From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

Abstract

We derive the linear acoustic and Stokes–Fourier equations as the limiting dynamics of a system of N hard spheres of diameter ${\varepsilon }$ in two space dimensions, when $N\rightarrow \infty$, ${\varepsilon }\rightarrow 0$, $N{\varepsilon }=\alpha \rightarrow \infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford’s strategy (Time evolution of large classical systems, Springer, Berlin, 1975), and on the pruning procedure developed in Bodineau et al. (Invent Math 203:493–553, 2016) to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform $L^2$ a priori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.

Appendices

Appendix A: The Linearized Boltzmann Equation and Its Fluid Limits

For the sake of completeness, we recall here some by now classical results about the linearized Boltzmann equation (1.14)

(A.1)

and its hydrodynamic limits as $\alpha \rightarrow \infty$ (for $q=0,1$). The results below are valid in any dimension $d \ge 2$, thus contrary to the rest of this article, we assume the space dimension to be d.

Because of the scaling invariance of the collision kernel, we shall actually restrict our attention in the sequel to the case where $M_\beta$ is the reduced centered Gaussian, i.e. $\beta = 1$ (and we omit the subscript $\beta$ in the following). The collision operator (A.1) will be denoted by $\mathcal {L}$.

A.1: The Functional Setting

The linearized Boltzmann operator $\mathcal {L}$ has been studied extensively (since it governs small solutions of the nonlinear Boltzmann equation). In the case of non singular cross sections, its spectral structure was described by Grad [11]. The main result is that it satisfies the Fredholm alternative in a weighted $L^2$ space. In the following we define the collision frequency

$$\begin{aligned} a(|v|) := \int M(v_1) \big ((v_{1}-v) \cdot \nu \big )_+ d\nu dv_{1} \end{aligned}$$

which satisfies, for some $C>1$,

$$\begin{aligned} 0<a_- \le a(|v|)\le C(1+|v|) \,. \end{aligned}$$

Proposition A.1

The linear collision operator $\mathcal {L}$ defined by (A.1) is a nonnegative unbounded self-adjoint operator on $L^2(Mdv)$ with domain

$$\begin{aligned} \mathcal {D}(\mathcal {L})=\{g\in L^2(Mdv)\,|\, a g\in L^2(Mdv)\}=L^2(\mathbb {R}^d;a M(v)dv) \end{aligned}$$

and nullspace

$$\begin{aligned} {\text {Ker}}(\mathcal {L})={\text {span}}\{1,v_1,\dots ,v_d,|v|^2\}\,. \end{aligned}$$

Moreover the following coercivity estimate holds: there exists $C>0$ such that, for each g in $\mathcal {D}(\mathcal {L})\cap ({\text {Ker}}(\mathcal {L}))^\perp$

$$\begin{aligned} \int g\mathcal {L}g(v)M(v)dv\ge C\Vert g\Vert _{L^2(a M dv)}^2\,. \end{aligned}$$

Sketch of proof

• The first step consists in characterizing the nullspace of $\mathcal {L}$. It must contain the collision invariants since the integrand in $\mathcal {L}g$ vanishes identically if $g(v) =1,v_1,v_2 , \dots ,v_d$ or $|v|^2$. Conversely, from the identity,

  $$\begin{aligned} \int \psi \mathcal {L}g Mdv= \frac{1}{4}\int (\psi +\psi _1-\psi '-\psi '_1)(g+g_1-g'-g'_1) \big ((v_{1}-v) \cdot \nu \big )_+Mdvdv_1d\nu \,, \end{aligned}$$

where we have used the classical notation

$$\begin{aligned} g_1:= g(v_1) \, , \quad g' = g(v') \ , \quad g'_1=g(v'_1) \, , \end{aligned}$$

we deduce that, if g belongs to the nullspace of $\mathcal {L}$, then

$$\begin{aligned} g+g_1=g'+g'_1 \, , \end{aligned}$$

which entails that g is a linear combination of $1,v_1$, $v_2,\dots ,v_d$ and $|v|^2$ (see for instance [20]).

Note that the same identity shows that $\mathcal {L}$ is self-adjoint.

• In order to establish the coercivity of the linearized collision operator $\mathcal {L}$, the key step is then to introduce Hilbert’s decomposition [15], showing that $\mathcal {L}$ is a compact perturbation of a multiplication operator :

  $$\begin{aligned} \mathcal {L}g (v)=a(|v|)g(v)-\mathcal {K}g (v) \, . \end{aligned}$$

Proving that $\mathcal {K}$ is a compact integral operator on $L^2(Mdv)$ relies on intricate computations using Carleman’s parametrization of collisions (which we also use in this paper for the study of recollisions). We shall not perform them here (see [15]).

Because a is bounded from below, $\mathcal {L}$ has a spectral gap, which provides the coercivity estimate. $\square$

Proposition A.1, along with classical results on maximal accretive operators, imply the following statement.

Proposition A.2

Let $g_0 \in L^2(M dvdx)$. Then, for any fixed $\alpha$, there exists a unique solution $g_\alpha \in C(\mathbb {R}^+, L^2(Mdvdx))\cap C^1( \mathbb {R}^+_*, L^2(M dvdx)) \cap C( \mathbb {R}^+_*, L^2(M a dvdx))$ to the linearized Boltzmann equation (A.1). It satisfies the scaled energy inequality

$$\begin{aligned} \Vert g_\alpha (t)\Vert ^2_{L^2(Mdvdx)} +\alpha ^{1+q} \int _0^t \int g_\alpha \mathcal {L}g_\alpha (t') Mdvdx dt'\le \Vert g_0\Vert _{L^2(Mdv)}^2\,. \end{aligned}$$

(A.2)

A.2: The Acoustic and Stokes Limit

The starting point for the study of hydrodynamic limits is the energy inequality (A.2). The uniform $L^2$ bound on $(g_\alpha )$ implies that, up to extraction of a subsequence,

$$\begin{aligned} g_\alpha \rightharpoonup g \hbox { weakly in } L^2_{loc} (dt,L^2(Mdvdx))\,. \end{aligned}$$

(A.3)

Let $\Pi$ be the orthogonal projection on the kernel of $\mathcal {L}$. The dissipation, together with the coercivity estimate in Proposition 1, further provides

$$\begin{aligned} \Vert g_\alpha -\Pi g_\alpha \Vert _{L^2(Ma dvdxdt)} = O(\alpha ^{-(q+1)/2}) \, , \end{aligned}$$

from which we deduce that

$$\begin{aligned} g (t,x,v) =\Pi g(t,x,v) \equiv \rho (t,x) + u(t,x)\cdot v +\theta (t,x) {|v|^2-d\over 2}. \end{aligned}$$

(A.4)

If the Mach number $\alpha ^q$ is of order 1, i.e. for $q=0$, one obtains asymptotically the acoustic equations. Denoting by $\langle \cdot \rangle$ the average with respect to the measure Mdv, we indeed have the following conservation laws

From (A.3) and (A.4) we then deduce that $(\rho , u , \theta )$ satisfy

(A.5)

By uniqueness of the limiting point, we get the convergence of the whole family $(g_\alpha )_{\alpha >0}$.

Since the limiting distribution g satisfies the energy equality

$$\begin{aligned} \Vert g\Vert ^2_{L^2(Mdvdx)} = \Vert g_0\Vert _{L^2(Mdv dx)}^2, \end{aligned}$$

or equivalently

$$\begin{aligned} \Vert g\Vert ^2_{L^2(Mdvdx)} +\alpha \int _0^t \int g \mathcal {L}g Mdvdx = \Vert \Pi g_0\Vert _{L^2(Mdv dx)}^2\,, \end{aligned}$$

convergence is strong as soon as $g_0 =\Pi g_0$. We thus have the following result (see [9] and references therein).

Proposition A.3

Let $g_0\in L^2(Mdvdx)$. For all $\alpha$, let $g_\alpha$ be a solution to the scaled linearized Boltzmann equation (A.1) with $q=0$. Then, as $\alpha \rightarrow \infty$, $g_\alpha$ converges weakly in $L^2_{loc} (dt,L^2(Mdvdx))$ to the infinitesimal Maxwellian $\displaystyle g=\rho +u\cdot v +\frac{1}{2} \theta (|v|^2-d)$ where $(\rho , u,\theta )$ is the solution of the acoustic equations (A.5) with initial datum $\displaystyle (\langle g_0\rangle , \langle g_0 v\rangle , \langle g_0 \frac{1}{d}(|v|^2-d)\rangle )$.

The convergence holds strongly in $L^\infty _t(L^2(Mdvdx))$ provided that $g_0=\Pi g_0$.

In the diffusive regime, i.e. for $q=1$, the moment equations state

$$\begin{aligned}&{1\over \alpha } {\partial }_t \langle g_\alpha \rangle + \nabla _x \cdot \langle g_\alpha v\rangle =0 \, ,\\&{1\over \alpha } {\partial }_t \langle g_\alpha v\rangle + \nabla _x \cdot \langle g_\alpha v\otimes v\rangle =0 \, ,\\&{1\over \alpha } {\partial }_t \langle g_\alpha |v|^2 \rangle + \nabla _x \cdot \langle g_\alpha v|v|^2\rangle =0\,. \end{aligned}$$

From (A.3) and (A.4) we deduce that

$$\begin{aligned} \nabla _x \cdot u = 0 \, , \quad \nabla _x (\rho +\theta ) =0\,, \end{aligned}$$

referred to as incompressibility and Boussinesq constraints.

To characterize the mean motion, we then have to filter acoustic waves, i.e. to project on the kernel of the acoustic operator

$$\begin{aligned} {\partial }_t P \langle g_\alpha v\rangle + \alpha P\nabla _x \cdot \Big \langle g_\alpha \left( v\otimes v-\frac{1}{2} |v|^2 Id\right) \Big \rangle =0 \, ,\\ {\partial }_t \Big \langle g_\alpha \left( {|v|^2 \over d+2} - 1 \right) \Big \rangle +\alpha \nabla _x \cdot \Big \langle g_\alpha v\left( {|v|^2 \over d+2} - 1 \right) \Big \rangle =0\,, \end{aligned}$$

where P is the Leray projection on divergence free vector fields. Define the kinetic momentum flux $\displaystyle \Phi (v) = v\otimes v-\frac{1}{d} |v|^2 Id$ and the kinetic energy flux $\displaystyle \Psi (v) = \frac{1}{d+2} v(|v|^2-d-2)$. As $\Phi , \Psi$ belong to $({\text {Ker}}\mathcal {L})^\perp$, and $\mathcal {L}$ is a Fredholm operator, there exist pseudo-inverses ${\tilde{\Phi }}, {\tilde{\Psi }} \in ({\text {Ker}}\mathcal {L})^\perp$ such that $\Phi = \mathcal {L}{\tilde{\Phi }}$ and $\Psi = \mathcal {L}{\tilde{\Psi }}$. Then,

$$\begin{aligned}&{\partial }_t P \langle g_\alpha v\rangle + \alpha P\nabla _x \cdot \langle \mathcal {L}g_\alpha {\tilde{\Phi }} \rangle =0 \, ,\\&{\partial }_t \langle g_\alpha \left( {|v|^2 \over d+2} - 1 \right) \rangle +\alpha \nabla _x \cdot \langle \mathcal {L}g_\alpha {\tilde{\Psi }}\rangle =0\,. \end{aligned}$$

Using the equation

$$\begin{aligned} \alpha \mathcal {L}g_\alpha = -v\cdot \nabla _x g_\alpha -\frac{1}{\alpha }{\partial }_t g_\alpha \end{aligned}$$

the Ansatz (A.4), and taking limits in the sense of distributions, we get

$$\begin{aligned} \begin{aligned} \nabla _x \cdot u = 0, \quad \nabla _x (\rho +\theta ) =0 \, ,\\ {\partial }_t u - \mu \Delta _x u=0 \, ,\\ {\partial }_t \theta - \kappa \Delta _x \theta =0\,. \end{aligned} \end{aligned}$$

(A.6)

These are exactly the Stokes–Fourier equations with

$$\begin{aligned} \mu = \frac{1}{(d-1)(d+2)} \langle \Phi : {\tilde{\Phi }} \rangle \quad \text {and} \quad \kappa = \frac{2}{d(d+2)} \langle \Psi \cdot {\tilde{\Psi }} \rangle . \end{aligned}$$

As previously, the limit is unique and the convergence is strong provided that the initial datum is well-prepared, i.e. if

$$\begin{aligned} g_0 (x,v) =u_0\cdot v +\frac{1}{2} \theta _0(|v|^2-(d+2)) \hbox { with } \nabla _x \cdot u_0 = 0 \,. \end{aligned}$$

(A.7)

One can therefore prove the following result.

Proposition A.4

Let $g_0\in L^2(Mdvdx)$. For all $\alpha$, let $g_\alpha$ be a solution to the scaled linearized Boltzmann equation (A.1) with $q=1$. Then, as $\alpha \rightarrow \infty$, $g_\alpha$ converges weakly in  $L^2_{loc} (dt,L^2(Mdvdx))$ to the infinitesimal Maxwellian $\displaystyle g=u\cdot v +\frac{1}{2} \theta (|v|^2-(d+2) )$ where $( u,\theta )$ is the solution of (A.6) with initial datum $(P \langle g_0 v\rangle , \langle g_0 ({|v|^2 \over d+2} - 1 )\rangle )$.

The convergence holds in $L^\infty _t(L^2(Mdvdx))$ provided that the initial datum is well-prepared in the sense of (A.7).

Remark A.5

In both cases, the defect of strong convergence for ill-prepared initial data can be described precisely.

If the initial profile in v is not an infinitesimal Maxwellian, i.e. if $g_0\ne \Pi g_0$, one has a relaxation layer of size $\alpha ^{-(1+q)}$ governed essentially by the homogeneous equation

$$\begin{aligned} {\partial }_t \Pi _\perp g_\alpha = -\alpha ^{q+1} \mathcal {L}g_\alpha \,. \end{aligned}$$

In the incompressible regime $q=1$, if the initial moments do not satisfy the incompressibility and Boussinesq constraints, one has to superpose a fast oscillating component (with a time scale $\alpha ^{-1}$). For each eigenmode of the acoustic operator, the slow evolution is given by a diffusive equation.

A straightforward energy estimate then shows that the asymptotic behavior of $g_\alpha$ is well described by the sum of these three contributions (main motion, relaxation layer and acoustic waves in incompressible regime).

Appendix B: Geometrical Lemmas

In this appendix, we prove several technical lemmas (namely Lemmas B.1, B.2, B.3 and B.4) which were key steps in Sections 3 and 6 in proving Propositions 3.5 and 6.2.

In the following we adopt the notation of those sections.

B.1: A Preliminary Estimate

Recall Equation (3.9) which is associated to the recollision of particles i, j

$$\begin{aligned} v_i - v_{j} = \frac{1}{\tau _{rec} } \delta x_\perp - \frac{\tau _1}{\tau _{rec}} ({\bar{v}}_{i} - v_{j})- \frac{1}{\tau _{rec} } \nu _{rec} \, , \end{aligned}$$

(B.1)

with the notations (3.8)

$$\begin{aligned} \tau _1 :=- \frac{1}{{\varepsilon }}(t_{1^*} - t_{2^*}+\lambda ) \, ,\qquad \tau _{rec} :=-\frac{1}{{\varepsilon }}(t_{rec} - t_{1^*}) \, , \end{aligned}$$

(B.2)

where

$$\begin{aligned} \frac{1}{{\varepsilon }}( x_{i} - x_{j}-q) = {\lambda \over {\varepsilon }} ({\bar{v}}_{i} - v_{j}) + \delta x_\perp \quad \hbox { with } \quad \delta x_\perp \cdot ({\bar{v}}_i - v_{j}) =0 \, . \end{aligned}$$

The distance between particles i, j at the collision time $t_{1^*}$ is given by

$$\begin{aligned} \big | x_i( t_{1^*}) - x_j ( t_{1^*}) \big | = {\varepsilon }\big | \delta x_\perp - \tau _1 ({\bar{v}}_{i} - v_{j}) \big | = {\varepsilon }\sqrt{\big | \delta x_\perp \big |^2 + \big | \tau _1 ({\bar{v}}_{i} - v_{j}) \big |^2 } \, . \end{aligned}$$

The distance between the particles varies with the collision time $t_{1^*}$ and the closer they are, the easier it is to aim (at the collision time $t_{1^*}$) to create a recollision at the later time $t_{rec}$. The key idea is that for relative velocities ${\bar{v}}_{i} - v_{j} \not = 0$, the particles will never remain close for a long time so that integrating over $t_{1^*}$ allows us to recover some smallness uniformly over the initial positions at time $t_{2^*}$.

