Rigorous derivation of the Landau equation in the weak coupling limit

We examine a family of microscopic models of plasmas, with a parameter $\alpha$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $\alpha$, for the whole range (0, 1/2], by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusion coefficient of the Landau equation and show it to be independent of the parameter.


Introduction
Particles in a plasma experience grazing collisions because they are ionized, interacting even at long distances as described by the Coulomb potential. So far, the full Coulomb model has been impossible to handle rigorously in a scaling limit (see [L] for a heuristic argument, and [DP] for a rigorous partial result), and the strategy has been to use an approximation by soft-sphere models with their bump-function potentials.
Microscopically, soft-sphere models consist of a lightweight particle traveling through a random configuration of large stationary particles, called "obstacles" or "scatterers," whose shape and density are determined by a parameter. The lightweight particle grazes obstacles when it gets within their ranges of influence, called "protection" disks. In two dimensions, this can be visualized as a ball rolling through a random field of hills. It is desirable to understand these microscopic models in the weak coupling limit, when the radius of the obstacles goes to zero, and to derive rigorously a macroscopic description in terms of a linear PDE.
More precisely, we introduce a parameter α ∈ (0, 1/2]. The i-th obstacle, centered at r i (see figure 1), is described by a suitably smooth and compactly supported (say, in the unit ball) radial potential V whose rescaling is: We use the convention of [DR]; another convention for rescaling has ε 2 , as in [DGL].
Then the microscopic dynamics for an obstacle configuration, ω, are Newtonian: x v r 1 r 2 r 3 r Figure 1. Example of the soft-sphere model: a configuration of obstacles and the corresponding trajectory of the light particle.
The case α = 0, where the particle travels a relatively long distance between collisions and each obstacle has a relatively large influence, corresponds to the Boltzmann-Grad limit of the hard-sphere model of a dilute (or Lorenz) gas, with the macroscopic evolution given by the Boltzmann equation. (See, for instance, [S1] and [BBS]; and for the quantum Lorenz gas, [EY1].) For α ∈ (0, 1/2], the number of obstacles must be about ε −2α−d+1 per unit of volume in order to have a net effect that is nonzero and finite (this is more than in the Boltzmann-Grad limit, where the number is about ε 1−d ). This large number of obstacles balances the small factor of ε α in the rescaled potential, and as ε → 0 in this, the weak coupling limit, the macroscopic dynamics of these plasma models are given by the linear Landau equation.
Previously, the results for this family of models were incomplete. Desvillettes and Ricci proved a weak version (convergence in expectation) of the two-dimensional weak coupling limit for 0 < α < 1/8 [DR]. They approximated the Landau equation by the Boltzmann equation in order to use a modification of Gallavotti's technique for the Boltzmann-Grad limit. From the outset, however, they needed α to be small, so that the radius of each obstacle is much smaller than the expected free flight time. The method is further limited to the regime 0 < α < 1/8 by an estimate of the probability of self-intersection, but can be improved to include 0 < α < 1/4 (see Appendix).
On the other hand, Kesten and Papanicolaou proved a stronger convergence (in law) of the weak coupling limit at the upper endpoint of the parameter range, α = 1/2, for dimensions three and higher and with a general random field, called the stochastic acceleration problem [KP1]. Then Dürr, Goldstein, and Lebowitz proved a two-dimensional version, with the Poisson distribution of obstacles [DGL]. More recently, Komorowski and Ryzhik handled the stochastic acceleration problem in two dimensions [KR2]. (See also [EY2] and [HLW] for the quantum weak coupling limit.) The difficulty in the middle range of α is that obstacles overlap a great deal more than for small α, but their individual influence is greater than for α = 1/2. There's an additional difficulty in the two-dimensional case because the probability of selfintersections is nontrivial. Fortunately, it turns out that "bad" self-intersections (ones that are repeated or almost tangential) are negligible in the scaling limit. This is important because such self-intersections cause a correlation between the past and the present (see Figures 2 and 3), and controlling memory effects is the main difficulty in these problems. In higher dimensions, this is unnecessary, because the probability of any self-intersection is negligible in the limit.   Theorem 1.1. For α ∈ (0, 1/2), the family of stochastic processes (v ε α (t)) t≥0 converges as ε → 0 in law to the velocity diffusion (v(t)) t≥0 generated by ∆ v , the Laplace-Beltrami operator on S 1 := {w ∈ R 2 : |w| = |v 0 |}.
In particular, if f 0 (x, v) is an initial distribution of positions and velocities, and Φ t α,ω,ε is the flow for the microscopic dynamics (1.1), then where h is the solution of the linear Landau equation: Moreover, the microscopic distinctions of the obstacles' steepness and density all disappear in the scaling limit, and the models have the same macroscopic behavior: Proposition 1. The diffusion coefficient in (1.3), ζ, is independent of α ∈ (0, 1/2] and can be expressed by the following formula: The main novelties in the present paper are better estimates on clusters of overlapping obstacles and the amount of time spent interacting with each cluster. By controlling these quantities, we show that the total influence of clusters is negligible in the scaling limit. The outline of the paper is as follows: • To prove convergence in law, we must show that the measures ν ε α induced by the microscopic processes (v ε α (t)) on C([0, T ] 2 ; R 2 × R 2 ) converge weakly to ν = ν α induced by the diffusion (v(t)).
-To this end, we first define some stopping times to eliminate wild behavior (e.g., too many self-crossings, or at too small an angle), which is negligible in the limit (Section 2). -Next we prove that the family of stopped processes is tight (Section 3). -And we identify the limit as a velocity diffusion (Section 4).
• Then we examine, in Section 5, the particular case of convergence in expectation, using a Gallavotti-type method and a combinatorial argument about clusters of obstacles. • Finally, in Section 6, we prove the Proposition about the diffusion coefficient being constant in α, and show that the formulas of [DR] and [DGL] agree. • The Appendix contains a narrower extension of the techniques of [DR].

