ T Υ T Π exp z Α z z

6 6 lecture-06.nb Here, IΑ Π exp Α z z Π exp Α Π Α, Α exp y Α z y y y y Α IΑ Α Α Α Α Α Α, so Σ k t p k Υ T t exp k Υ T t It is seen that the phase ixing occurs on the tie scale k Υ T, as anticipated Σ k tσ k k T t Using the above result, we can also readily calculate the conductivity in the spectral representation, Σ, k p 0 k Υ T t exp t k Υ T t t. By changing the integration variable to Τ t, we can cast this as follows: Σ, k p 0 where we introduced Ζ. k Υ T k Υ T Τ exp Τ Τ k Υ T Τ p 0 Τ Ζ exp Τ Τ 4 Ζ Τ, This integral can also be expressed in ters of the so-called plasa dispersion function that is defined as follows (see the figure below):

7 lecture-06.nb 7 ZΖ 0 exp Ζ z z 4 z Ζ 0 exp p p 4 Ζ p Ζ Π Ζ. (We assued k 0, otherwise the upper liit would have changed its sign). Here, Ζ is the so-called Dawson function, Ζ expζ 0 Ζexpy y Ζ Π erfiζ, which is an entire function. In order to express Σ through Z, notice that Z ' Ζ ZΖ This gives Ζ Σ, k p Τ Τ Ζ exp Τ Τ 0 Ζ 4 Ζ Ζ Τ exp Τ Τ Τ. 0 Ζ 4 Ζ Ζ Z ' Ζ. Accordingly, the susceptibility is given by Σ, k Χ, k p Ζ Z ' Ζ p k Υ T Z ' Ζ Z ' Ζ k Λ D, where Λ D Υ T p is the Debye length. Since Z is an entire function of its arguent, Χ is also an entire function of. Figure