Suppose $|\tau _1| |{\bar{v}}_i - v_j| \le M$. Since $v_{1^*}$ is in a ball of size R, and $\nu _{1^*}$ belongs to ${{\mathbb {S}}}$, we have

$$\begin{aligned} \int \mathbf{1}_{ \{ |\tau _1| |{\bar{v}}_i - v_j | \le M \}} |(v_{1^*} - {\bar{v}}_i) \cdot \nu _{1^*}| |{\bar{v}}_i-v_j| d \tau _1 dv_{1^*} d\nu _{1^*} \le C R^2 M \,. \end{aligned}$$

(B.3)

For later purposes, it will be useful to evaluate the integral (B.3) in terms of the integration parameter $t_{1^*}$: we get by the change of variable $\tau _1 = ( t_{1^*} - t_{2^*} - \lambda )/{\varepsilon }$

$$\begin{aligned} \int \, \mathbf{1}_{ \{ |\tau _1| |{\bar{v}}_i - v_j| \le M \}} \, \, |(v_{1^*} - {\bar{v}}_i) \cdot \nu _{1^*}| d t_{1^*} dv_{1^*} d\nu _{1^*} \le C R^2 M \frac{{\varepsilon }}{|{\bar{v}}_i-v_j|}. \end{aligned}$$

The singularity in $|{\bar{v}}_i-v_j|$ translates the fact that the distance between the particles may remain small during a long time if their relative velocity is small. This singularity can then be integrated out, up to a loss of a $|\log {\varepsilon }|$, using an additional parent of i or j thanks to (C.4) in Lemma C.2: we obtain

$$\begin{aligned} \int \mathbf{1}_{ \{ |\tau _1| |{\bar{v}}_i - v_j| \le M \}} \prod _{k \in \{1^*,2^*\}} |(v_k - v_{a(k)}) \cdot \nu _{k}| d t_{k} dv_{k} d\nu _{k }\le C MR^5 t {\varepsilon }|\log {\varepsilon }|\,.\quad \end{aligned}$$

(B.4)

To get rid of the logarithmic loss, one may use two extra degrees of freedom associated with the parents $2^*, 3^*$ of i or j: from (C.5) and (C.8), we obtain the upper bound

$$\begin{aligned} \int \mathbf{1}_{ \{ |\tau _1| |{\bar{v}}_i - v_j| \le M \}} \prod _{k \in \{1^*,2^*,3^*\}} |(v_k - v_{a(k)}) \cdot \nu _{k}| d t_{k} dv_{k} d\nu _{k }\le C MR^8 t^2 {\varepsilon }\,. \end{aligned}$$

(B.5)

Note that in the case when i and j are colliding at time $t_{2^*}$, Lemma C.1 shows that only two integrations are necessary.

B.2: A Recollision with a Constraint on the Outgoing Velocity

The following lemma deals with the cost of the first recollision when one of the outgoing velocities is constrained to lie in a given ball. More precisely, if the first recollision occurs between particles i, j, and k is a given label, we will impose that (see Figure 8)

$$\begin{aligned} |v'_i - v_k| \le {\varepsilon }^{3/4} \quad \text {or} \quad |v'_j - v_k| \le {\varepsilon }^{3/4}. \end{aligned}$$

(B.6)

Fig. 8

Small relative velocities $|v'_i - v_k| \le {\varepsilon }^{3/4}$ after the first recollision

Lemma B.1

Fix a final configuration of bounded energy $z_1 \in \mathbb {T}^2 \times B_R$ with $1 \le R^2 \le C_0 |\log {\varepsilon }|$, a time $1\le t \le C_0|\log {\varepsilon }|$ and a collision tree $a \in \mathcal {A}_s$ with $s \ge 2$.

There exist sets of bad parameters $\mathcal {P}_2 (a, p,\sigma )\subset \mathcal {T}_{2,s} \times {{\mathbb {S}}}^{s-1} \times \mathbb {R}^{2(s-1)}$ for $4 \le p \le p_1$ (for some integer $p_1$) and $\sigma \subset \{2,\dots , s\}$ of cardinal $|\sigma |\le 4$ such that

• $\mathcal {P}_2 (a, p,\sigma )$ is parametrized only in terms of $( t_m, v_m, \nu _m)$ for $m \in \sigma$ and $m < \min \sigma$;

  $$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2 (a,p,\sigma ) } \displaystyle \prod _{m\in \sigma } \, \big | \big (v_{m}-v_{a(m)} ( t_{{m}} ) )\cdot \nu _{{m}} \big | d t_{{m}} d \nu _{{m}}dv_{{m}} \le C(Rt)^r s^2 {\varepsilon }\,, \end{aligned}$$

  (B.7)

  for some constant r,

• and any pseudo-trajectory starting from $z_1$ at t, with total energy bounded by $R^2$ and such that the first recollision produces a small relative velocity as in (B.6) is parametrized by

  $$\begin{aligned} (t_n, \nu _n, v_n)_{2\le n\le s }\in \bigcup _{4 \le p \le p_1} \bigcup _\sigma \mathcal {P}_2(a, p,\sigma )\,. \end{aligned}$$

Before launching into the details of the proof, we first give the gist of it. Since the first recollision occurs between i, j, we get a constraint as in (B.1) on the velocity

$$\begin{aligned} v_i - v_{j} = \frac{1}{\tau _{rec} } \delta x_\perp - \frac{\tau _1}{\tau _{rec}} ({\bar{v}}_{i} - v_{j})- \frac{1}{\tau _{rec} } \nu _{rec} \, , \end{aligned}$$

meaning that $v_i$ belongs to a rectangle of width $\frac{R}{|\tau _1| |{\bar{v}}_{i} - v_{j}|}$ thanks to (3.10), which after integration over two parents leads to estimate of the type ${\varepsilon }| \log {\varepsilon }|^3$ (see Lemma 3.7). Imposing an extra condition on the velocity $v'_i$ after the first recollision means that the recollision angle $\nu _{rec}$ can take values only in a small set. Thus the constraint above will be stronger and $v_i$ has to take values in a reduced set, much thinner than the rectangle considered in Lemma 3.7. The core of the proof of Lemma B.1 is to identify this reduced set and to show that after integrating over some parents its measure is less than $O({\varepsilon })$: here and in the following, we do not try to keep track of the powers of R and t coming up in the estimates.

Proof

Throughout the proof, we suppose that the parameters associated with the first recollision between i, j satisfy

$$\begin{aligned} | ({\bar{v}}_{i} - v_{j}) \tau _1| \ge R^{4 \over 4 - 5 \gamma } \ge R^3 \quad \text {for some} \ \gamma \in ]\frac{2}{3}, \frac{4}{5}[ \ \text {to be fixed later}. \end{aligned}$$

(B.8)

Otherwise, the estimate (B.5) applied with $M = R^{4 \over 4 - 5 \gamma }$ leads to a suitable upper bound of order ${\varepsilon }$. As a consequence of (B.8), we deduce that $| \tau _1|$ is large enough

$$\begin{aligned} | \tau _1| \ge R^2 . \end{aligned}$$

(B.9)

After the first recollision, $v'_i$ is given by one of the following formulas

(B.10)

Note that the second choice is the value $v_j'$ and we use this abuse of notation to describe the case when $|v'_j - v_k| \le {\varepsilon }^{3/4}$.

We expect the condition (B.6) to impose a strong constraint on the recollision angle $\nu _{rec}$. We indeed find from (B.10) that this condition implies

$$\begin{aligned} \begin{aligned} \text{ either } \qquad v_k - v_j = (v_i-v_j) \cdot \nu _{rec}^\perp \, \nu _{rec}^\perp +O({\varepsilon }^{3 /4}) \, ,\\ \text{ or } \qquad v_k - v_j = (v_i-v_j) \cdot \nu _{rec} \, \nu _{rec} +O({\varepsilon }^{3 /4}) \, . \end{aligned} \end{aligned}$$

(B.11)

We consider now three different cases according to the label k. Each different case listed below will be associated with scenarios, labelled by some p which will take values in $4, \dots , p_1$ for some $p_1$ we shall not attempt to compute.

$\underline{\text {Case} \, k\ne j \text {and} \, k\ne 1^*}$

• If $| v_j-v_k |> {\varepsilon }^{5/8} \gg {\varepsilon }^{3/4}$, we deduce from the constraint (B.11) that the recollision angle is in a small angular sector

  $$\begin{aligned} \nu _{rec} =\pm {(v_j - v_k)^\perp \over |v_k - v_j|} + O( {\varepsilon }^{1/8}) \quad \hbox { or }\quad \nu _{rec} =\pm { v_k - v_j \over |v_k - v_j |} + O( {\varepsilon }^{1/8}) \,.\qquad \end{aligned}$$

  (B.12)

Plugging this Ansatz in (B.1), we get

$$\begin{aligned} v_i - v_{j} = \frac{1}{\tau _{rec} } \delta x_\perp - \frac{\tau _1}{\tau _{rec}} ({\bar{v}}_{i} - v_{j})- \frac{1}{\tau _{rec} }{\mathcal {R}_{n' \pi /2}(v_k - v_j ) \over |v_k - v_j |} + O\left( {{\varepsilon }^{1/8}\over \tau _{rec}}\right) ,\quad \end{aligned}$$

denoting by $\mathcal {R}_\theta$ the rotation of angle $\theta$ and $n' = 0,1,2,3$ depending on the identity in (B.12). This implies that $v_i - v_{j}$ lies in a finite union of thin rectangles of size  $2R \times 4R{\varepsilon }^{1/8} \min \left( 1,\frac{1}{|\tau _1||{\bar{v}}_i-v_j|} \right)$, recalling again (3.10). We thus conclude by integrating in $(t_{1^*}, v_{1^*}, \nu _{1^*})$ and $(t_{2^*}, v_{2^*}, \nu _{2^*})$, exactly as in the proof of Proposition 3.5, that these configurations are encoded in a set $\mathcal {P}_2(a,p, \sigma )$ (with $|\sigma |\le 2$) of size $O( R^7 s t^3 {\varepsilon }^{9/8} |\log {\varepsilon }|^3)$. $\square$

• If $|v_j-v_k| \le {\varepsilon }^{5/8}$, we forget about all other constraints: we conclude by combining (C.3)–(C.4), as in the case 1.2(c) in Section 6.2, that these pseudodynamics are encoded in a set $\mathcal {P}_2(a,p,\sigma )$ (with $|\sigma |\le 2$) of size $O( R^5 s t^2 {\varepsilon }^{5/4} |\log {\varepsilon }|)$.

$\underline{\text {Case} \,k= j}$

If $k= j$, then $|v_k - v_i' |=| v_j'- v_i'| = |v_i- v_j| \le {\varepsilon }^{3/4}$, and we conclude exactly as in the previous case by combining (C.3)–(C.4), that these pseudodynamics are encoded in a set $\mathcal {P}_2(a,p,\sigma )$ (with $|\sigma |\le 2$) of size $O( R^5 s t^2 {\varepsilon }^{3/2} |\log {\varepsilon }|)$.

$\underline{\text {Case}\, k= 1^*}$

This is the most delicate case as $v_i'$ and $v_k$ are linked through the same collision. We stress the fact that the label i refers to a pseudo particle, thus many cases have to be considered (see Figure ).

Fig. 9

The different scenarios associated with the case $k= 1^*$ are depicted. In all cases, i refers to the pseudo particle recolliding with j; thus the labels i, k can be switched after the collision. We used the notation (C.9) for the velocities $V',V'_*$ after scattering

We start by the case depicted in Figure  (i). Denote by $v_k = V'_*$ the velocity of particle $1^*$ after collision at $t_{1^*}^-$. The constraint (B.6) that $|v_k - v'_i| \le {\varepsilon }^{3/4}$ or that $|v_k - v'_j| \le {\varepsilon }^{3/4}$ implies that one of the following identities then holds

$$\begin{aligned}&\displaystyle v_i - (v_i-v_j) \cdot \nu _{rec}\, \nu _{rec} = v_k + O({\varepsilon }^{3 /4})\, ,\\&\displaystyle \text{ or }\quad v_j + (v_i-v_j) \cdot \nu _{rec}\, \nu _{rec}=v_i - (v_i-v_j) \cdot \nu ^\perp _{rec} \nu ^\perp _{rec} = v_k + O({\varepsilon }^{3/ 4}) \, , \end{aligned}$$

and we further have that $|v_i - v_k| = | {\bar{v}}_i - v_{1^*}|$.

If $|{\bar{v}}_i - v_{1^*}|\le {\varepsilon }^{5/8}$, then $v_{1^*}$ has to be in a ball of radius ${\varepsilon }^{5/8}$ so we find a bound $O(R{\varepsilon }^{5/4}t )$ on integration over $1^*$.

If $|{\bar{v}}_i - v_{1^*}|\ge {\varepsilon }^{5/8}$, then

$$\begin{aligned} \nu _{rec} =\pm {v_i- v_k \over |v_i - v_k | }+O({\varepsilon }^{1/8}) \qquad \hbox {or} \qquad \nu _{rec} =\pm {(v_i- v_k )^\perp \over |v_i - v_k |}+O({\varepsilon }^{1/8}) \,. \end{aligned}$$

Plugging this Ansatz in (B.1), we get

$$\begin{aligned} v_{i} - v_{j} = \frac{1}{\tau _{rec} }\delta x_\perp - \frac{\tau _1}{\tau _{rec}} ({\bar{v}}_{i} - v_{j})- \frac{1}{\tau _{rec} }\mathcal {R}_{n \frac{\pi }{2} } { v_i - v_k \over | v_i- v_k |} + O\left( {{\varepsilon }^{1/8} \over \tau _{rec}}\right) ,\qquad \end{aligned}$$

(B.13)

with $n\in \{0,1,2,3\}$. Compared with the formulas of the same type encountered in the proof of Proposition 3.5, this one has the additional difficulty that the “unknown” $v_i$ is on both sides of the equation. Furthermore the direction of $v_k -v_i$ may have very fast variations when $|v_k - v_i| = |v_{1^*} - {\bar{v}}_i|$ is small. To take this into account, we will consider different cases.

Using the notation (B.2), we define

$$\begin{aligned} w:= \delta x_\perp - ({\bar{v}}_{i} - v_{j}) \tau _1, \quad \text { and} \quad u:=|w|/\tau _{rec} \, . \end{aligned}$$

By construction

$$\begin{aligned} |w| \ge | ({\bar{v}}_{i} - v_{j}) \tau _1| \quad \text {and} \quad u\le 4R\, , \end{aligned}$$

(B.14)

where the latter inequality follows from (B.13). Recall that we can restrict to values such that $|w| \ge | ({\bar{v}}_{i} - v_{j}) \tau _1| \ge R^{4 \over 4 - 5 \gamma }$ due to (B.8). With these new variables, the condition (B.13) may be rewritten

$$\begin{aligned} v_{i} - {\bar{v}}_i = v_j-{\bar{v}}_i +u {w\over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { v_i - {v_k}\over | v_i- v_k|} + O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ),\nonumber \\ \end{aligned}$$

(B.15)

where the error term

$$\begin{aligned} F( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) = {{\varepsilon }^{1/8} R\over | {\bar{v}}_{i} - v_{j}| |\tau _1|} \end{aligned}$$

has been estimated thanks to (B.14).

The other cases depicted in Figure  obey the same equation (see (B.16) for $(ii \; a)$ and (B.26) for $(ii \; b)$). We will analyze the solutions of this equation for all cases of Figure .

$\blacktriangleright$ The easiest case is $(ii\, a)$ when the collision at $t_{1^*}$ has no scattering, i.e. $v_i = v_{1^*}$ and $v_k = {\bar{v}}_i$. We split the analysis into two more subcases.

• If $| v_{1^*} - {\bar{v}}_i | \ge {1\over |w|^\gamma }$, the condition (B.15) reads

  $$\begin{aligned} v_{1^*} - {\bar{v}}_i = v_j-{\bar{v}}_i +u {w\over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { v_{1^*} - {\bar{v}}_i\over | v_{1^*} - {\bar{v}}_i|} + O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ) \,,\nonumber \\ \end{aligned}$$

  (B.16)

with $n \in \{ 0,1,2,3 \}$. To analyse this equation, we implement a fixed point method and consider $u \in [-4R,4R]$ as a parameter, forgetting its dependency on $\tau _{rec}$ (i.e. on $v_i$). Due to the assumption that $| v_{1^*} - {\bar{v}}_i | \ge {1/ |w|^\gamma }$, this imposes a constraint on u since one needs to ensure that $| v_j-{\bar{v}}_i +u {w\over |w|} | \ge c/ |w|^\gamma$. Depending on the angle between $v_j-{\bar{v}}_i$ and w and depending on the size of $|v_j-{\bar{v}}_i|$, this implies that u should belong to one or two intervals in $u \in [-4R,4R]$. Given u in one of those admissible intervals, we first look for a solution of the equation without the error term. The mapping

$$\begin{aligned} \Theta : B_R \setminus B_{|w|^{-\gamma }} ({\bar{v}}_i)&\rightarrow {{\mathbb {S}}}\nonumber \\ v_{1^*}&\mapsto {v_{1^*} - {\bar{v}}_i\over | v_{1^*} - {\bar{v}}_i|} \end{aligned}$$

(B.17)

is Lipschitz continuous with constant $|w|^\gamma$. We deduce by a fixed point argument that, for any admissible u

$$\begin{aligned} v_{1^*} - {\bar{v}}_i = v_j-{\bar{v}}_i +u {w\over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { v_{1^*} - {\bar{v}}_i\over | v_{1^*} - {\bar{v}}_i|} \end{aligned}$$

has a unique solution ${\hat{v}}_{1^*}(u)$ (which is clearly Lipschitz in u). Thus for a given u, any solution of (B.16) satisfies

$$\begin{aligned} | v_{1^*} - {\hat{v}}_{1^*}(u) | \le O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ) . \end{aligned}$$

(B.18)

Note that among the solutions $v_{1^*}$ of (B.16), we are looking only for the solutions which are compatible with the constraint $u = |w|/\tau _{rec}$ where $\tau _{rec}$ is a function of $v_{1^*}$. In particular ${\hat{v}}_{1^*}(u)$ will not correspond to a velocity compatible with this constraint. Nevertheless it is enough use the bound (B.18) as a sufficient condition and to retain only the information that $v_{1^*}$ has to belong to a tube $T( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1)$ located around the curve $u \rightarrow {\hat{v}}_{1^*}(u)$ with $|u| \le 4 R$ and of width $O \big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \big )$. Integrating first with respect to the collision with $1^*$, we get

$$\begin{aligned}&\int \mathbf{1}_{ \{ v_{1^*}\in T( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1) \} } \,b(\nu _{1^*},v_{1^*}) \, dv _{1^*} d\nu _{1^*} \le C R^2 F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \nonumber \\&\quad = C {R^3 {\varepsilon }^{1/8} \over |\tau _1| \, |v_j-{\bar{v}}_i| } , \end{aligned}$$