The cut-offs
We prove the theorem for the processes with stopping times that can be removed for the limiting process. These cut-offs prevent wild behavior that would lead to a correlation (the regimes of dependence in figures 2 and 3) between the past of a trajectory and its future. First, for p ∈ C([0, T ] 2 ; R 2 × R 2 ), define: Then we define the stopping times as follows. The first time the trajectory approaches its past within distance a and with angle less than φ is cut off by τ φ,a : p(s) · p(u) ≤ 0, and |p(t) · p(s)| |p(t)||p(s)| ≥ cos φ}. (2.2) The first time trajectory crosses itself more than K times is cut off by τ K : τ K := inf{t ≥ 0 :∃s i < t i , for i = 1, . . . , K, and t 1 < · · · < t K = t, Additionally there is a velocity cut-off, to prevent the trajectory getting "stuck" somewhere or going too fast: We call the overall stopping time τ : (2.5) τ := min{τ φ,a , τ K , τ v }.
These stopping times actually depend on ε, so we write the stopped processes as: . And their induced measures are written asν ε α . To see that the cut-offs can be removed, namely thatν ε α and the measure induced by the original process, ν ε α , have the same limit, ν, we claim that, for arbitrary T > 0: The first statement is immediate. The second follows from a self-crossing lemma in [DGL,, which says roughly that tangential self-intersections are negligible. The third follows as a corollary, because the trajectory is almost surely a continuously differentiable curve.

Tightness
We will show that the family of processes v ε,α t is tight, to get the weak convergence of the induced measuresν ε α . First we need some generalizations of the lemmas in [DGL,§ 4].
3.1. The Martingale-compensator decomposition. We can view the Poisson point process of obstacles from the particle's perspective (a locally Poisson process denoted N ε α , realizations of which are measures on S × R + ), and the difference (denoted M ε α ) turns out to be a martingale. This is expressed in the following equation, where the first term on the right-hand side is a martingale, and the second is a measurable left-continuous process called the compensator.
Then we write the potential as F ε α (y) := − ∇V ε α (y), the stopped position process x ε,α 3.2. Bounding clusters of overlapping obstacles. To control the probability of λ scatterers being found within distance nε of the particle at time t, we introduce some notation: Lemma 3.1. For any λ, ε, T > 0, Proof. The proof follows [DGL] with appropriate modifications. We start from a result of the conservation of energy, with v 0 = 1: Hence That is to say, if the particle does not come close to λ scatterers at once before time If the particle is close to λ scatterers simultaneously before time T , then Tile Γ by squares γ ε i with side length 2nε α and i = 1, . . . , For each i, cover γ ε i symmetrically by the larger squareγ ε i of side-length 4nε; this implies that Hence, by the Poisson field's translation invariance, Markov's inequality with an exponential finishes the proof.
3.3. Bounding the crossing times. We would like to control, again for α ∈ (0, 1/2), the time that the particle takes to cross the ball S ε n (x ε,α s ), compared to a trajectory crossing the ball with constant velocity. Define: We will call the first set in this intersection Λ ε n+1 , and we write B ′ := sup|F (x)|.
Lemma 3.2. If t ε n (s) andt ε n (s) are the first entrance and exit times of the trajectory with respect to the ball S ε n (x ε s ), then for trajectories in U λ and for ε < ( Proof. If a particle enters S ε n (x ε,α s ) with constant velocity v ε,α 1 = v ε α (t ε n (s))/2, position x ε α (t ε n (s)), at time t ε n (s), then its exit timet ε n (s) will satisfŷ By the point process description, (3.2), we have for t ≤t ε n (s): Putting these two facts together, we have for t <t ε n (s): So on U λ , and for ε < (

