1. Introduction to Landau damping
1.3. Range of validity
The following issues are addressed in the literature [42], [46], [61], [95] and slightly con-troversial:
• Does Landau damping really hold for gravitational interaction? The case seems thinner in this situation than for plasma interaction, all the more as there are many instability results in the gravitational context; up to now there has been no consensus among mathematical physicists [79]. (Numerical evidence is not conclusive because of the difficulty of accurate simulations in very large time—even in one dimension of space.)
• Does the damping hold for unbounded systems? Counterexamples from [30] and [31] show that some kind of confinement is necessary, even in the electrostatic case. More precisely, Glassey and Schaeffer show that a solution of the linearized Vlasov–Poisson
(6) “Angular” here refers to action-angle variables, and applies even for straight trajectories in a torus.
4·10−5 3·10−5 2·10−5 10−5 0
−10−5
−2·10−5
−3·10−5
−4·10−5
−6 −4 −2 0 2 4 6
v h(v)
t=0.16
4·10−5 3·10−5 2·10−5 10−5 0
−10−5
−2·10−5
−3·10−5
−4·10−5
−6 −4 −2 0 2 4 6
v h(v)
t=2.00
Figure 1. A slice of the distribution function (relative to a homogeneous equilibrium) for gravitational Landau damping, at two different times.
equation in the whole space (linearized around a homogeneous equilibriumf0of infinite mass) decays at best likeO(t−1), modulo logarithmic corrections, forf0(v)=c/(1+|v|2);
and likeO((logt)−α) iff0 is a Gaussian. In fact, Landau’s original calculations already indicated that the damping isextremely weak at large wavenumbers; see the discussion in [54,§32]. Of course, in the gravitational case, this is even more dramatic because of the Jeans instability.
• Does convergence hold in infinite time for the solution of the “full” non-linear equation? This is not clear at all since there is no mechanism that would keep the distribution close to the original equilibriumfor all times. Some authors do not believe that there is convergence as t!∞; others believe that there is convergence but argue that it should be very slow [42], sayO(1/t). In the first mathematically rigorous study of the subject, Backus [4] notes that in general the linear and non-linear evolution break
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−130 10 20 30 40 50 60 70 80 90 100
t logE(t)
electric energy in log scale
−12
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−230 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
t logE(t)
gravitational energy in log scale
Figure 2. Time-evolution of the norm of the field, for electrostatic (left) and gravitational (right) interactions. Notice the fast Langmuir oscillations in the electrostatic case.
apart after some (not very large) time, and questions the validity of the linearization.(7) O’Neil [75] argues that relaxation holds in the “quasilinear regime” on larger time scales, when the “trapping time” (roughly proportional the inverse square root of the size of the perturbation) is much smaller than the damping time. Other speculations and arguments related to trapping appear in many sources, e.g. [61] and [64]. Kaganovich [44] argues that non-linear effects may quantitatively affect Landau damping related phenomena by several orders of magnitude.
The so-called “quasilinear relaxation theory” [54, §49], [1,§9.1.2], [49, Chapter 10]
uses second-order approximation of the Vlasov equation to predict the convergence of the spatial average of the distribution function. The procedure is most esoteric, involving
av-(7) From the abstract: “The linear theory predicts that in stable plasmas the neglected term will grow linearly with time at a rate proportional to the initial disturbance amplitude, destroying the validity of the linear theory, and vitiating positive conclusions about stability based on it.”
eraging over statistical ensembles, and diffusion equations with discontinuous coefficients, acting only near the resonance velocity for particle-wave exchanges. Because of these dis-continuities, the predicted asymptotic state is discontinuous, and collisions are invoked to restore smoothness. Linear Fokker–Planck equations(8) in velocity space have also been used in astrophysics [58, p. 111], but only on phenomenological grounds (the ad-hoc addition of a friction term leading to a Gaussian stationary state); and this procedure has been exported to the study of 2-dimensional incompressible fluids [15], [16].
Even if it were more rigorous, quasilinear theory only aims at second-order cor-rections, but the effect of higher-order perturbations might be even worse. Think of something like
e−t
n∈N0
ε√ntn n!
(whereN0={0,1,2, ...}), then truncation at any order inε converges exponentially fast ast!∞, but the whole sum diverges to infinity.
Careful numerical simulation [95] seems to show that the solution of the non-linear Vlasov–Poisson equation does converge to a spatially homogeneous distribution, but only as long as the size of the perturbation is small enough. We shall call this phenomenon non-linear Landau damping. This terminology summarizes the problem well, still it is subject to criticism since (a) Landau himself sticked to the linear case and did not discuss the large-time convergence of the distribution function; (b) damping is expected to hold when the regime is close to linear, but not necessarily when the non-linear term domi-nates;(9) and (c) this expression is also used to designate related but different phenomena [1,§10.1.3]. It should be kept in mind that in the present paper, non-linearity does mani-fest itself, not because there is a significant initial departure from equilibrium (our initial data will be very close to equilibrium), but because we are addressing very large times, and this is all the more tricky to handle, as the problem is highly oscillating.
• Is Landau damping related to the more classical notion of stability in orbital sense? Orbital stability means that the system, slightly perturbed at initial time from an equilibrium distribution, will always remain close to this equilibrium. Even in the favorable electrostatic case, stability is not granted; the most prominent phenomenon being the Penrose instability [77] according to which a distribution with two deep bumps may be unstable. In the more subtle gravitational case, various stability and instability criteria are associated with the names of Chandrasekhar, Antonov, Goodman, Doremus, Feix, Baumann, etc. [10,§7.4]. There is a widespread agreement (see e.g. the comments
(8) These equations act on some ensemble average of the distribution; they are different from the Vlasov–Landau equation.
(9) Although phase mixing might still play a crucial role in violent relaxation or other unclassified non-linear phenomena.
in [95]) that Landau damping and stability are related, and that Landau damping cannot be hoped for if there is no orbital stability.