(B.19)

where we replaced $F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon })$ by its value. Next we integrate with respect to $|\tau _1|$ in the set $[R, \frac{T R}{{\varepsilon }}]$ (see (B.9)) and we get after a change of variable in $t_1^*$

$$\begin{aligned} \int \mathbf{1}_{ \{ v_{1^*}\in T( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1) \}} \,b(\nu _{1^*},v_{1^*}) \, dv _{1^*} d\nu _{1^*} dt_{1^*} \le C R^3{{\varepsilon }^{9/8} | \log {\varepsilon }|\over |v_j-{\bar{v}}_i| }. \end{aligned}$$

(B.20)

It remains then to integrate the singularity $|v_j-{\bar{v}}_i|$. As the first recollision involves i, j, there exists always an additional parent of (i, j) to provide a degree of freedom. Thus after integration, the singularity $|v_j-{\bar{v}}_i|$ can be controlled up to a loss $O(|\log {\varepsilon }|)$ by application of Lemma C.2 (see also page 24). This situation will be referred to as a new scenario $\mathcal {P}_2(a,p,\sigma )$ for some p, and $|\sigma | \le 2$. Summing over all possible q and all possible j provides the estimate

$$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2(a,p,\sigma ) } \prod _{m\in \sigma } \,\big | \big (v_{m}-v_{a(m)} ( t_{{m}} ) )\cdot \nu _{{m}} \big | d t_{{m}} d \nu _{{m}}dv_{{m}} \le C s R^5 t^2 {\varepsilon }^{9/8} |\log {\varepsilon }|^2 \, . \end{aligned}$$

• If $| v_{1^*} - {\bar{v}}_i | \le {1\over |w|^\gamma }$, then

  $$\begin{aligned} \int \mathbf{1}_{ \{ | v_{1^*} - {\bar{v}}_i |\le \frac{1}{|w|^\gamma } \}} \,b(\nu _{1 ^*},v_{1 ^*}) \, dv _{1^*} d\nu _{1^*} \le {C R \over |w|^{2 \gamma }} \le {C R \over \tau _1^{2 \gamma } |v_j-{\bar{v}}_i|^{2 \gamma }}, \end{aligned}$$

by (B.14). By (B.9), we know that $| \tau _1| \ge R^2$, thus the singularity is integrable in $\tau _1$. Changing to the variable $t_{1^*}$ by using (B.2), we find that

$$\begin{aligned} \int \mathbf{1}_{\{ | v_{1^*} - {\bar{v}}_i |\le \frac{1}{|w|^\gamma } \} } \,b(\nu _{1^*},v_{1^*}) \, dv_{1^*} d\nu _{1^*} dt_{1^*} \le {\varepsilon }{C \over |v_j-{\bar{v}}_i|^{2 \gamma } }. \end{aligned}$$

It remains then to integrate the singularity $|v_j-{\bar{v}}_i|^{- 2 \gamma }$, which can be done with two additional parents of (i, j) since $\gamma <1$ (see Lemma C.2). This situation will be referred to as a new scenario $\mathcal {P}_2(a,p,\sigma )$ for some p, and $|\sigma | \le 3$. We have

$$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2(a,p,\sigma ) } \prod _{m\in \sigma } \, \big | \big (v_{m}-v_{a(m)} ( t_{{m}} ) )\cdot \nu _{{m}} \big | d t_{{m}} d \nu _{{m}}dv_{{m}} \le C s (Rt)^r {\varepsilon }\, . \end{aligned}$$

(B.21)

Fig. 10

Carleman’s parametrization $(V',V_*')$ can be evaluated in terms of the measure $dV' d\mu$ or alternatively by parametrizing $V' - {\bar{v}}_i$ in polar coordinates by the measure $\lambda d \lambda \, d \psi d\mu$ with $\lambda = | V' - {\bar{v}}_i |$. The direction ${V'-V_*'\over |V'-V_*'| }$ can be recovered by a rotation from $V' - {\bar{v}}_i$ by an angle $\theta$ (B.23) or $V_*' - {\bar{v}}_i$ by an angle $\theta '$ (B.27)

$\blacktriangleright$ If there is scattering at time $t_{1^*}$ of the type (i) such that the velocities are given by $v_i = V'$ and $v_k = v'_{1^*} = V'_*$, with the notation in (C.9), then we consider two cases depending as above on the size of $| v_i - {\bar{v}}_i |$ compared to $|w|^{-\gamma }$.

• If $| v_i - {\bar{v}}_i | = | V' - {\bar{v}}_i | \ge {1/ |w|^\gamma }$, then Identity (B.15) leads to

  $$\begin{aligned} V' - {\bar{v}}_i= & {} v_j-{\bar{v}}_i +u {w\over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { V'- V'_*\over | V'-V'_*|} \nonumber \\&+ O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ) \,. \end{aligned}$$

  (B.22)

We will use Carleman’s parametrization and denote $\mu = | V'_* - {\bar{v}}_i|$ (see Figure 10). For a given $V'_*$, we have to compute the regularity of the map which associates to $V'$ the direction of $V'- V'_*$. From the mapping $V' \rightarrow \Theta (V')$ defined in (B.17), we first determine the direction of $V'- {\bar{v}}_i$ and then rotate this direction by $\theta$ to get

$$\begin{aligned} { V'- V'_*\over | V'-V'_*|} = \mathcal {R}_\theta [ \Theta (V')] \quad \hbox { with } \quad \theta = \arctan { \mu \over \lambda }. \end{aligned}$$

(B.23)

If $\mu > {1 \over |w|^{\gamma /4 } }$, the mapping $V' \rightarrow \mathcal {R}_\theta [\Theta (V')]$ is continuous with Lipschitz constant less than

$$\begin{aligned} |w|^\gamma \times \frac{\mu }{\mu ^2 + \lambda ^2} \le \frac{|w|^\gamma }{\mu } \le |w|^{5 \gamma /4}. \end{aligned}$$

(B.24)

As $\gamma < \frac{4}{5}$ and $|w| \ge R^{4 \over 4 - 5 \gamma }$ (B.8), we deduce by a fixed point argument that, for any $u\le 4R$

$$\begin{aligned} V' - {\bar{v}}_i = v_j-{\bar{v}}_i +u {w \over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { V'- V'_* \over | V'-V'_*|} \end{aligned}$$

has a unique solution ${\hat{V}}' (u)$ (which is Lipschitz in u). Thus for a given u, any solution of (B.22) satisfies

$$\begin{aligned} | V'-{\hat{V}}' (u) | \le O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ). \end{aligned}$$

In other words, $V'$ has to belong to a tube $T( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1)$ of width $O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big )$ around the curve $u \rightarrow {\hat{V}}' (u)$. By Carleman’s parametrization, we can then integrate over $d V' d\mu$ and get an estimate of the form (B.19) when replacing F by its value

$$\begin{aligned}&\int \mathbf{1}_{ \{ \mu > {1\over |w|^{\gamma /4} } \} } \mathbf{1}_{ \{ V' \in T( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1) \}} \, d V' d \mu \le C R^2 F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon })\nonumber \\&\quad = C {R^3 {\varepsilon }^{1/8} \over |\tau _1| \, |v_j-{\bar{v}}_i| } . \end{aligned}$$

(B.25)

Integrating then over $\tau _1$ leads to an upper bound analogous to (B.20)

$$\begin{aligned} \int \mathbf{1}_{ \{ \mu > {1\over |w|^{\gamma /4} } \} } \mathbf{1}_{ \{ V' \in T( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1) \}} \, d V' d \mu dt_{1^*} \le C R^3{{\varepsilon }^{9/8} | \log {\varepsilon }| \over |v_j-{\bar{v}}_i| }. \end{aligned}$$

We then conclude by integrating as usual the singularity $1/ |v_j-{\bar{v}}_i|$ thanks to Lemma C.2. This provides a new scenario $\mathcal {P}_2(a,p,\sigma )$ for some p and $|\sigma | \le 2$.

If $\mu < {1\over |w|^{\gamma /4} }$, we only use the condition (B.1) which reads in this case

$$\begin{aligned} V' - {\bar{v}}_i = v_{j} - {\bar{v}}_i + \frac{1}{\tau _{rec} } w - \frac{1}{\tau _{rec} } \nu _{rec} \quad \text{ with } \quad \Big | \frac{1}{\tau _{rec} } \Big | \le { 4 R \over |\tau _1| \, |v_j-{\bar{v}}_i| }. \end{aligned}$$

As a consequence, $V'$ has to be in the rectangle $\mathcal {R}( \delta x_\perp , {\bar{v}}_i-v_j, q,\tau _1)$ of size $R \times { 4 R \over |\tau _1| \, |v_j-{\bar{v}}_i| }$. Together with the condition on $\mu$, this leads to

$$\begin{aligned} \int \mathbf{1}_{\{ V' \in \mathcal {R}( \delta x_\perp , v_j - {\bar{v}}_i, q,\tau _1) \} } \mathbf{1}_{ \{ \mu <\frac{1}{|w|^{\gamma /4} } \} } \, d V' d \mu \le { C R^2 \over |\tau _1|^{1+\gamma /4} \; |v_j-{\bar{v}}_i|^{1+\gamma /4} } . \end{aligned}$$

Since $1 + \frac{\gamma }{4} >1$, we can integrate with respect to $t_{1^*}$ and gain a factor ${\varepsilon }$. It remains then to integrate the singularity $|v_j-{\bar{v}}_i|^{-(1+\gamma /4)}$. Since $1+\frac{\gamma }{4} < 2$, this can be done by using (C.6) and (C.7) with two additional parents of (i, j). This provides another contribution to $\mathcal {P}_2(a,p,\sigma )$ with $|\sigma | \le 3$.

• If $| V' - {\bar{v}}_i | \le {1\over |w|^\gamma }$. This can be dealt as in the case (B.21).

$\blacktriangleright$ If there is scattering at time $t_{1^*}$ of type $(ii \; b)$ such that the velocities are given by $v_i = V'$ and $v'_{1^*} = V'_*$ with the notation in (C.9), then as above we separate the analysis into two sub-cases, depending on the relative size of $| v_i - {\bar{v}}_i |$ and $|w|^{-\gamma }$.

• Suppose that $| V'_* - {\bar{v}}_i | \ge {1\over |w|^\gamma }$. The analogue of identity (B.15) reads

  $$\begin{aligned} V' _*- {\bar{v}}_i = v_j-{\bar{v}}_i +u {w\over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { V'- V'_*\over | V'-V'_*|} + O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ),\nonumber \\ \end{aligned}$$

  (B.26)

with $n \in \{ 0,1,2,3 \}$. Denote by $\lambda = | V' - {\bar{v}}_i|$ and consider two cases.

If $\lambda > {1\over |w|^{\gamma /4} }$, we proceed as in (B.23) and recover the direction of $V'- V'_*$ by

$$\begin{aligned} { V'- V'_*\over | V'-V'_*|} = \mathcal {R}_{\theta '} [\Theta (V'_*)] \qquad \hbox { with } \qquad \theta ' = \arctan {\lambda \over \mu }. \end{aligned}$$

(B.27)

Given $\lambda > {1 \over |w|^{\gamma /4} }$, the map $V'_* \rightarrow \mathcal {R}_{\theta '} [\Theta (V'_*)]$ has a Lipschitz constant $|w|^{5 \gamma /4}$ as $| V'_* - {\bar{v}}_i | \ge {1\over |w|^\gamma }$ (see (B.24)). Since $\gamma < \frac{4}{5}$, we deduce by a fixed point argument that, for any $u\le 4R$

$$\begin{aligned} V'_* - {\bar{v}}_i = v_j-{\bar{v}}_i +u {w\over |w|} - {u\over |w|} \mathcal {R}_{n \frac{\pi }{2} } { V'- V'_*\over | V'-V'_*|} \end{aligned}$$

has a unique solution ${\hat{V}}'_*(u)$ (which is Lipschitz in u) and that any solution of (B.26) takes values close to this solution

$$\begin{aligned} | V'_*-{\hat{V}}'_* (u) | \le O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big ). \end{aligned}$$

In other words, $V'_*$ has to belong to a tube $T( \delta x_\perp , v_j- {\bar{v}}_i, q,\tau _1)$ of size $\delta = O \Big ( F ( |{\bar{v}}_{i} - v_{j}|, \tau _1, {\varepsilon }) \Big )$ around the smooth curve $u \rightarrow {\hat{V}}'_*(u)$ which stretches in the direction ${w \over |w|}$. Mimicking the proof of (C.10), we can decompose the tube $T( \delta x_\perp , v_j- {\bar{v}}_i, q,\tau _1)$ into small blocks of side length $\delta$. Summing over these blocks, we recover that the measure of $\{ V'_* \in T( \delta x_\perp , v_j- {\bar{v}}_i, q,\tau _1)\}$ is less than

$$\begin{aligned} R^2 \delta \big | \log \delta \big | \le {R^3 {\varepsilon }^{1/8} \, \big | \log {\varepsilon }\big | \over | {\bar{v}}_{i} - v_{j}| |\tau _1|}. \end{aligned}$$

Integrating with respect to $t_{1^*}$ and then integrating the singularity $|{\bar{v}}_i - v_j|$, we get a contribution of order $R^3 {\varepsilon }^{1/8} |\log {\varepsilon }|^2$ which controls the occurence of a new scenario $\mathcal {P}_2 (a,p,\sigma )$.

If $\lambda < {1 \over |w|^{\gamma /4} }$, we only use the condition (B.1) which implies that $V'_*$ has to be in the rectangle $\mathcal {R}( \delta x_\perp , v_j - {\bar{v}}_i, q,\tau _1)$ of size $R \times \frac{4R}{|w|}$. With the notation of Figure 10, Carleman’s parametrization $(V',V_*')$ can be evaluated in terms of the measure $\lambda d \lambda \, d \psi d\mu$. Thus we get

$$\begin{aligned} \int \mathbf{1}_{ \{ V' _* \in \mathcal {R}( \delta x_\perp , v_j - {\bar{v}}_i , q,\tau _1) \} } d \mu d\psi \le {R|\log w| \over |w|}\,, \end{aligned}$$

together with the condition on $\lambda$ which is independent

$$\begin{aligned} \int \mathbf{1}_{ \{ | \lambda |< \frac{1}{|w|^{\gamma /4}} \} } \lambda d \lambda \le {1\over |w|^{\gamma /2} }. \end{aligned}$$

As a consequence

$$\begin{aligned} \int \mathbf{1}_{ \{ V' \in \mathcal {R}( \delta x_\perp , v_j - {\bar{v}}_i, q,\tau _1) \}} \mathbf{1}_{ \{ | V'_* - {\bar{v}}_i|< \frac{1}{|w|^{\gamma /4}} \} } \,b(\nu _{1 ^*},v_{1 ^*}) \, dv _{1^*} d\nu _{1 ^*} dt_{1^*} \le C { {\varepsilon }\log |v_j-{\bar{v}}_i| \over |v_j-{\bar{v}}_i|^{1+\gamma /2} }. \end{aligned}$$

It remains then to integrate the singularity $|v_j-{\bar{v}}_i|^{-(1+\gamma /2)}$, which can be done by using (C.6) and (C.7) with two additional parents of (i, j). This leads to another scenario to $\mathcal {P}_2(a,p,\sigma )$ for some p.

• Suppose that $| V'_* - {\bar{v}}_i | \le {1\over |w|^\gamma }$. By integration with respect to $b(\nu _{1 ^*},v_{1 ^*}) \, dv _{1^*} d\nu _{1 ^*}$, we will gain only one power of $|w|^{-\gamma }$ due to the scattering (see Lemma C.2). This is not integrable with respect to time $\tau _1$ and we have therefore to use also the fact that there is a recollision between (i, j) to regain some control. Because i has only a very small deflection at time $t_{1^*}$, this implies that a “kind of recollision” has to be triggered already before the collision with $1^*$, i.e. at time $t_{2^*}$.

Fig. 11

In the case $| V'_* - {\bar{v}}_i | \le {1\over |w|^\gamma }$, one has to use as well the degree of freedom from the collision with $2^*$. In this figure, the particle j is not deviated and $u_j = v_j$. The parameter w stands for the distance between i, j at time $t_{1^*}$

Denote by $(y_i, u_i)$ and $(y_j,u_j)$ the positions and velocities of the pseudo-particles i and j at time $t_{3^*}$ (see Figure 11) and set

$$\begin{aligned} \frac{y_i - y_j}{{\varepsilon }} = \delta y _\perp + \frac{\lambda '}{{\varepsilon }} ( u_i - u_j) \quad \text {with} \quad \delta y _\perp \cdot ( u_{i} - u_{j}) = 0 \, . \end{aligned}$$

We define also

$$\begin{aligned} \tau _2 = - \frac{1}{{\varepsilon }} ( t_{2^*} - t_{3^*} + \lambda ')\,, \qquad {\tilde{\tau }}_{rec} = \frac{t_{2^*}-t_{rec}}{{\varepsilon }}\,, \qquad \tau _{rec} = \frac{t_{1^*}-t_{rec}}{{\varepsilon }} \le {\tilde{\tau }}_{rec}\,. \end{aligned}$$

By analogy with Eq. (B.1), we get

$$\begin{aligned} {\bar{v}}_i - v_{j} = \frac{1}{{\tilde{\tau }}_{rec} } \delta y _\perp - \frac{\tau _2}{{\tilde{\tau }}_{rec}} ( u_{i} - u_{j}) - \frac{\tau _{rec}}{{\tilde{\tau }}_{rec}} ( v_i - {\bar{v}}_i) - \frac{1}{{\tilde{\tau }}_{rec} } \nu _{rec} \, , \end{aligned}$$

(B.28)

where the additional term comes from the small deflection at time $t_1^*$. By assumption, this term is less or equal than $|w|^{-\gamma }$. As previously, we have that

$$\begin{aligned} 4 R \ge |{\bar{v}}_i - v_j| + \frac{\tau _{rec}}{{\tilde{\tau }}_{rec}} |v_i - {\bar{v}}_i| \ge \frac{1}{{\tilde{\tau }}_{rec}} \Big ( |\tau _2| |u_{i} - u_{j}| - 1 \Big ) \ge \frac{1}{ 2 {\tilde{\tau }}_{rec}} |\tau _2| |u_{i} - u_{j}| \,, \end{aligned}$$

as it is enough to consider $|\tau _2| |u_{i} - u_{j}| \gg 1$ (see (B.5)). Thus we get

$$\begin{aligned} {\tilde{\tau }}_{rec} \ge \frac{1}{8 R} |\tau _2| |u_i-u_j| \,. \end{aligned}$$

In the remaining of the proof, we fix the parameter $\gamma$ and a new parameter $\alpha$ as follows

$$\begin{aligned} \gamma = \frac{3}{4}\,, \quad \alpha =\frac{4}{7} \end{aligned}$$

(B.29)

and consider two cases according to large and small values of $\tau _1$.