Tightness.
We now turn to showing tightness forν ε α , the family of measures induced by the cut-off processes v ε,α Proof. We use the point process description (3.2) to write: i.e., the particle's first exit time after u. Terms B, C, and D concern scatterers that the particle encounters at self-crossings, at time s ∧ τ ε , and at time 0, respectively. By the overlap and crossing-time lemmas, 3.1 and 3.2, we can bound these terms: If λ = ε −β , with β < 2α, then these terms are negligible in the limit ε → 0.
To handle A, on the other hand, requires the martingale decomposition, (3.1). However, we run into a problem trying to implement this decomposition, because the function f (σ, u) is not adapted to the family of sigma algebras, F t , generated by N ε α -instead, it's anticipative. To replace f by an adapted function, we do a Taylor expansion of F ε α around the linex ε,α u, where III is the linear term, and IV is the remainder, with mean value y ε (u, σ, t ′ ): We observe that term I is F u -adapted and continuous in u. A useful estimate from the crossing time lemma and sup u≤τ ε |t ε u,σ − u| ≤ 4ε is: sup The other terms are still anticipative, so they need to be analyzed further. For terms II and IV , it suffices (by sending λ → ∞ in the overlap lemma) to show that they decay asymptotically on the set In order to bound II and IV , we use the crossing time lemma, 3.2, to show that on Λ ε n , both of the following are of order λ 2 ε 1+α : These terms go to zero as ε → 0, for λ = ε −β , with β < 1/3. Hence, for term IV , we can estimate: |e 1 · ∇e 2 · ∇F |.
Next we handle tightness for y ε 0 , showing that for some positive constant C, Let K be the number of self-crossings in the trajectory, and let T ε be the tube around the trajectory up to the stopping time τ ε . At the i-th self-crossing, let T ε i andT ε i be the times that the particle enters and exits the tube. Set U : Then 3.2 implies that there is a number n = n(φ, a) independent of ε and such that {sup s≤T N ε,α n(φ,a) (s) ≤ λ}.

Characterization of the limit
To identify the limiting process as the velocity diffusion associated to the linear Landau equation, we use the Stroock-Varadhan martingale formulation, thereby showing that the tight family of processes converges to the velocity diffusion, we show that two quantities involving the process and the infinitesimal generator of the diffusion are martingales; the desired result will follow by Levy's lemma.

Lemma 4.1. Let s, t ∈ [0, T ], f be a smooth test function, and φ s be a smooth and bounded test function that depends only on
Proof. It suffices to prove (4.3) for linear and quadratic functions f ; we restrict our attention here to the linear case. Using the decomposition (3.5) we can see that the B, C, and D terms are negligible in the limit, as follows. We cut their expectation into two pieces, S := {sup s≤T N ε,α n(φ,a) (s) ≤ λ} and S c ; and on the first term we use the bound on the supremum of B + C + D from (3.10), and on the second term we use the overlap lemma, (3.1): Here, h involves terms like λ 2 T 2 ε 2α−2 , and λ can be chosen to be a small enough (for example, ε −α+3/2+β T −1−β , with β small and positive) to get the right-hand side to vanish with ε. Hence only the term A matters: Next recall the decomposition of (3.13), v ε,α t ∼ y ε + (t) + y ε − (t) + y ε 0 (t). We definê y ε 0 (t) andŷ ε ± (t) by replacing N ε α byN ε α , the point process that ignores self-crossings, in their formulas; this results in negligible errors. Then we can combine this with the Taylor expansion in (3.11), and use the martingale property ofŷ ε 0 to arrive at: Then we can use the formulas from the previous section, (3.14) and (3.16), to reduce the proof of the lemma to the following two limits: (4.5) The first of these two limits can be handled by manipulations similar to [DGL], For the second of the two limits in (4.5), we examine the part without the infinitesimal operator. Using the martingale decomposition ofN ε α and the definitions of ρ, F ε α , and ∇F ε α , we arrive at: Now we manipulate the inner integrals to make them look like the desired operator: (4.6)