• If $|\tau _1| \ge {1 \over |{\bar{v}}_i - v_j|^{\gamma /(\gamma -\alpha )}}$, then we get a control on the size of $1/ |w|^\gamma$

$$\begin{aligned} \frac{1}{|w|^\gamma } \le \frac{1}{(|\tau _1| |{\bar{v}}_i - v_j|)^\gamma } \le \frac{1}{ |\tau _1|^\alpha } , \end{aligned}$$

as $|w| \ge |\tau _1| |{\bar{v}}_i - v_j|$ from (B.14).

Equation (B.28), imposes the condition that, at time $t_{2^*}$, ${\bar{v}}_i - v_j$ has to be in a domain which is a kind of rectangle $\mathcal {K}$ with axis $\delta y _\perp - \tau _2 ( u_{i} - u_{j})$ and varying width

$$\begin{aligned} \frac{1}{|w|^\gamma } + \frac{1}{|\tau _2| |u_i-u_j|} \le \frac{1}{|\tau _1|^\alpha } + \frac{1}{|\tau _2| |u_i-u_j|}. \end{aligned}$$

Recall from (B.9) that $|\tau _1|\ge R$. Combined with the condition $| V'_* - {\bar{v}}_i | \le {1\over |w|^\gamma }$ at time $t_{1^*}$, we get thanks to Lemma C.2

$$\begin{aligned}&\int \mathbf{1}_{ \{ |\tau _1|\ge R \} } \mathbf{1}_{\{ | V'_* - {\bar{v}}_i | \le {1\over |w|^\gamma } \}} \mathbf{1}_{\{ {\bar{v}}_i -v_j \in \mathcal {K}\}} \prod _{\ell = 1^*,2^*}b(\nu _\ell ,v_\ell ) \, dv_\ell d\nu _\ell d t_{\ell } \\&\quad \le \int \mathbf{1}_{ \{ |\tau _1|\ge R \} } \, {\mathbf{1}_{\{ {\bar{v}}_i -v_j \in \mathcal {K}\}} \over |\tau _1|^\alpha } b(\nu _{2^*},v_{2^*}) \, dv_{2^*} d\nu _{2^*} d t_{2^*} d t_1. \end{aligned}$$

At this stage, one has to be careful as $\tau _1$ was defined in (B.2) by

$$\begin{aligned} \tau _1 :=- \frac{1}{{\varepsilon }}(t_{1^*} - t_{2^*}+\lambda ) \quad \text {with} \quad \lambda = \big ( x_i (t_{2^*}) - x_j (t_{2^*}) - q \big ) \cdot \frac{{\bar{v}}_{i} - v_{j}}{|{\bar{v}}_{i} - v_{j}|}, \end{aligned}$$

thus $|\tau _1|$ depends on ${\bar{v}}_{i} - v_{j}$, i.e. also on $v_{2^*}$. In order to simplify the dependency between the variables $v_{2^*}$ and $t_1$, we replace $t_{1^*}$ with the variable $\tau _1$. This boils down to integrating with respect to ${\varepsilon }d\tau _1$. The geometric structure implies that $\tau _1$ takes now values in a complicated domain which we will estimate from above by keeping only the constraint $|\tau _1| \in [ R, \frac{R^2}{{\varepsilon }}]$. This decouples the variables in the integral and we finally get

$$\begin{aligned}&\int \mathbf{1}_{ \{ |\tau _1|\ge R \} } \mathbf{1}_{\{ | V'_* - {\bar{v}}_i | \le {1\over |w|^\gamma } \}} \mathbf{1}_{\{ {\bar{v}}_i -v_j \in \mathcal {K}\}} \prod _{\ell = 1^*,2^*}b(\nu _\ell ,v_\ell ) \, dv_\ell d\nu _\ell d t_{\ell } \nonumber \\&\quad \le {\varepsilon }\int \mathbf{1}_{ \{ |\tau _1| \in [ R , \frac{R^2}{{\varepsilon }}]\} } \, {\mathbf{1}_{\{ {\bar{v}}_i -v_j \in \mathcal {K}\}} \over |\tau _1|^\alpha } b(\nu _{2^*},v_{2^*}) \, dv_{2^*} d\nu _{2^*} d t_{2^*} d \tau _1 \nonumber \\&\quad \le CR^2 {\varepsilon }\int \mathbf{1}_{ \{ |\tau _1| \in [ R , \frac{R^2}{{\varepsilon }}] \} } \, \left( \frac{1}{|\tau _1|^{2 \alpha }} + \frac{1}{|\tau _1|^\alpha |\tau _2| |u_i-u_j|} \right) d t_{2^*} d \tau _1 \, . \end{aligned}$$

(B.30)

The first term is integrable in $|\tau _1|\ge R$ as $2 \alpha >1$. The second term can be integrated first with respect to $R \le |\tau _1| \le \frac{R^2}{{\varepsilon }}$ which provides a factor ${\varepsilon }^\alpha$, then with respect to $t_{2^*}$ which provides an additional ${\varepsilon }| \log {\varepsilon }|$. The singularity with respect to small relative velocities can be controlled by two additional integrations. Thus the second term leads to an upper bound less than ${\varepsilon }$ and the corresponding scenarios are indexed by sets $\sigma$ with cardinal 4.

• If $|\tau _1| \le {1 \over |{\bar{v}}_i - v_j|^{\gamma /(\gamma -\alpha )}}$, then we can forget about (B.28). We indeed have that

$$\begin{aligned}&\int \mathbf{1}_{\{ | V'_* - {\bar{v}}_i | \le {1\over |w|^\gamma } \}} \mathbf{1}_{ \{ |\tau _1| \le {1 \over |{\bar{v}}_i - v_j|^{\gamma /(\gamma -\alpha )}} \} } b(\nu _{1^*} ,v_{1^*}) \, dv_{1^*} d\nu _{1^*} d\tau _1\\&\quad \le \int \mathbf{1}_{ \{ |\tau _1| \le {1 \over |{\bar{v}}_i - v_j|^{\gamma /(\gamma -\alpha )}} \} } {1\over (\tau _1 |{\bar{v}}_i - v_j|) ^{ \gamma }} d\tau _1 \\&\quad \le {1\over |{\bar{v}}_i - v_j| ^{ \gamma }} \int \mathbf{1}_{ \{ |\tau _1| \le {1 \over |{\bar{v}}_i - v_j|^{\gamma /(\gamma -\alpha )}} \} } \frac{1}{ |\tau _1|^\gamma } d\tau _1 \le {1\over |{\bar{v}}_i - v_j| ^{ \gamma + \frac{\gamma (1-\gamma )}{\gamma -\alpha } }}. \end{aligned}$$

As $\frac{ \gamma (1-\alpha )}{\gamma -\alpha } <2$, the singularity at small relative velocities is integrable by using Lemma C.2. Thus the change of variable to $t_{1^*}$ allows us to recover an upper bound of order ${\varepsilon }$.

Throughout the proof, the bad sets were analyzed in terms of the recolliding particles, thus we have to reindex these sets in terms of the labels $\sigma$ of the parents. A similar procedure has been done already at the end of the proof of Proposition 3.5. Given a set $\sigma$ of parents, it may only determine the particle i, so that an extra factor $s^2$ has to be added in (B.7) to take into account the choice of j, k. $\square$

B.3: Parallel Recollisions

The following result was used in Section 6.2.2 page 50 to deal with parallel recollisions when $t_{1^* } = t_{{\tilde{1}}}$. The setting is recalled in Figure 12.

Fig. 12

Parallel recollisions

Lemma B.2

Fix a final configuration of bounded energy $z_1 \in \mathbb {T}^2 \times B_R$ with $1 \le R^2 \le C_0 |\log {\varepsilon }|$, a time $1\le t \le C_0|\log {\varepsilon }|$ and a collision tree $a \in \mathcal {A}_s$ with $s \ge 2$.

There exist sets of bad parameters $\mathcal {P}_2 (a, p,\sigma )\subset \mathcal {T}_{2,s} \times {{\mathbb {S}}}^{s-1} \times \mathbb {R}^{2(s-1)}$ for $p_1 < p\le p_2$ and $\sigma \subset \{2,\dots , s\}$ of cardinal $|\sigma |\le 5$ such that

• $\mathcal {P}_2 (a, p,\sigma )$ is parametrized only in terms of $( t_m, v_m, \nu _m)$ for $m \in \sigma$ and $m < \min \sigma$;

  $$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2 (a,p,\sigma ) } \displaystyle \prod _{m\in \sigma } \, \big | \big (v_{m}-v_{a(m)} ( t_{{m}} ) )\cdot \nu _{{m}} \big | d t_{{m}} d \nu _{{m}}dv_{{m}} \le C(Rt)^r s^2 {\varepsilon }\,,\qquad \end{aligned}$$

  (B.31)

  for some constant $r \ge 1$,

• and any pseudo-trajectory starting from $z_1$ at t, with total energy bounded by $R^2$ and such that the first two recollisions involve two disjoint pairs of particles having the same first parent is parametrized by

  $$\begin{aligned} (t_n, \nu _n, v_n)_{2\le n\le s }\in \bigcup _{p_1 < p\le p_2} \bigcup _\sigma \mathcal {P}_2(a, p,\sigma )\,. \end{aligned}$$

Proof

As in the previous section, we suppose from now on that the parameters associated with the first recollision are such that $|\tau _1| |{\bar{v}}_i - v_j| \ge R$. Otherwise, the estimate (B.5) applied with $M = R$ leads to the expected upper bound.

In the following, the parents of i, j will be denoted by the superscript $\ ^*$ and those of $k, \ell$ by the superscript $\tilde{\ }$. Denote by $t_* := \min (t_{2^*}, t_{{\tilde{2}}})$ the first time (before $t_{1^*}$) when one of the particles i, j or k has been deviated. Without loss of generality (up to exchanging j and k), we can assume that i and k are not colliding together at time $t_*$.

We describe the recollision between (i, j) by the identity

$$\begin{aligned} v_{i} - v_{j} = \frac{1}{t_{rec}-t_{1^*}} \big ( x_i(t_{1^*}) -x_j(t_{1^*})+q+{\varepsilon }\nu _{rec} \big ) \, , \end{aligned}$$

(B.32)

with q an element in $\mathbb {Z}^2$ which we fix from now on (in the end the estimates will be multiplied by $R^2t^2$ to take this fact into account). Similarly the recollision between $(k,\ell )$ can be written

$$\begin{aligned} v_\ell - v_{k} = \frac{1}{{\tilde{t}}_{rec}-t_{1^*}} \big ( x_i(t_{1^*})+{\varepsilon }\nu _{1^*} -x_k(t_{1^*})+{\tilde{q}}+{\varepsilon }{\tilde{\nu }}_{rec} \big ), \end{aligned}$$

(B.33)

with ${\tilde{q}}$ an element in $\mathbb {Z}^2$ which again we fix from now on, up to mutiplying again the estimates by $R^2t^2$ at the end.

We introduce the notation

$$\begin{aligned} {\tilde{x}}_{i,k}(t_{1^*}):= x_i(t_{1^*}) -x_k(t_{1^*})+{\tilde{q}} \quad \text{ and } \quad x_{i,j}(t_{1^*}):= x_i(t_{1^*}) -x_j(t_{1^*})+ q \, . \end{aligned}$$

Equation (B.32) implies that $v_i - v_j$ lies in a rectangle $\mathcal {R}_1$ of main axis $x_{i,j}(t_{1^*})$, and of size $CR\times ( R{\varepsilon }/|x_{i,j}(t_{1^*})| )$. We recall that an integration of this constraint in the collision parameters of particle $1^*$ gives a bound of the type $\min (1, {\varepsilon }|\log {\varepsilon }|^2 / |{\bar{v}}_i - v_j|)$. On the other hand, Equation (B.33) implies that $v_\ell - v_k$ lies in a rectangle $\mathcal {R}_2$ of main axis ${\tilde{x}}_{i,k}(t_{1^*})$ and of size $CR\times (R{\varepsilon }/ |{\tilde{x}}_{i,k}(t_{1^*})|)$.

Let us give the main ideas of the argument. We can rewrite these conditions with Carleman’s parametrization (C.9), with either $(v_i,v_\ell ) = (V', V'_*)$ or $(v_i, v_\ell ) = (V'_*, V')$. We will actually focus on the second situation which is the worst one. We will use the parametrization in polar coordinates as in Figure 10.

The first condition states that $V'_*$ lies in a small rectangle of size $CR\times (R{\varepsilon }/|x_{i,j}(t_{1^*})| )$, which we shall eventually integrate with the measure $d\mu d \psi$. We can show that this integral provides a contribution $(R{\varepsilon }/|x_{i,j}(t_{1^*})| )(1+ |\log ({\varepsilon }/|x_{i,j}(t_{1^*})| ) |)$.

The second condition tells us that $V'$ has to be in the intersection of the line orthogonal to $(V'_*-{\bar{v}}_i)$ passing through ${\bar{v}}_i$ and the rectangle $v_k +\mathcal {R}_2$. We have therefore to evaluate the length of this intersection which appears when we integrate with respect to $\lambda d\lambda$. $\square$

Fig. 13

The dashed line represents the main axis of the rectangle $\mathcal R_2$, oriented in the direction ${\tilde{x}}_{i,k}(t_{1^*})$. The angle $\theta$ is the smallest angle between the axis of $\mathcal R_2$ and any line passing through $v_i$ and intersecting the axis of $\mathcal R_2$

Denote by u the distance from ${\bar{v}}_i$ to the rectangle $v_k + \mathcal R_2$ :

$$\begin{aligned} u := \big |( {\bar{v}}_i - v_k) \wedge {\tilde{x}}_{i,k}(t_{1^*}) |/ | {\tilde{x}}_{i,k}(t_{1^*})|\, \end{aligned}$$

• if this distance is large enough, we expect the length of the intersection to be small;

• if the distance u is small, this imposes an additional constraint on ${\bar{v}}_i - v_k$, that we will analyse with different arguments depending on the size of ${\tilde{x}}_{i,k}(t_{1^*})$.

$\underline{\text {Case}\, u \ge {\varepsilon }^{3/4} }$ The intersection of the line orthogonal to $(V'_*-{\bar{v}}_i)$ passing through ${\bar{v}}_i$ and the rectangle $v_k +\mathcal {R}_2$ of width $\frac{C R{\varepsilon }}{ |{\tilde{x}}_{i,k}(t_{1^*})|}$ (see Figure 13) is a segment of size at most

$$\begin{aligned} d \le \min \Big ( \frac{C{\varepsilon }R }{|{\tilde{x}}_{i,k}(t_{1^*})| \sin \theta } , CR\Big ), \end{aligned}$$

(B.34)

where $\theta$ is the minimal angle between the axis of $\mathcal {R}_2$ and any line passing through ${\bar{v}}_i$ and intersecting $v_k+ \mathcal {R}_2$. We have

$$\begin{aligned} \sin \theta \ge \frac{u}{2 R} \ge \frac{{\varepsilon }^\frac{3}{4}}{2 R}. \end{aligned}$$

It follows from (B.34) that

$$\begin{aligned} d \le \frac{C{\varepsilon }^\frac{1}{4} R^2 }{ |{\tilde{x}}_{i,k}(t_{1^*})| }. \end{aligned}$$

Multiplying this estimate by the size of $\mathcal {R}_1$, we get the following upper bound for the measure in $|(v_{1^*}-{\bar{v}}_i) \cdot \nu _{1^*} | \, dv_{1^*} d\nu _{1^*}$

$$\begin{aligned} \frac{C R^5 {\varepsilon }^{\frac{5}{4}}|\log {\varepsilon }| }{ \, \big |{\tilde{x}}_{i,k}(t_{1^*})\big | \, \big |x_{i,j}(t_{1^*})\big |} \Big (1+ \Big |\log \frac{{\varepsilon }}{ |x_{i,j}(t_{1^*})|} \Big |\Big ). \end{aligned}$$

• If $|{\tilde{x}}_{i,k}(t_{1^*})\big | \ge {\varepsilon }^{1/8}$ then the bound becomes

  $$\begin{aligned} \frac{C R^5 {\varepsilon }^{\frac{9}{8}}|\log {\varepsilon }| }{ \big |x_{i,j}(t_{1^*})\big |} \, , \end{aligned}$$

  and we are back to the usual computations as in the proof of Proposition 3.5: we integrate over $t_{1^*}$ then over one parent of (i, j) to kill the singularity at small relative velocities, and this gives rise in the end to

  $$\begin{aligned} C R^5 (R^3t)^2 {\varepsilon }^{\frac{9}{8}} |\log {\varepsilon }|^3 \, . \end{aligned}$$

• If $|{\tilde{x}}_{i,k}(t_{1^*})\big | \le {\varepsilon }^{1/8}$, we have a kind of “recollision” between particles i and k at time $t_1^*$. Denote by ${\tilde{2}}, {\tilde{3}}$ the first two parents of (i, k). We therefore get that ${\bar{v}}_i - v_k$ has to belong to the union of $(Rt)^2$ rectangles $\mathcal {R}_3$ of size $CR\times ( R{\varepsilon }^{1/8} /|{\tilde{x}}_{i,k}(t_{{\tilde{2}} })| )$, with

  $$\begin{aligned} {\tilde{x}}_{i,k}(t_{{\tilde{2}}}) : = x_i(t_{{\tilde{2}}}) -x_k(t_{{\tilde{2}}}) +{\tilde{q}} \,. \end{aligned}$$

Combined with the condition that $v_i \in \mathcal {R}_1$, one has to integrate

$$\begin{aligned} R^3 \mathbf{1}_{ \{ {\bar{v}}_i- v_k \in \mathcal {R}_3 \}} \min \left( {{\varepsilon }\, |\log {\varepsilon }|^2\over |{\bar{v}}_i - v_j|}, 1 \right) . \end{aligned}$$

Denote by $\sigma =\{1^*, 2^*, 3^*\} \cup \{{\tilde{2}}, {\tilde{3}} \}$ so that the cardinal $|\sigma |$ can be 3, 4 or 5. Integrating over $1^*$ leads to an inequality involving constraints on the pairs (i, k) and (i, j)

$$\begin{aligned}&\int \mathbf{1}_{\{ {\bar{v}}_i -v_k \in \mathcal {R}_3 \}} \mathbf{1}_{ \{ v_i - v_j \in \mathcal {R}_1 \} } \, \prod _{m \in \sigma } b(\nu _m,v_m) \, dv_m d\nu _m dt_m \nonumber \\&\quad \le R^3 {\varepsilon }|\log {\varepsilon }|^2 \int \frac{ \mathbf{1}_{\{ {\bar{v}}_i -v_k \in \mathcal {R}_3 \} } }{ | {\bar{v}}_{i}- v_{j}|} \, \prod _{m \in \{ 2^*, 3^*, {\tilde{2}}, {\tilde{3}} \} } b(\nu _m,v_m) \, dv_m d\nu _m dt_m \nonumber \\&\quad \le R^3 {\varepsilon }|\log {\varepsilon }|^2 \left( (R^3t)^2 \int \mathbf{1}_{ \{ {\bar{v}}_i -v_k \in \mathcal {R}_3 \} } \prod _{m= {\tilde{2}}, {\tilde{3}}} b(\nu _m,v_m) \, dv_m d\nu _m dt_m \right) ^{1/4} \nonumber \\&\qquad \times \left( (R^3t)^2 \int \frac{ 1 }{ | {\bar{v}}_{i}- v_{j}|^{4/3}} \prod _{m= 2^*,3^*} b(\nu _m,v_m) \, dv_m d\nu _mdt_m\right) ^{3/4} \nonumber \\&\quad \le C(Rt)^r {\varepsilon }^{33/32} |\log {\varepsilon }|^{11/4} \,, \end{aligned}$$

(B.35)

where we used Hölder’s inequality in order to decouple both terms: the first one provides as previously a bound ${\varepsilon }^{\frac{1}{8}}|\log {\varepsilon }|^{3}$ and the second one is bounded thanks to (C.6)–(C.7) as the singularity in the relative velocities is less than 2. Note that Hölder’s inequality was performed over the 4 variables $\{ 2^*, 3^*, {\tilde{2}}, {\tilde{3}} \}$, but only two variables are relevant for each integral, thus the contribution of the two others is bounded from above by the factor $(R^3t)^2$.

$\underline{\text {Case}\, u \le {\varepsilon }^{3/4}}$

We recall that

$$\begin{aligned} {\tilde{x}}_{i,k}(t_{1^*})&: =&x_i(t_{1^*}) -x_k(t_{1^*})+{\tilde{q}} = x_i(t_{{\tilde{2}}}) -x_k(t_{{\tilde{2}}}) +{\tilde{q}}+ ( {\bar{v}}_i - v_k) (t_{1^*}-t_{{\tilde{2}}}) \, . \end{aligned}$$

Recalling that

$$\begin{aligned} {\tilde{x}}_{i,k}(t_{{\tilde{2}}}) : = x_i(t_{{\tilde{2}}}) -x_k(t_{{\tilde{2}}}) +{\tilde{q}} \, , \end{aligned}$$

the constraint $u = \frac{ |( {\bar{v}}_i - v_k) \wedge {\tilde{x}}_{i,k}(t_{1^*}) |}{| {\tilde{x}}_{i,k}(t_{1^*})| } \le {\varepsilon }^\frac{3}{4}$ implies

$$\begin{aligned} \big |( {\bar{v}}_i - v_k) \wedge {\tilde{x}}_{i,k}(t_{{\tilde{2}}}) \big | \le C{\varepsilon }^\frac{3}{4} Rt \, . \end{aligned}$$

(B.36)

Recall that the constraint (B.32) on the rectangle ${\mathcal R}_1$ produces a singularity $\frac{R{\varepsilon }}{|x_{i,j}(t_{1^*})|} (1+ |\log ( \frac{{\varepsilon }}{|x_{i,j}(t_{1^*})| }) |)$, and we argue as follows:

• If $| {\tilde{x}}_{i,k}(t_{{\tilde{2}}})| \le {\varepsilon }^{\frac{5}{8}}$, we have a kind of “recollision” between particles i and k at time $t_{{\tilde{2}}}$. We thus proceed as in Case 1 of Section 6.

  • For small relative velocities, we integrate the constraint $|{\bar{v}}_i-v_j | \le {\varepsilon }^\frac{9}{16}$ over two parents of $\{ i, j \}$ using (C.3), (C.4) and we find directly a bound ${\varepsilon }^\frac{9}{8} |\log {\varepsilon }|^2$.

  • When the relative velocities are bounded from below $|{\bar{v}}_i-v_j | \ge {\varepsilon }^\frac{9}{16}$, the contribution of rectangle ${\mathcal R}_1$ gives a bound of the order $CR^2 {\varepsilon }^\frac{7}{16} |\log {\varepsilon }|^2$. By integrating the “recollision” (i, k) over ${\tilde{2}}, {\tilde{3}}$, we find a bound $CR^7 t^3 {\varepsilon }^\frac{5}{8} |\log {\varepsilon }|^3$ so finally this case produces as usual (see Proposition 3.5), after integration over three parameters, the error $CR^9 t^3 {\varepsilon }^\frac{17}{16} |\log {\varepsilon }|^5$.

• If  $| {\tilde{x}}_{i,k}(t_{{\tilde{2}}}) | \ge {\varepsilon }^{\frac{5}{8}}$ then according to (B.36), ${\bar{v}}_i-v_k$ must lie in the union of $(Rt)^2$ rectangles ${\mathcal R}_4$ with axis ${\tilde{x}}_{i,k}(t_{{\tilde{2}}})$ and size $CR \times CRt{\varepsilon }^{\frac{1}{8}}$. This condition has to be coupled with the singularity ${\varepsilon }|\log {\varepsilon }|^2/|{\bar{v}}_i-v_j|$ due to the constraint from the rectangle $\mathcal {R}_1$. We therefore have to integrate

  $$\begin{aligned} R^3 \mathbf{1}_{ \{ {\bar{v}}_i- v_k \in \mathcal {R}_4 \} } \min \left( {{\varepsilon }|\log {\varepsilon }|^2 \over |{\bar{v}}_i - v_j|}, 1 \right) . \end{aligned}$$

  Denote by $\sigma =\{1^*, 2^*, 3^*\} \cup \{{\tilde{2}}\}$ where ${\tilde{2}}$ is the first parent of (i, k). In this case, the cardinal of $\sigma$ is 3 or 4. Integrating first over $1^*$ and then using Hölder’s inequality as in (B.35), we have

  $$\begin{aligned}&\int \mathbf{1}_{ \{ {\bar{v}}_i -v_k \in \mathcal {R}_4 \} } \mathbf{1}_{ \{ v_i - v_j \in \mathcal {R}_1 \} } \, \prod _{m \in \sigma }b(\nu _m,v_m) \, dv_m d\nu _mdt_m \\&\quad \le R^3 {\varepsilon }|\log {\varepsilon }|^2 \int { \mathbf{1}_{ \{ {\bar{v}}_i- v_k \in \mathcal {R}_4 \} } \over |{\bar{v}}_i - v_j|} \, \prod _{m \in \sigma \setminus \{ 1^*\} }b(\nu _m,v_m) \, dv_m d\nu _mdt_m \\&\quad \le R^3 {\varepsilon }|\log {\varepsilon }|^2 \left( (R^3t)^2 \int \mathbf{1}_{ \{ {\bar{v}}_i -v_k \in \mathcal {R}_4 \}} b(\nu _{{\tilde{2}}},v_{{\tilde{2}}}) \, dv_{{\tilde{2}}} d\nu _{{\tilde{2}}} dt_{{\tilde{2}}} \right) ^{1/4} \\&\qquad \times \left( (R^3t) \int \frac{ 1 }{ | {\bar{v}}_{i}- v_{j}|^{4/3}} \prod _{m= 2^*,3^*} b(\nu _m,v_m) \, dv_m d\nu _mdt_m \right) ^{3/4}\\&\quad \le C(Rt)^r {\varepsilon }^{33/32} |\log {\varepsilon }|^{9/4}. \end{aligned}$$

Given a set $\sigma$ of parents, it may only determine the particle i, so that an extra factor $s^2$ has to be added in (B.31) to take into account the choice of j, k. This completes the proof of Lemma B.2. $\square$

B.4: Recollisions in Chain

The following Lemma was used in Section 6.2.2 to deal with the case when recollisions occur in chain, with $t_{{\tilde{1}}} = t_{1^*}$, i.e. both recollisions occur without any intermediate collisions as depicted in Figure 14.

Fig. 14

Recollisions in chain

Lemma B.3

Fix a final configuration of bounded energy $z_1 \in \mathbb {T}^2 \times B_R$ with $1 \le R^2 \le C_0 |\log {\varepsilon }|$, a time $1\le t \le C_0|\log {\varepsilon }|$ and a collision tree $a \in \mathcal {A}_s$ with $s \ge 2$.

There exist sets of bad parameters $\mathcal {P}_2 (a, p,\sigma )\subset \mathcal {T}_{2,s} \times {{\mathbb {S}}}^{s-1} \times \mathbb {R}^{2(s-1)}$ for $p_2 < p\le p_3$ and $\sigma \subset \{2,\dots , s\}$ of cardinal $|\sigma |\le 3$ such that

• $\mathcal {P}_2 (a, p,\sigma )$ is parametrized only in terms of $( t_m, v_m, \nu _m)$ for $m \in \sigma$ and $m < \min \sigma$;

  $$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2 (a,p,\sigma ) } \displaystyle \prod _{m\in \sigma } \, \big | \big (v_{m}-v_{a(m)} ( t_{{m}} ) )\cdot \nu _{{m}} \big | d t_{{m}} d \nu _{{m}}dv_{{m}} \le C(Rt)^r s^2 {\varepsilon }\,,\qquad \end{aligned}$$

  (B.37)

  for some constant $r \ge 1$,

• and any pseudo-trajectory starting from $z_1$ at t, with total energy bounded by $R^2$ and such that the first two recollisions occur in chain as in Figure 14 is parametrized by

  $$\begin{aligned} (t_n, \nu _n, v_n)_{2\le n\le s }\in \bigcup _{p_2 < p\le p_3} \bigcup _\sigma \mathcal {P}_2(a, p,\sigma )\,. \end{aligned}$$

Proof

Recall that the condition for the first recollision states

$$\begin{aligned} v_{i} - v_{j} = \frac{1}{\tau _{rec}} \delta x_\perp - \frac{\tau _1}{\tau _{rec}} ({\bar{v}}_{i} - v_{j})- \frac{1}{\tau _{rec}} \nu _{rec} \, , \end{aligned}$$

(B.38)

with $x_i, x_j$ the positions at time $t_{2^*}$

$$\begin{aligned} \begin{aligned} \delta x&:= \frac{1}{{\varepsilon }}(x_i-x_j-q)= \lambda ( {\bar{v}}_{i} - v_{j}) + \delta x_\perp \qquad \hbox {with} \quad \delta x_\perp \cdot ( {\bar{v}}_i - v_{j}) =0 \, ,\qquad \\ \tau _1&:= \frac{1}{{\varepsilon }}(t_{1^*} - t_{2^*} - \lambda ) \, ,\qquad \tau _{rec}: =\frac{1}{{\varepsilon }}(t_{rec} - t_{1^*}) \, , \end{aligned} \end{aligned}$$

(B.39)

for some q in $\mathbb {Z}^2$ of norm smaller than O(Rt) to take into account the periodicity.

When $|\tau _1| | {\bar{v}}_i-v_j| \le R^2$, estimate (B.5) is enough to obtain an upper bound of order ${\varepsilon }$ without taking into account the second recollision. Our goal here is to prove that the constraint of having a second recollision produces an integrable function of $|\tau _1| \, | {\bar{v}}_i-v_j| \ge R^2$, hence a bound $O({\varepsilon })$ after integration over $1^*$.

From (B.38), we deduce as in (3.10) that

$$\begin{aligned} \frac{1}{|\tau _{rec} |} \le {4R\over |\tau _1| | {\bar{v}}_{i} - v_{j}|} \quad \text {which implies that} \quad |\tau _{rec} | \ge R/4 \gg 1\, . \end{aligned}$$

(B.40)

Two cases have to be considered: $k= 1^*$ and $k\ne 1^*$.

$\underline{\text {Case} \, k=1^* .}$

The equation for the second recollision states

$$\begin{aligned} \tau '_{rec} (v'_{i} - {v_{1^*}'})=\pm \nu _{1^*} - \tau _{rec} ( v_i - {v_{1^*}'}) (+\nu _{rec})-{\tilde{\nu }}_{rec}, \end{aligned}$$

where

$$\begin{aligned} \tau '_{rec} :=\frac{1}{{\varepsilon }}({\tilde{t}}_{rec} - t_{rec}) \, , \end{aligned}$$

and where the ± and the translation by $\nu _{rec}$ depend on the possible exchanges in the labels of the particles at collision times. It can be rewritten, thanks to (B.10),

$$\begin{aligned} \begin{aligned} \tau '_{rec} (v_j - v_i )\cdot \nu _{rec} \, \nu _{rec}&=\pm \nu _{1^*} - (\tau _{rec}+ \tau '_{rec}) ( v_i - {v'_{1^*}}) (+\nu _{rec})-{\tilde{\nu }}_{rec} \\ \hbox { or } \qquad \tau '_{rec} (v_j - v_i )\cdot \nu _{rec}^\perp \, \nu _{rec}^\perp&=\pm \nu _{1^*} - (\tau _{rec}+ \tau '_{rec}) ( v_i - {v'_{1^*}}) (+\nu _{rec})-{\tilde{\nu }}_{rec}\,. \end{aligned}\nonumber \\ \end{aligned}$$

(B.41)

We further know that $|v_i - {v'_{1^*}}| = | {\bar{v}}_i - v_{1^*}|$.

• If $| {\bar{v}}_i - v_{1^*}| \ge R \, |\tau _{rec}|^{-3/4}$, then the vector in the right-hand side of (B.41) has a magnitude of order

  $$\begin{aligned} |\tau _{rec}+ \tau '_{rec}| \, | v_i - {v'_{1^*}}| \ge |\tau _{rec}| \, | v_i - {v'_{1^*}}| \ge R \, |\tau _{rec}|^{1/4} \, . \end{aligned}$$

It follows that the vector $\nu _{rec}$ has to be aligned in the direction of $v_i - v_{1^*}'$ with a controlled error

$$\begin{aligned} \nu _{rec} = \mathcal {R}_{n\pi /2} { v_i - {v_{1^*}'}\over | {\bar{v}}_i- v_{1^*}|} + O \left( {1\over | \tau _{rec}|^{1 /4}} \right) \,, \end{aligned}$$

for $n=0,1,2,3$, recalling that $\mathcal {R}_\theta$ is the rotation of angle $\theta$.

Plugging the formula for $\nu _{rec}$ into (B.38) and using (B.40), we get

$$\begin{aligned} v_{i} - v_{j}= & {} \frac{1}{\tau _{rec} } \delta x_\perp - \frac{\tau _1}{\tau _{rec}} ( {\bar{v}}_{i} - v_{j}) - \frac{1}{\tau _{rec}} \mathcal {R}_{n\pi /2}{ v_i - {v_{1^*}'}\over | {\bar{v}}_i- v_{1^*}|} \\&+ O \left( {R^{5 /4} \over |\tau _1|^{5 /4} \; |{\bar{v}}_{i} - v_{j}|^{5 /4}} \right) \,.\qquad \end{aligned}$$

This equation has the same structure as (B.13). Thus using the same arguments as in the proof of Lemma B.1, we get

• a contribution of size $O( |\tau _1|^{-5 /4} \; |{\bar{v}}_{i} - v_{j}|^{-5 /4}|\log | \tau _1 ({\bar{v}}_i - v_j)| )$ when the mapping $v_i \mapsto {v_i - v_{1^*}' \over |v_i - v_{1^*}'|}$ is Lipschitz with constant strictly less than $|w|^{\gamma }$ for some $\gamma \in (0,1)$;

• the same integrable contribution as in Lemma B.1 in degenerate cases when some velocities are close to each other (typically at a distance $O(|w|^{-\gamma })$).