Convergence in expectation
We will show that, in particular, for an initial distribution f 0 , where h is the solution of the linear Landau equation , (1.3). Although this is a weaker mode of convergence than that just proved, the following argument illustrates the key intuition behind the result for the entire range of α: Although there are significant numbers of clusters of obstacles, their total influence is actually negligible. The initial set-up is similar to [DR] but differs after the first two steps.
First, observe that f ε α (t, x, v) can be written as (using |B(x)| to denote the measure of the ball of radius t around the initial position x): Define a cut-off χ 1 , killing configurations that have an obstacle at the initial position: . Making this cut-off introduces an asymptotically vanishing error. That is, there exists a function φ 1 (ε) → 0 as ε → 0 such that f ε α can be written as: Next we define another cut-off, χ 2 , killing configurations with obstacles that are not encountered by the light particle and thus have no effect on the trajectory: : ∀i = 1, . . . , M, r i ∈ T (t)}). Here T (t) is the tube of radius ε around the light particle's trajectory: Then we have again an asymptotically vanishing error: Next we make the key observation that single obstacles dominate the trajectory's path. First, we note that there is a sequence of thresholds, α n converging to 1/2 from below, such that for α < α n , there is a negligible number of clusters of n obstacles. For instance, α 2 = 1/4, that is, only for α ≥ 1/4 do we need to worry about the numbers (if not influence) of doublets (clusters of 2 obstacles). Let n be the number of internal doublets up to time t; n is a random variable with an expected value of order ε 1−4α . Let θ j be the deflection angle for the collision with the j-th doublet.
Then the key observation is that E ε [ n j=1 θ j ] → 0 as ε → 0, and can be seen as follows. We introduce an expansion of the deflection angle at the j-th doublet (which comes from (6.7) in the next section): . We note that Eθ j = 0, even when the test particle hits a cluster of n j scatterers overlapping to form one large obstacle. This can be seen easily for the cluster which is actually just one scatterer, because θ j is an odd function of the impact parameter. For n j > 1, we assume without loss of generality that after leaving the previous cluster, the light particle's position is x j = 0, and its velocity, v j = e 1 (the first coordinate vector). Let r i be the centers of the scatterers in the order that they are encountered, with i between 1 and n j . For each such configuration , there is an opposite configurationR j , obtained by reflecting across the first coordinate axis, with corresponding scattering angleθ j = −θ j . So θ j is an odd function in this sense, and vanishes when integrated against an even measure.
Then we compute: And this last quantity goes to zero as ε → 0 because α ∈ (0, 1/2).  Figure 4. The changes of variables, from r j to t j and b j , and from b j to θ j .
Resuming along the lines of [DR], we introduce a change of variables L, replacing obstacle locations by hitting times and impact parameters: The difference is that here, we define the domain of L to be the set of all trajectories that do not start on a scatterer; that do not have extraneous scatterers; and that have stopping time τ ε > t (i.e., they have no almost-tangential self-intersections, not too many self-intersections, and velocity that does not change too much). In particular, we handle the change of variables at a doublet as illustrated in figure 5. Then we can write f ε α using the change of variables and (2.7), with an error φ 3 (ε) vanishing as ε → 0, as: Heret stands in for (t 1 , . . . , t k ), and △ := {t : t 1 ∈ (0, t), t 2 ∈ (t 1 , t), . . . , t k ∈ (t k−1 , t)}. Similarly,b := (b 1 , . . . , b k ), and ε := (−ε, ε) × · · · × (−ε, ε).
Then the procedure of [DR] can be followed and a change of variables made (with a slight modification for doublets, described below), from impact parameters {b i } to deflection angles {θ i } (see figure 4), which has Jacobian determinant: Here, Γ ε (θ i ) is the rescaled scattering cross section. We use the fact that the deflection angle θ through a doublet can be approximated: θ = θ 1 + θ 2 + φ(ε). Here θ 1 and θ 2 are the deflection angles corresponding to b 1 and b 2 in figure 5, and φ(ε) is a small error vanishing as ε → 0.
Using also R θ (v) to denote the rotation of the vector v by angle θ, we make the following polygonal approximation to the trajectory: Additionally, we approximate |T (t)| by 2εt. Then we can rewrite f ε α : We then recognize this expansion as the series form of a solution to the family of Boltzmann equations: Finally, the h ε converge in the appropriate sense to h, the solution of the Landau equation (1.3), because the scattering cross sections Γ ε concentrate on grazing collisions. We note that Eθ j = 0, even when the test particle hits a cluster of n j scatterers overlapping to form one large obstacle. This can be seen easily for the case of hitting a lone scatterer, because θ j is an odd function of the impact parameter. For n j > 1, we assume without loss of generality that after leaving the previous cluster of scatterers, the light particle's position is x j = 0, and its velocity, v j = e 1 (the first coordinate vector). Let r i be the centers of the scatterers in the order that they are encountered, with i between 1 and n j . For each such configuration , there is an opposite configurationR j , obtained by reflecting across the first coordinate axis, with corresponding scattering angleθ j = −θ j . So θ j is an odd function in this sense, and when integrated against an even measure, vanishes.