Thus integrating with respect to $t_{1^*}$ we recover the factor ${\varepsilon }$ and the singularity in $|{\bar{v}}_{i} - v_{j}|$ is removed as usual by integration over the parents of i, j.

• If $| {\bar{v}}_i - v_{1^*}| \le R|\tau _{rec} |^{-3/4}$, we find that $v_{1^*}$ has to belong to a domain of size less than $( |\tau _1| \, | {\bar{v}}_i-v_j| )^{-3/2}$ as $|\tau _{rec}| \ge |\tau _1| \, | {\bar{v}}_i-v_j|$. Hence again, we obtain an integrable function of $|\tau _1| \, | {\bar{v}}_i-v_j|$, with no extra gain in ${\varepsilon }$.

$\underline{\text {Case} \, k\ne 1^*.}$ In the following we denote by $1^*,2^*\dots$ the parents of the set $(i,j,k,\ell )$ at time $t_{rec}$: contrary to previous cases, and since they both have the same first parent we do not distinguish the parents of (i, j) and $(k,\ell )$ but consider them as a whole.

The position of particle k at the time ${\tilde{t}}_{rec}$ of the second recollision is given by

$$\begin{aligned} x_k (t_{rec}) = x_k + v_k ({\tilde{t}}_{rec} -t_{2^*}) \, . \end{aligned}$$

We have written $x_k$ for the position of particle k at time $t_{2^*}$. We end up with the condition for the second recollision

$$\begin{aligned} ({\tilde{t}}_{rec} - t_{rec}) (v'_{i} - v_{k}) = (x_j-x_k )(t_{rec}) -{\varepsilon }{\tilde{\nu }}_{rec}(+{\varepsilon }\nu _{rec})+ {\tilde{q}} \, , \end{aligned}$$

(B.42)

for some ${\tilde{q}} \in \mathbb {Z}^2$ not larger than O(Rt), and where the translation ${\varepsilon }\nu _{rec}$ arises only if the labels of particles are exchanged at $t_{rec}$. In the following, we fix q and ${\tilde{q}}$ and will multiply the final estimate by $(R^2 t^2)^2$ to take into account the periodicity in both recollisions. Using the notation (B.39), we then rescale in ${\varepsilon }$ and write

$$\begin{aligned} \tau _{rec}:=\frac{t_{rec} - t_{1^*}}{{\varepsilon }}\, , \quad \tau '_{rec}:=\frac{ {\tilde{t}}_{rec} - t_{rec}}{{\varepsilon }}. \end{aligned}$$

Then Eq. (B.42) for the second recollision becomes

$$\begin{aligned} \tau '_{rec}(v'_i - v_k) = {\tilde{x}}_{jk} (t_{rec}) - {\tilde{\nu }}_{rec} ( + \nu _{rec}) \, , \end{aligned}$$

(B.43)

where ${\varepsilon }{\tilde{x}}_{jk} (t_{rec})$ stands for the relative position between j, k at time $t_{rec}$.

As in the proof of Lemma B.1, the Eq. (B.43) implies that $v'_i - v_k$ belongs to a rectangle $\mathcal {R}$ of size $2R \times \frac{2R}{|{\tilde{x}}_{jk} (t_{rec})|}$ and axis ${\tilde{x}}_{jk} (t_{rec})$. Furthermore $v'_i$ belongs as well to the circle of diameter $[v_j, v_i]$ by definition. Computing the intersection of the rectangle and of the circle, we obtain a constraint on the angle $\nu _{rec}$. Then plugging this constraint in the equation for the first recollision, we will conclude as in Lemma B.1 that $v_i$ has to belong to a very small set (Figure 15). $\square$

Fig. 15

The velocity $v_i'$ belongs to the rectangle of axis ${\tilde{x}}_{jk} (t_{rec})$ as well as to the circle of diameter $[v_j, v_i]$

This strategy can be applied in most situations. We have however to deal separately with the two following geometries :

• if the relative velocity $v_i- v_j$ is small, the rectangle can contain a macroscopic part of the circle : we forget about the second recollision and just study the constraint of small relative velocities;

• if the distance $| {\tilde{x}}_{jk} (t_{rec}) |$ is small, then i will be close to k at the first recollision time and this will facilitate the second recollision : we then forget about the second recollision and write two independent constraints at the first recollision time.

• Suppose that

  $$\begin{aligned} | v_i- v_j | \ge {1 \over |\tau _{rec}|^{5 /8}} \quad \text {and} \quad | {\tilde{x}}_{jk} (t_{rec}) | \ge | \tau _{rec} |^{ 3 / 4} . \end{aligned}$$

  (B.44)

From (B.43), we deduce that a necessary condition for the second recollision to hold is that

$$\begin{aligned} (v_i' - v_k) \cdot {{\tilde{x}}_{jk}^\perp (t_{rec})\over | {\tilde{x}}_{jk} (t_{rec})|} = {(v_i - \big ( (v_i-v_j) \cdot \nu _{rec} \big ) \, \nu _{rec} - v_k ) \cdot {\tilde{x}}_{jk}^\perp (t_{rec})\over | {\tilde{x}}_{jk} (t_{rec})|} = O\Big ( {1\over |\tau '_{rec}|}\Big ) , \end{aligned}$$

where $|\tau '_{rec}|$ can be bounded from below thanks to (B.44)

$$\begin{aligned} |\tau '_{rec}| \ge {| {\tilde{x}}_{jk} (t_{rec})|\over 4R} \ge {|\tau _{rec}|^{3/4} \over 4R}\,. \end{aligned}$$

Using the bound from below on the relative velocity $|v_i - v_j|$, we finally get

$$\begin{aligned} \Big ( {v_i-v_j\over |v_i - v_j|} \cdot \nu _{rec} \Big ) \Big ( {{\tilde{x}}_{jk}^\perp (t_{rec})\over |{\tilde{x}}_{jk} (t_{rec})|} \cdot \nu _{rec} \Big ) ={(v_i-v_k) \cdot {\tilde{x}}_{jk}^\perp (t_{rec}) \over |v_i-v_j| |{\tilde{x}}_{jk}^\perp (t_{rec})|} + O\Big ( {1\over |\tau _{rec}|^{1/8} }\Big )\,. \end{aligned}$$

Define the angles $\theta = < {\tilde{x}}_{jk}^\perp (t_{rec}), \nu _{rec} >$ and $\alpha =< {\tilde{x}}_{jk}^\perp (t_{rec}), v_i-v_j>$. We have

$$\begin{aligned} \cos \theta \cos (\theta - \alpha )= & {} \frac{1}{2} \Big ( \cos (2\theta - \alpha ) +\cos \alpha \Big ) \\= & {} {(v_i-v_k) \cdot {\tilde{x}}_{jk}^\perp (t_{rec}) \over |v_i-v_j| |{\tilde{x}}_{jk}^\perp (t_{rec})|}+ O\Big ( {1\over |\tau _{rec}|^{1/8} }\Big ), \end{aligned}$$

so that

$$\begin{aligned} \cos (2\theta - \alpha ) = {(v_i + v_j - 2v_k) \cdot {\tilde{x}}_{jk}^\perp (t_{rec}) \over |v_i-v_j| |{\tilde{x}}_{jk}^\perp (t_{rec})|}+ O\Big ( {1\over |\tau _{rec}|^{1/8} }\Big ). \end{aligned}$$

Recall the notation of the proof of Lemma B.1

$$\begin{aligned} w:= \delta x^\perp _{i,j} - ({\bar{v}}_{i} - v_{j}) \tau _1, \quad \text { and} \quad u:={|w| \over \tau _{rec}} \le 4R, \end{aligned}$$

where ${\varepsilon }w$ is the distance between $x_i,x_j$ at time $t_{1^*}$ and it is enough to consider $|w| \ge R^2$ thanks to (B.5). As the derivative of $\arccos$ is singular at $\pm 1$, we will consider an approximation $\arccos _{|w|}$ which coincides with $\arccos$ on $[-1 + \frac{1}{|w|^{2 \delta }} ,1- \frac{1}{|w|^{2 \delta }} ]$ (for a given $\delta \in (0,\frac{1}{16})$) and is constant in the rest of $[-1,1]$ so that

$$\begin{aligned} \big | \partial _x \arccos _{|w|} (x) \big | \le |w|^\delta \quad \text {and} \quad \Vert \arccos _{|w|} - \arccos \Vert _\infty \le \frac{1}{|w|^\delta } . \end{aligned}$$

Thus the angle $\theta$ can be approximated by

$$\begin{aligned} \theta = {\bar{\theta }} _\pm + O\left( \left( \frac{u}{|w|} \right) ^{1/8}|w|^\delta + \frac{1}{|w|^\delta } \right) , \end{aligned}$$

with

$$\begin{aligned} {\bar{\theta }} _\pm = \pm \frac{1}{2} \arccos _{|w|} \left( {(v_i + v_j - 2v_k) \cdot {\tilde{x}}_{jk}^\perp (t_{rec}) \over |v_i-v_j| |{\tilde{x}}_{jk}^\perp (t_{rec})|}\right) + \frac{1}{2} < {\tilde{x}}_{jk}^\perp (t_{rec}), v_i-v_j> .\nonumber \\ \end{aligned}$$

(B.45)

Plugging this constraint in the equation for the first recollision, we get

$$\begin{aligned} v_{i} -v_j = u {w\over |w|} - {u\over |w|} \mathcal {R}_{{\bar{\theta }} _\pm } {{\tilde{x}}_{jk}^\perp (t_{rec}) \over |{\tilde{x}}_{jk} (t_{rec})|} + O \Big ( u^\delta \left( \frac{u}{|w|} \right) ^{9/8-\delta } + \frac{u}{|w|} \, \frac{1}{|w|^\delta } \Big ) .\qquad \end{aligned}$$

(B.46)

As $|w| \gg 1$, the leading term of this equation is $v_{i} -v_j \simeq u {w\over |w|}$, but we have to analyse carefully the corrections. Compared with the formulas of the same type encountered in the proof of Lemma B.1, this one has the additional difficulty that the dependence with respect to u is very intricate. Instead of solving (B.46), we are going to look at sufficient conditions satisfied by the solutions of (B.46). In particular, u will be considered as a parameter independent of $|\tau _{rec}|$. For a given u, we are going to solve the equation

$$\begin{aligned} v_{i} -v_j = u {w\over |w|} - {u\over |w|} \mathcal {R}_{{\bar{\theta }} _\pm } {{\tilde{x}}_{jk}^\perp (t_{rec}) \over |{\tilde{x}}_{jk} (t_{rec})|} \quad \text {with} \quad |v_{i} -v_j| \ge \frac{1}{2} \left( {u \over |w|} \right) ^{5/8},\quad \end{aligned}$$

(B.47)

where ${\tilde{x}}_{jk} (t_{rec})$ was originally defined in (B.43) as the relative position between j, k at time $t_{rec}$, but is now simply a function of u

$$\begin{aligned} {\tilde{x}}_{jk} (t_{rec}) = x_{jk} (t_{1^*} ) - {|w| \over u} (v_j-v_k). \end{aligned}$$

The solutions of (B.46) are such that $u \simeq |v_{i} -v_j|$, thus from the condition (B.47) on the relative velocities, it is enough to restrict the range of the parameter u to $u \ge \frac{1}{4} \left( {u \over |w|} \right) ^{5/8}$, i.e. $u \in [\frac{1}{4 |w|^{5/3}}, 4R]$. We will first show that for any such u, there is a unique solution ${\hat{v}}_i (u)$ of (B.47). The solution of (B.46) will be located close to the curve $u \rightarrow {\hat{v}}_i(u)$, thus we will then need to control the regularity of the curve $u \rightarrow {\hat{v}}_i(u)$ to estimate the size of the tubular neighborhood around this curve.

• For fixed u, note that ${\tilde{x}}_{jk}^\perp (t_{rec})$ is also fixed. The only dependence with respect to $v_i$ in the right-hand side of (B.47) is via ${\bar{\theta }}_\pm$:

  $$\begin{aligned} d{\bar{\theta }}_\pm \le&\frac{1}{2} |w|^\delta \, { |v_i-v_j|^2 {\tilde{x}}_{jk}^\perp (t_{rec}) \cdot dv_i - ((v_i+v_j - 2v_k) \cdot {\tilde{x}}_{jk}^\perp (t_{rec})) \;(v_i-v_j)\cdot dv_i \over |v_i-v_j|^3 | {\tilde{x}}_{jk} (t_{rec})|} \nonumber \\&- \frac{1}{2} d< {\tilde{x}}_{jk}^\perp (t_{rec}), v_i-v_j>. \end{aligned}$$

  (B.48)

where we used the Lipschitz bound satisfied by $\arccos _{|w|}$. Note that second term in (B.48) controls the variation of the angle $< {\tilde{x}}_{jk}^\perp (t_{rec}), v_i-v_j>$ and has Lipschitz constant less than $\frac{1}{|v_i-v_j|}$. Together with the bounds (B.44), this implies that $v_i \mapsto {\bar{\theta }}_\pm (v_i,u)$ is Lipschitz continuous with constant

$$\begin{aligned} C |w|^{ \delta } \; \max \frac{1}{|v_i-v_j|} \le C\left( {|w| \over u} \right) ^{5/8 + \delta } u^\delta \ll {|w| \over u} . \end{aligned}$$

We therefore conclude by Picard’s fixed point theorem that there is a unique solution ${\hat{v}}_i = {\hat{v}}_i (u)$. As $\delta < 1/16$, we further have that any solution to (B.46) satisfies

$$\begin{aligned} v_i = {\hat{v}}_i (u) + O \Big ( u^\delta \left( \frac{u}{|w|} \right) ^{9/8-\delta } + \frac{u}{|w|} \, \frac{1}{|w|^\delta } \Big ), \end{aligned}$$

for $u \in [\frac{1}{4 |w|^{5/3}}, 4R]$.

• Let us now study the regularity of $u \mapsto {\hat{v}}_i (u)$. In (B.47), we have both an explicit dependence with respect to u and a dependence via the direction of ${\tilde{x}}_{jk} (t_{rec})$. To take into account the condition (B.44), we further restrict the range of u to

  $$\begin{aligned} u \in \left[ \frac{1}{4 |w|^{5/3}}, 4R\right] \quad \text {and} \quad | {\tilde{x}}_{jk} (t_{rec}) | \ge \left( {|w| \over u} \right) ^{ 3 / 4} . \end{aligned}$$

  (B.49)

The derivative of ${ {\tilde{x}}_{jk} (t_{rec}) \over | {\tilde{x}}_{jk} (t_{rec})|}$ with respect to u is controlled by

$$\begin{aligned} {|w| |v_j-v_k| \over u^2| {\tilde{x}}_{jk} (t_{rec})|} \le C(R) \left( {|w| \over u} \right) ^{1+5/8-3/4} = C(R)\left( {|w| \over u} \right) ^{7/8}, \end{aligned}$$

(B.50)

as $u \ge {1 \over 2} \left( u \over {|w|} \right) ^{5/8}$ thanks to (B.49). Thus the Lipschitz constant of $u\mapsto {\bar{\theta }}_\pm (v_i,u)$ is less than $\big ( |w| / u \big )^{7/8} \, |w|^\delta$. Gathering both estimates, we finally get by differentiating (B.47) with respect to $v_i$ and u that $u\mapsto {\hat{v}}_i(u)$ is Lipschitz continuous with constant $1+ C \big ( |w| / u \big )^{-1/8 + \delta } u^\delta$, which is bounded as $\delta <1/16$. The solutions of (B.46) are at a distance at most ${R \over |w|^{1+\delta } }$ from the curve $u\mapsto {\hat{v}}_i(u)$. Thus under the condition (B.44), any recollision in chain will belong to the tubular neighborhood of $u\mapsto {\hat{v}}_i(u)$.

Fig. 16

In the case $| {\tilde{x}}_{j,k} (t_{rec}) | \le | \tau _{rec}| ^{ 3 / 4}$, we will forget about the recollision between i, k and use instead that j and k are close at time $t_{rec}$

In order to estimate the measure that $v_i$ belongs to this tubular neighborhood, we proceed as in (C.10) and cover this tube by $O( |w|^{1+\delta } )$ balls of size ${R \over |w|^{1+\delta } }$. Integrating with respect to $dv_{1^*} d\nu _{1^*}$, we get an estimate $O \left( \frac{|\log |w||}{|w|^{1+\delta } } \right)$. By construction $|w| \ge |\tau _1| |{\bar{v}}_i - v_j|$ so that the remainder can be integrated with respect to $\tau _1$. Changing to the variable $t_{1^*}$, we obtain an upper bound of order ${\varepsilon }$. We then kill the singularity $\frac{\big | \log |{\bar{v}}_i - v_j| \big | }{ |{\bar{v}}_i - v_j|^{1+\delta }}$ at small relative velocities by integrating with respect to two additional parents, applying (C.6) and then (C.7).

• Suppose that $| v_i- v_j | \le |\tau _{rec}|^{-5 /8}$. We obtain by (C.3) that

  $$\begin{aligned}&\int \mathbf{1}_{ \{ | v_i- v_j | \le | \tau _{rec}|^{-5 /8} \}} |(v_{1^*} - v_i)\cdot \nu _{1^*}| d\nu _{1^*} dv_{1^*} \nonumber \\&\quad \le \int \mathbf{1}_{\{ | v_i- v_j | \le \frac{1}{|\tau _1|^{5 /8} \, |{\bar{v}}_i - v_j|^{5 /8}} \}} |(v_{1^*} - v_i)\cdot \nu _{1^*}| d\nu _{1^*} dv_{1^*} \nonumber \\&\quad \le \frac{C R^2}{|\tau _1|^{5 /8} \, |{\bar{v}}_i - v_j|^{5 /8} } \min \left( \frac{1}{|\tau _1|^{5 /8} \, |{\bar{v}}_i - v_j|^{13 /8}}, 1\right) \le \frac{C R^2}{|\tau _1|^{9 /8} \, |{\bar{v}}_i - v_j|^{77 /40}},\nonumber \\ \end{aligned}$$

  (B.51)

where in the last inequality, we used that $\min (\delta , 1) \le \delta ^{4/5}$. This produces an integrable function of $|\tau _1|$ and leads to an upper bound of order ${\varepsilon }$. The singularity in $|{\bar{v}}_i-v_j|$ can be integrated out by applying (C.6) and then (C.7) on the parents of i, j.

• Suppose that $| {\tilde{x}}_{j,k} (t_{rec}) | \le | \tau _{rec}| ^{ 3 / 4}$, this condition can be interpreted as a “kind of recollision” between j and k at time $t_{rec}$. Note that this situation is similar to the last case studied in Lemma B.1, where the size of the error depends on $|\tau _1| |{\bar{v}}_i - v_j|$ (Figure 16).

The first recollision between i, j imposes that $v_i - v_j$ belongs to a rectangle $\mathcal {R}$. Integrating first the condition for the recollision between (i, j) with respect to $b(\nu _{1 ^*},v_{1 ^*}) \, dv _{1^*} d\nu _{1 ^*}$, we gain a factor $(\tau _1 |{\bar{v}}_i - v_j|)^{-1}$. We will not use the recollision between i, k and focus on the additional constraint that the distance between j, k is less than ${\varepsilon }|\tau _{rec}| ^{ 3 / 4}$ at time $t_{rec}$.

Denote by ${\tilde{1}}$ the first parent of (j, k). By analogy with equation (B.1), the constraint $| {\tilde{x}}_{j,k} (t_{rec}) | \le | \tau _{rec}| ^{ 3 / 4}$ reads

$$\begin{aligned} (x_j - x_k) (t_{{\tilde{1}}} )+ ( t_{rec} - t_{{\tilde{1}}}) ( v_j - v_k) = {\varepsilon }\eta + q, \end{aligned}$$

with $|\eta | \le |\tau _{rec}|^{ 3 / 4}$ and a given $q \in \mathbb {Z}^2$ with modulus less than Rt. With the notation ${\tilde{\tau }}_{rec} = \frac{t_{rec} - t_{{\tilde{1}}}}{{\varepsilon }}$, this can be rewritten

$$\begin{aligned} v_j - v_k = \frac{(x_j - x_k) (t_{{\tilde{1}}} + q)}{ {\varepsilon }{\tilde{\tau }}_{rec}} + \frac{ \eta }{{\tilde{\tau }}_{rec}} . \end{aligned}$$

(B.52)

Since ${\tilde{\tau }}_{rec} \ge \tau _{rec}$, we get

$$\begin{aligned} \frac{ \eta }{|{\tilde{\tau }}_{rec}|} \le \frac{ 1}{ |\tau _{rec}|^{1/4}} \le \frac{1}{( |\tau _1| |{\bar{v}}_i - v_j|)^{1/4}} , \end{aligned}$$

so that $v_j - v_k$ has to belong to a rectangle ${\tilde{\mathcal {R}}}$ of width less than $\frac{1}{( |\tau _1| |{\bar{v}}_i - v_j|)^{1/4}}$.

As in the last case of Lemma B.1, we split the proof according to the size of $|\tau _1|$.

• If $|\tau _1| \ge {1 \over |{\bar{v}}_i - v_j|^6}$, we deduce that

  $$\begin{aligned}\frac{1}{|\tau _1| |{\bar{v}}_i - v_j|} \le \frac{1}{|\tau _1|^{5/6}}. \end{aligned}$$

Then, we compute the cost of satisfying the previous constraints

$$\begin{aligned}&\int \mathbf{1}_{\{ v_i - v_j \in \mathcal {R}\}} \mathbf{1}_{\{ v_j -v_k \in {\tilde{\mathcal {R}}} \}} \prod _{\ell =1^*,{\tilde{1}}} b(\nu _\ell ,v_\ell ) \, dv_\ell d\nu _\ell d t_\ell \\&\quad \le \int {\mathbf{1}_{\{ v_j -v_k \in {\tilde{\mathcal {R}}} \}} \over |\tau _1| |{\bar{v}}_i - v_j| } b(\nu _{{\tilde{1}}},v_{{\tilde{1}} }) \, dt_{1^*} \, dv_{{\tilde{1}}} d\nu _{{\tilde{1}}} d t_{{\tilde{1}}} \\&\quad \le {\varepsilon }\int {\mathbf{1}_{\{ v_j -v_k \in {\tilde{\mathcal {R}}} \}} \over |\tau _1| |{\bar{v}}_i - v_j| } b(\nu _{{\tilde{1}}},v_{{\tilde{1}} }) \, d \tau _1 \, dv_{{\tilde{1}}} d\nu _{{\tilde{1}}} d t_{{\tilde{1}}}. \end{aligned}$$

As in the case of (B.30), the change of variable from $t_{1^*}$ to $\tau _1$ leads to a factor ${\varepsilon }$ and decouples the dependence between the variable $t_{1^*}$ and $v_{{\tilde{1}}}$ by keeping only the constraint $|\tau _1| \ge R$. We can then complete the upper bound as usual

$$\begin{aligned}&\int \mathbf{1}_{\{ v_i - v_j \in \mathcal {R}\}} \mathbf{1}_{\{ v_j -v_k \in {\tilde{\mathcal {R}}} \}} \prod _{\ell =1^*,{\tilde{1}}} b(\nu _\ell ,v_\ell ) \, dv_\ell d\nu _\ell d t_\ell \\&\quad \le {\varepsilon }\int {\mathbf{1}_{\{ v_j -v_k \in {\tilde{\mathcal {R}}} \}} \over |\tau _1|^{5/6} } b(\nu _{{\tilde{1}}},v_{{\tilde{1}} }) \, d \tau _1 dv_{{\tilde{1}}} d\nu _{{\tilde{1}}} d t_{{\tilde{1}}} \\&\quad \le {\varepsilon }C(R) \int {\log |\tau _1| \over |\tau _1|^{25/24} } d \tau _1, \end{aligned}$$

where the singularity is integrable in $|\tau _1| \in [ R, + \infty ]$.

• If $|\tau _1| \le {1 \over |{\bar{v}}_i - v_j|^6}$, we forget about (B.52). We indeed have that

  $$\begin{aligned} \int {1\over |\tau _1| \, |{\bar{v}}_i - v_j| } d\tau _1 \le {1\over |{\bar{v}}_i - v_j| } \int \frac{1}{|\tau _1|} d\tau _1 \le {C |\log |{\bar{v}}_i - v_j||\over |{\bar{v}}_i - v_j| }\,. \end{aligned}$$

The singularity at small relative velocities is controlled with two additional integration.

Given a set $\sigma$ of parents, it may only determine the particle i, so that an extra factor $s^2$ has to be added in (B.37) to take into account the choice of j, k. This concludes the proof of Lemma B.3. $\square$

Fig. 17

On the left, two recollisions in chain due to periodicity. On the right, the symmetry argument (B.56)

B.5: Two Particles Recollide Twice in Chain Due to Periodicity

We have seen in Proposition 3.5 that a self-recollision between two particles created at the same collision has a cost ${\varepsilon }$. It may happen also that two particles have a recollision and then a second self-recollision due to periodicity (see Figure 17). This is a very constrained case which is treated in the following Lemma.

Lemma B.4

Fix a final configuration of bounded energy $z_1 \in \mathbb {T}^2 \times B_R$ with $1 \le R^2 \le C_0 |\log {\varepsilon }|$, a time $1\le t \le C_0|\log {\varepsilon }|$ and a collision tree $a \in \mathcal {A}_s$ with $s \ge 2$.

There exists a set of bad parameters $\mathcal {P}_2 (a, p_4,\sigma )\subset \mathcal {T}_{2,s} \times {{\mathbb {S}}}^{s-1} \times \mathbb {R}^{2(s-1)}$ and $\sigma \subset \{2,\dots , s\}$ of cardinal $|\sigma | \le 3$ such that

• $\mathcal {P}_2 (a, p_4 ,\sigma )$ is parametrized only in terms of $( t_m, v_m, \nu _m)$ for $m \in \sigma$ and $m < \min \sigma$;

  $$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2 (a,p_4 ,\sigma ) } \displaystyle \prod _{m\in \sigma } \, \big | \big (v_{m}-v_{a(m)} ( t_{{m}} ) )\cdot \nu _{{m}} \big | d t_{{m}} d \nu _{{m}}dv_{{m}} \le C(Rt)^r s {\varepsilon }\,,\qquad \end{aligned}$$

  (B.53)

  for some constant r,

• and any pseudo-trajectory starting from $z_1$ at t, with total energy bounded by $R^2$, and such that the first two recollisions involve the same two particles which recollide twice in chain is parametrized by

  $$\begin{aligned} (t_n, \nu _n, v_n)_{2\le n\le s }\in \bigcup _\sigma \mathcal {P}_2(a, p_4 ,\sigma )\,. \end{aligned}$$

Proof

We recall the equation (3.9) on the first recollision

$$\begin{aligned} v_i-v_j = \frac{1}{\tau _{rec}} (\delta x_\perp - \tau _1 ({\bar{v}}_i-v_j) - \nu _{rec}) \quad \text {with} \quad \frac{1}{|\tau _{rec} |} \le {4R\over |\tau _1| | {\bar{v}}_{i} - v_{j}|}\, .\nonumber \\ \end{aligned}$$

(B.54)

The equation on the second recollision is

$$\begin{aligned} (v'_i - v'_j) ({\tilde{t}}_{rec}- t_{rec}) = {\varepsilon }{\tilde{\nu }}_{rec} + {\varepsilon }\nu _{rec} + {\tilde{q}} \end{aligned}$$

(B.55)

for some ${\tilde{t}}_{rec}\ge 0$, ${\tilde{\nu }}_{rec} \in {{\mathbb {S}}}$, and ${\tilde{q}} \in \mathbb {Z}^2 \setminus \{ 0 \}$. Note that ${\tilde{q}} \not = 0$ as the second recollision occurs from the periodicity. As usual we fix ${\tilde{q}}$ and multiply the estimates in the end by $O(R^2t^2)$ to take that into account.

The condition (B.55) implies that the vector $v'_i - v'_j$ is located in a cone of axis ${\tilde{q}}$ and angular sector $2{\varepsilon }$. By definition, we have

$$\begin{aligned} v_i'- v_j'=( v_i-v_j ) - 2 (v_i- v_j)\cdot \nu _{rec} \; \nu _{rec}, \end{aligned}$$

(B.56)

which means that $\nu _{rec}^\perp$ is the bisector of $v_i- v_j$ and $v_i'- v_j'$ (see Figure 17).

From (B.54), we deduce that the direction of $v_i- v_j$ is

$$\begin{aligned} {\delta x_\perp - \tau _1 ({\bar{v}}_i-v_j) \over |\delta x_\perp - \tau _1 ({\bar{v}}_i-v_j)|} + O\left( {1\over |\tau _1 ({\bar{v}}_i-v_j)|}\right) \,. \end{aligned}$$

From (B.55), we deduce that the direction of $v'_i-v'_j$ is

$$\begin{aligned} {{\tilde{q}}\over |{\tilde{q}}|} + O( {\varepsilon })\,. \end{aligned}$$

Finally we get that $\nu _{rec}^\perp$ is known up to an error term which can be bounded by

$$\begin{aligned} \eta = {\varepsilon }+ {1\over \sqrt{ |\tau _1 ( {\bar{v}}_i-v_j)|}}. \end{aligned}$$

Note that we have introduce the square root as in the proof of Lemma B.3 for integrability purposes of the singularity $|{\bar{v}}_i-v_j|$.

Plugging this constraint on $\nu _{rec}$ in (B.54), we get that $v'_i- v_j$ has to belong, for each given $q, {\tilde{q}}$, to a rectangle $\mathcal {R}$ of axis $\delta x_\perp - \tau _1 ({\bar{v}}_i-v_j)$ and size $R \times R \frac{\eta }{|\tau _1( {\bar{v}}_i-v_j)|}$. By Lemma C.4, we obtain

$$\begin{aligned} \int \mathbf{1}_{ \{ v_i - v_j \in \mathcal {R}\} } \; \big | \big ( v_1^*-v_j ) \cdot \nu _1^*\big ) \big | dv_1^* d\nu _1^* \le CR^3 {{\varepsilon }|\log {\varepsilon }| \over \tau _1| {\bar{v}}_i-v_j|} + { CR^3 \over \tau _1^{3/2} | {\bar{v}}_i-v_j|^{3/2}}. \end{aligned}$$

Taking the union of the previous rectangles for the different choices of $q,{\tilde{q}}$, we define the set $\mathcal {P}_2(a, p_4 ,\sigma )$ associated with the scenario of two particles recolliding twice in chain due to periodicity. By integration with respect to time, we then get

$$\begin{aligned} \int \mathbf{1}_{\mathcal {P}_2(a, p_4 ,\sigma )} \; \big | \big ( v_1^*-v_j ) \cdot \nu _1^*\big ) \big |dv_1^* d\nu _1^*dt_1^* \le CR^3 {{\varepsilon }^2 |\log {\varepsilon }| ^2\over |{\bar{v}}_i-v_j|} + CR^3 { {\varepsilon }\over |{\bar{v}}_i-v_j|^{3/2}}. \end{aligned}$$

We then apply twice Lemma C.2 on two parents of i, j to integrate the singularities at small relative velocities.

Given $\sigma$, there are at most s choices for the pair (i, j) as $\sigma$ determines at least one of the labels. Thus the previous scenario leads to the set $\mathcal {P}_2(a, p_4 ,\sigma )$ with measure controlled by (B.53). $\square$

Appendix C: Carleman’s Parametrization and Scattering Estimates

In Sections 3, 6 and Appendix B, we were faced with integrals containing singularities in relative velocities $v_i-v_j$ and with a multiplicative factor of the type $(v^* - {\bar{v}}_{i}) \cdot \nu ^*$ where $v_i$ is recovered from $v^*$, $\nu ^*$ and ${\bar{v}}_{i}$ through a scattering condition. This appendix is devoted to the proof of “tool-box” lemmas for computing these singular integrals. These lemmas are used many times in this paper.

Lemma C.1

Fix a velocity ${\bar{v}}_i$ and let $v_i, v_j$ be the velocities after a collision (with or without scattering)

$$\begin{aligned} (v_i, v_j) = ({\bar{v}}_i, v^*) \quad \text {or} \quad {\left\{ \begin{array}{ll} v_i = {\bar{v}}_{i} + (v^*- {\bar{v}}_{i}) \cdot \nu ^* \nu ^*, \\ v_j = v^*- (v^* - {\bar{v}}_{i}) \cdot \nu ^* \nu ^*, \end{array}\right. } \end{aligned}$$

Fig. 18

Scattering relations

with $\nu ^* \in {{\mathbb {S}}}$ and $v^* \in \mathbb {R}^2$ (see Figure 18). Assume all the velocities are bounded by R then

$$\begin{aligned} \int { 1\over |v_i - v_j| } \, \big | \big ( v ^* - {\bar{v}}_i)\cdot \nu ^* \big |\, dv^* d\nu ^* \le CR^2 . \end{aligned}$$

(C.1)

Proof

In both cases, the velocities before and after the collision are related by $|v_i - v_j| = |v ^* - {\bar{v}}_i|$. Inequality (C.1) follows from the fact that the singularity $1/ |v ^* - {\bar{v}}_i|$ is integrable. $\square$

Lemma C.2

Fix ${\bar{v}}_i$ and $v_j$, and define $v_i$ to be one of the following velocities

$$\begin{aligned} v_i = v^*- (v^* - {\bar{v}}_{i}) \cdot \nu ^* \nu ^*,\nonumber \\ \text{ or } \quad v_i = {\bar{v}}_{i} + (v^*- {\bar{v}}_{i}) \cdot \nu ^* \nu ^*\,, \end{aligned}$$

(C.2)

with $\nu ^* \in {{\mathbb {S}}}$ and $v^* \in B_R \subset \mathbb {R}^2$ (see Figure 18). Assume all the velocities are bounded by $R>1$ and fix $\delta \in ]0,1[$. Then the following estimates hold, denoting $b(\nu ^*,v ^*):= |(v ^* - {\bar{v}}_i) \cdot \nu ^*|$:

$$\begin{aligned}&\int \mathbf{1}_{|v_i -v_j|\le \delta } \,b(\nu ^*,v ^*) \, dv ^* d\nu ^* \le C R^2 \delta \min \left( {\delta \over |v_j-{\bar{v}}_i|}, 1 \right) \, , \end{aligned}$$

(C.3)

$$\begin{aligned}&\int \min \left( { \delta \over |v_i - v_j| }, 1 \right) b(\nu ^* ,v ^*) \, dv^* d\nu ^* \le CR^2\delta |\log \delta | { + C R^3 \delta } \, , \end{aligned}$$

(C.4)

$$\begin{aligned}&\int { 1\over |v_i - v_j| } \, b(\nu ^*,v ^*) \, dv^* d\nu ^* \le CR^2 \Big ( \big |\log |{\bar{v}}_{i}- v_{j}|\big | +R\Big ) \, , \end{aligned}$$

(C.5)

$$\begin{aligned}&\int { 1\over |v_i - v_j|^\gamma } \, b(\nu ^*,v ^*) \, dv^* d\nu ^* \le {CR^2\over |{\bar{v}}_{i}- v_{j}|^{\gamma -1}} +CR^3 \quad \text{ for } \, \gamma \in ]1,2[ \,,\qquad \end{aligned}$$

(C.6)

$$\begin{aligned}&\int { 1\over |v_i - v_j|^\gamma } \, b(\nu ^*,v ^*) \, dv^* d\nu ^* \le CR^3 \quad \text{ for } \, \gamma \in ]0,1[ \,, \end{aligned}$$

(C.7)

$$\begin{aligned}&\int \big |\log |v_i - v_j|\big | \, b(\nu ^*,v ^*) \, dv^* d\nu ^* \le CR^3 \,. \end{aligned}$$

(C.8)

Proof

We start by recalling Carleman’s parametrization, which we shall be using many times in this Appendix: it is defined by

$$\begin{aligned} (v^*,\nu ^* )\in \mathbb {R}^2 \times {{\mathbb {S}}} \mapsto {\left\{ \begin{array}{ll} V' _* := v ^*- (v ^* - {\bar{v}}_{i}) \cdot \nu ^* \nu ^*\\ V' := {\bar{v}}_{i} + (v ^*- {\bar{v}}_{i}) \cdot \nu ^* \nu ^* \end{array}\right. } \end{aligned}$$

(C.9)

where $(V',V'_*)$ belong to the set ${\mathcal C}$ defined by

$$\begin{aligned} {\mathcal C}:=\Big \{(V',V'_*)\in \mathbb {R}^2\times \mathbb {R}^2 \,/\, (V'-{\bar{v}}_i)\cdot (V'_*-{\bar{v}}_i) =0\Big \} \, . \end{aligned}$$

This map sends the measure $b(\nu ^*,v ^*) \, dv^*d\nu ^*$ on the measure $dV'dS(V'_*)$, where dS is the Lebesgue measure on the line orthogonal to $(V'-{\bar{v}}_i)$ passing through ${\bar{v}}_i$.