The diffusion constant
We take the following as the definition of the diffusion constant: Proposition 2. The diffusion constant, ζ, defined above, is independent of α ∈ (0, 1/2) and can be expressed by the following formula: Proof. We do several expansions to compute the diffusion constant. First, we compare the deflection through one obstacle, centered at the origin, to the path that would be taken if the obstacle weren't there. Let the entry time, place, and velocity be 0, x − = x(0), and v − ; the deflected exit time, position, and velocity be τ , x + , v + ; and the non-deflected (straight-line) exit time, position, and velocity beτ ,x + , andv + . (See figure 6.) Figure 6. The straight-line trajectory exiting at timeτ and the deflected trajectory, at τ , with the obstacle's center at the origin.
We desire an estimate for |τ −τ |, so we examine the identity: ), so we have the estimate: Next, we estimate v + − v − , using the definition F ε α (y) := −∇V ε (y). We could make the following estimate (but we will actually do better): Accordingly, we estimate that for s between τ andτ , and some a between τ and s, and using the compact support of V : We now combine (6.4) with (6.3), the estimate for |τ − τ |. Instead of the previous remainder term, O(ε 2α ), we get: Thus: Expand F ε α (x(s)), usingx(s) = x − + tv − , and rename the first two terms: To analyze Y 2 further, observe that: Regarding the first term, Y 21 , we integrate out t: As for the second term in this decomposition, Y 22 : Then integrating, we get: Thus we have, combining the estimates for Y 1 , Y 21 , and Y 22 : Figure 7. Change of variables from s to ξ, which is the distance remaining to the center of the crossing (scaled by ε). Now we claim that X 1 := τ 0 Y 1 = τ 0 F ε α (x(s))ds is perpendicular to v − . Without loss of generality we may assume that v − = (1, 0). Making the following change of variables, (s, b) → (ξ, b), as in figure 7, we see that the claim is true: Then we introduce a variable u such that: |b| u = b 2 + ξ 2 , and thus |b| Using this change of variables, and u 0 := |b|/ η 2 + b 2 = |b|, (6.6) So if we examine the deflection angle, θ, as pictured in figure 6, we obtain: Then orthogonality of X 1 with v − implies: (6.7) We note that in particular, (6.7) gives us an expansion of the deflection angle with leading term of order ε α . In terms of the difference in velocities: (6.8) The proof is finished by putting this expression for θ 2 into the definition of the diffusion constant, (6.1), and using (6.6).

Appendix
Retaining the same essential argument, the result of [DR] (convergence in expectation of the evolution of the initial distribution f 0 ) can be stretched to include α ∈ [1/8, 1/4). The estimate that must be improved in order to achieve this is (40) in [DR,Lemma 1]. This section illustrates how to iterate their geometric method to get a tighter bound on J ii 1,ε , the error term estimating the probability of non-consecutive overlappings and recollisions, a bound that decays for α < 1/4. First fix n disjoint closed subintervals of (0, π), each of the form I m := [φ m , φ m + π/2n].
Our trajectory will have n gaps where the times t hm can vary. All other times t k and all angles θ k are fixed.
2 0 x(t ) Figure 8. A recollision (x(t j ) lying in the tube of radius ε around the trajectory) constrains how much h 1 and h 2 can vary in total; the variation for each is bounded by √ 2ε.
We claim that the recollision condition, r j ∈ s∈(ti,ti+1) B(x(s), 2ε), together with each I m being bounded away from the endpoints of [0, π], results in each t hm taking values in a set whose measure is O(ε). The claim can be seen by working backwards from h n : the height of the trajectory at time t hn can vary only in an interval of size ε, due to the recollision condition restricting the trajectory to a tube of width ε. Then the total variation of all the t hm , m = 1, . . . , n can be no larger than Cε. Here the constant C is chosen so that 1 sin φ ≤ C, for φ ∈ {φ 1 , φ n + π/2n}.
Hence the variation of any single t hm is bounded above by Cε, there being no negative values of t hm allowed and all angles pointing upwards at the variable times (i.e., being strictly between zero and π). Then J ii 1,ε can be estimated like before: · · · j hn=i+n n m=1 1 hm−1 k=1 θ k ∈ I m 1 β j ∈ s∈(ti,ti+1) The exponent is positive if α < n 4(n+1) , so taking n to infinity (as ε goes to zero) gives the result for α < 1/4.