Now let us consider the case when $|v_i-v_j| \le \delta$ and prove (C.3). What we need here is to estimate the measure of the pre-image of the small ball of center $v_j$ and radius $\delta$ by the scattering operator: let us study how for fixed $v_j$, the set $\{|v_i-v_j| \le \delta \}$ is transformed by the inverse scattering map. Notice that the most singular case concerns the case when $v_i = V' _*$ belongs to the small ball of radius $\delta$: indeed in the case when it is $V'$ then the measure $b(\nu ^*,v ^*) \, dv^*d\nu ^*$ will have support in a domain of size $O(\delta ^2)$. So now assume that $V' _*$ satisfies $|V' _* -v_j|\le \delta$.

Fig. 19

$V'_*$ has to belong to the ball of radius $\delta$ around $v_j$, thus it has to be in the cone with the dotted lines. By Carleman’s parametrization, this imposes constraints on the angular sector of $V'-{\bar{v}}_i$

• If $|v_j - {\bar{v}}_i| \le \delta$, meaning that ${\bar{v}}_i$ is itself in the same ball, then for any $V' \in B_R$, the intersection between the small ball and the line ${\bar{v}}_i + \mathbb {R}(V'-{\bar{v}}_i)^\perp$ is a segment, the length of which is at most $\delta$. We therefore find

  $$\begin{aligned} \int \mathbf{1}_{|V' _* -v_j|\le \delta } \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \le C R^2 \delta \,. \end{aligned}$$

• If $|v_j - {\bar{v}}_i| >\delta$, in order for the intersection between the ball and the line ${\bar{v}}_i + \mathbb {R}(V'-{\bar{v}}_i)^\perp$ to be non empty, we have the additional condition that $V'-{\bar{v}}_i$ has to be in an angular sector of size $\delta /| v_j-{\bar{v}}_i|$ (see Figure 19). We therefore have

  $$\begin{aligned} \int \mathbf{1}_{|V' _* -v_j|\le \delta } \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \le C R^2 {\delta ^2\over |v_j-{\bar{v}}_i|}. \end{aligned}$$

Thus (C.3) holds.

The other estimates provided in Lemma C.2 then come from Fubini’s theorem: let us start with (C.4). We write

$$\begin{aligned}&\int \min \big ( {\delta \over |v_i-v_j|} ,1\big ) \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \\&\quad = \int \mathbf{1}_{ |v_i-v_j| \le \delta } \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \\&\qquad + \int \mathbf{1}_{ |v_i-v_j|> \delta } {\delta \over |v_i-v_j|} \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*\\&\quad \le CR^2 \delta + \int \mathbf{1}_{ |v_i-v_j| > \delta } {\delta \over |v_i-v_j|} \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \end{aligned}$$

thanks to (C.3). The contribution of the velocities such that $|v_i - v_j| \ge 1$ can be bounded by $R^3 \delta$. Thus it is enough to consider

$$\begin{aligned}&\int {\delta \mathbf{1}_{ 1 \ge |v_i-v_j|> \delta }\over |v_i-v_j|} \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \\&\quad = \delta \int \Big ( \int _{|v_i-v_j|} ^1 \frac{dr}{r^{2}} +1 \Big ) \mathbf{1}_{ 1 \ge |v_i-v_j| > \delta } \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*\\&\quad \le \delta \int _{\delta }^1 \frac{dr}{r^2} \int \mathbf{1}_{ |v_i-v_j| \le r} \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*+CR^3 \delta \, , \end{aligned}$$

so using (C.3) again we get

$$\begin{aligned} \int \mathbf{1}_{ 1 \ge |v_i-v_j| > \delta } {\delta \over |v_i-v_j|} \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*&\le CR^2\delta \int _{\delta }^1 \frac{dr}{r} + CR^3 \delta \, , \end{aligned}$$

from which (C.4) follows.

Next let us prove (C.5)–(C.7). We have

$$\begin{aligned}&\int {1\over |v_i-v_j|^\gamma } \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \\&\quad = \gamma \int \Big ( \int _{|v_i-v_j|} ^1 \frac{1}{r^{1+\gamma }} dr +1 \Big ) \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*\\&\quad =\gamma \int _0^1 \frac{1}{r^{\gamma +1}} \Big ( \int \mathbf{1}_{|v_i -v_j|\le r} \, b(\nu ^*,v ^*) \, dv ^* d\nu ^* \Big ) dr + CR^3\\&\quad \le C_\gamma R^2 \Big ( \int _0^{ |v_j-{\bar{v}}_i|} \frac{1}{|v_j-{\bar{v}}_i|} r^{1-\gamma } dr + \int _{ |v_j-{\bar{v}}_i|} ^1 \frac{1}{r^{\gamma } } dr \Big ) + C R^3, \end{aligned}$$

which gives the expected estimates. Similarly

$$\begin{aligned}&\int \big |\log |v_i-v_j|\big | \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*\\&\quad = \int \int _{|v_i-v_j|} ^1 \frac{1}{r} dr \, b(\nu ^*,v ^*) \, dv ^* d\nu ^*\\&\quad \le CR^2 \left( \int _0^{ |v_j-{\bar{v}}_i|} \frac{r}{|v_j-{\bar{v}}_i|} dr + \int _{ |v_j-{\bar{v}}_i|} ^1 dr \right) \le CR^3 \, . \end{aligned}$$

This ends the proof of Lemma C.2. $\square$

Remark C.3

The proof of Lemma C.3 shows that in order to keep control on the collision integral the power $\gamma$ of the singularity must not be too large (namely smaller than 2).

Finally the following result describes the size of a collision integral when relative velocities are prescribed to lie in a given rectangle.

Lemma C.4

Consider two pseudo-particles i, j as well as their first parent $1^*$. Denote by $\nu _{1^*} \in {{\mathbb {S}}}$ and $v_{1^*} \in \mathbb {R}^2$ their scattering parameters. We assume also that all the velocities are bounded by $R>1$. Let $\mathcal {R}$ be a rectangle with sides of length $\delta , \delta '$, then

$$\begin{aligned}&\int \mathbf{1}_{v_i -v_j \in \mathcal {R}} \; \big | (v_{1^*} - v_{a(1^*)} ) \cdot \nu _{1^*} \big | \, dv_{1^*} d\nu _{1^*} \nonumber \\&\quad \le C R^2 \min (\delta , \delta ') \big ( |\log \delta | + |\log \delta ' | + 1\big ), \end{aligned}$$

(C.10)

Proof

Note that if i, j are generated by the same collision, then better estimates can be obtained from Lemma C.1. The case without scattering is also straightforward. Thus from now, we assume that $v_i$ is given by (C.2).

To derive (C.10), we suppose that $\delta \le \delta ' \le 1$ and that the collision with $1^*$ takes place with i which had a velocity ${\bar{v}}_i$. We cover the rectangle $v_j + \mathcal {R}$ into $\lfloor \delta '/\delta \rfloor$ balls of radius $2 \delta$. Let $\omega$ be the axis of the rectangle $v_j + \mathcal {R}$ and denote by $w_k = w_0 + \delta k \, \omega$ the centers of the balls which are indexed by the integer $k \in \{ 0, \dots , \lfloor \delta '/\delta \rfloor \}$. Applying (C.3) to each ball, we get

$$\begin{aligned} \int \mathbf{1}_{v_i -v_j \in \mathcal {R}} \; b(\nu _{1^*},v_{1^*}) \, dv_{1^*} d\nu _{1^*}\le & {} \sum _{k = 0}^{ \lfloor \delta '/\delta \rfloor } \int \mathbf{1}_{ |v_i - w_k| \le 2 \delta } \; b(\nu _{1^*},v_{1^*}) \, dv_{1^*} d\nu _{1^*} \\\le & {} C R^2 \sum _{k = 0}^{ \lfloor \delta '/\delta \rfloor } \delta \min \left( {\delta \over |w_k -{\bar{v}}_i|}, 1 \right) , \\\le & {} C R^2 \delta \sum _{k = 0}^{ \lfloor \delta '/\delta \rfloor } {\delta \over |w_k -{\bar{v}}_i| + \delta } \\\le & {} C R^2 \delta \left( \log \left( \frac{\delta '}{\delta }\right) +1 \right) , \end{aligned}$$

where the log divergence in the last inequality follows by summing over k. This completes the proof of (C.10).

This completes the proof of Lemma C.4. $\square$

Appendix D: Initial Data Estimates

This section is devoted to the proof of Proposition 2.6 stated page 14.

Using the notation $X_{k,N} := \{ x_k, \dots , x_N \}$, we write

$$\begin{aligned}&\Big | \left( f_{N }^{0(s)} - f^{(s)} _0 \right) (Z_s)\mathbf{1}_{{\mathcal D}_{{\varepsilon }}^s}(X_s) \Big | \\&\quad \le M_\beta ^{\otimes s}(V_s) \sum _{i=1}^s \big |g_{\alpha ,0}(z_i)\big | \, \Big | {\mathcal Z}_N^{-1} \int \mathbf{1}_{{\mathcal D}_{{\varepsilon }}^N}(X_N) \, dX_{s+1,N}-1\Big | \\&\qquad +{}{\mathcal Z}_N^{-1} M_\beta ^{\otimes s}(V_s) \sum _{i=s+1}^N \Big |\int M_\beta (v_i) g_{\alpha ,0}(z_i) \mathbf{1}_{{\mathcal D}_{{\varepsilon }}^N}(X_N) \, dv_idX_{s+1,N}\Big | \, , \end{aligned}$$

where ${\mathcal D}_{{\varepsilon }}^N$ stands for the exclusion constraint on the positions (with a slight abuse of notation compared to (1.4)). The first term is estimated as in the proof of Proposition 3.3 in [8]

$$\begin{aligned} M_\beta ^{\otimes s}(V_s) \sum _{i=1}^s \big |g_{\alpha ,0}(z_i)\big | \, \Big | {\mathcal Z}_N^{-1} \int \mathbf{1}_{{\mathcal D}_{{\varepsilon }}^N}(X_N) \, dX_{s+1,N}-1\Big | \le C ^s {\varepsilon }\alpha M_\beta ^{\otimes s} (V_s) \Vert g_{\alpha ,0}\Vert _{L^\infty } \, . \end{aligned}$$

The exchangeability of the variables allows us to rewrite the second term as

$$\begin{aligned} I(Z_s)&:= {\mathcal Z}_N^{-1} M_\beta ^{\otimes s}(V_s) \sum _{i=s+1}^N \Big |\int M_\beta (v_i) g_{\alpha , 0}(z_i) \mathbf{1}_{{\mathcal D}_{{\varepsilon }}^N}(X_N) \, dv_i dX_{s+1,N}\Big | \\&\quad \le (N-s) M_\beta ^{\otimes s}(V_s){\mathcal Z}_N^{-1} \\&\qquad \Big | \int M_\beta (v_{s+1}) g_{\alpha , 0}(z_{s+1}) \Big ( \prod _{k \ne s+1} \mathbf{1}_{|x_k-x_{s+1} |> {\varepsilon }} \Big ) \; \chi _{s+2}(X_N) \, dz_{s+1} dX_{s+2,N}\Big | \, , \end{aligned}$$

where we used the notation

$$\begin{aligned} \chi _{s+2}(X_N):= {\hat{\chi }}^+_{s+2}(X_{s+2,N}) \, {\hat{\chi }}^-_{s+2}(X_N) \end{aligned}$$

(D.1)

which distinguishes the interaction of the particles $X_{s+2,N}$ with themselves and with $X_s$, defining

$$\begin{aligned} {\hat{\chi }}^+_{s+2}(X_{s+2,N}):= \prod _{s+2 \le \ell < k \le N} \mathbf{1}_{|x_k-x_\ell |> {\varepsilon }} \quad \text {and} \quad {\hat{\chi }}^-_{s+2}(X_N):=\prod _{\begin{array}{c} s+2 \le \ell \le N \\ 1 \le k \le s \end{array}} \mathbf{1}_{|x_k-x_\ell |> {\varepsilon }} \,. \end{aligned}$$

The exclusion between $s+1$ and the rest of the system is also decomposed into a term for the interaction with $X_s$ and another one for the interaction with $X_{s+2,N}$. Defining

$$\begin{aligned} \chi ^-_{s+1}(X_{s+1}) := \prod _{k \le s} \mathbf{1}_{|x_k-x_{s+1} |> {\varepsilon }}\quad \text {and} \quad \chi ^+_{s+1}(X_{s+1,N}):=\prod _{k \ge s+2} \mathbf{1}_{|x_k-x_{s+1} | > {\varepsilon }} \end{aligned}$$

we have

$$\begin{aligned} \prod _{k \ne s+1} \mathbf{1}_{|x_k-x_{s+1} | > {\varepsilon }}&= \chi ^-_{s+1}(X_{s+1}) \, \chi ^+_{s+1}(X_{s+1,N}) \\&= \chi ^+_{s+1}(X_{s+1,N}) - \big ( 1- \chi ^-_{s+1}(X_{s+1}) \big ) \chi ^+_{s+1}(X_{s+1,N}) \,. \end{aligned}$$

We deduce that

$$\begin{aligned} I(Z_s) \le M_\beta ^{\otimes s}(V_s) \Big ( I_1 (Z_s) + I_2 (Z_s) \Big ) \end{aligned}$$

with

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle I_1 (Z_s): = {\mathcal Z}_N^{-1} N \int M_\beta (v_{s+1}) \big | g_{\alpha , 0}(z_{s+1}) \big | \big ( 1- \chi ^-_{s+1}(X_{s+1}) \big ) {\hat{\chi }}^+_{s+2}(X_{s+2,N}) \, dz_{s+1} dX_{s+2,N}\, , \\ \displaystyle I_2 (Z_s) := {\mathcal Z}_N^{-1}N \Big | \int M_\beta (v_{s+1}) g_{\alpha , 0}(z_{s+1}) \chi ^+_{s+1}(X_{s+2,N}) \; \chi _{s+2}(X_N) \, dz_{s+1} dX_{s+2,N}\Big |\,. \end{array}\right. } \end{aligned}$$

From (2.14) and the assumption $N {\varepsilon }= \alpha \ll 1 /{\varepsilon }$, we get

$$\begin{aligned} {\mathcal Z}_N^{-1}\int {\hat{\chi }}^+_{s+2}(X_{s+2,N}) \, dX_{s+2,N} = \frac{{\mathcal Z}_{N-s-2} }{ {\mathcal Z}_N} \le \exp \big (C s \alpha {\varepsilon }\big ) \le \exp \big (C s \big ) \,. \end{aligned}$$

We infer that the term $I_1$ is bounded by the fact that $x_{s+1}$ is close to $X_s$

$$\begin{aligned} I_1 (Z_s) \le s N {\varepsilon }^2 \exp \big (C s \big ) \Vert g_{\alpha ,0}\Vert _{L^\infty } \le s \exp \big (C s \big ) \alpha {\varepsilon }\Vert g_{\alpha ,0}\Vert _{L^\infty }\,. \end{aligned}$$

Using the assumption $\displaystyle \int _\mathbb {D}M_\beta g_{\alpha ,0} (z) dz = 0$, the second term is rewritten as

$$\begin{aligned} I_2 (Z_s) ={\mathcal Z}_N^{-1} N \big | \int M_\beta (v_{s+1}) g_{\alpha , 0}(z_{s+1}) \big ( 1 - \chi ^+_{s+1}(X_{s+1,N}) \big ) \; \chi _{s+2}(X_N) \, dz_{s+1} dX_{s+2,N} \big |\,. \end{aligned}$$

Plugging the identity (D.1)

$$\begin{aligned} \chi _{s+2}(X_N) = {\hat{\chi }}^+_{s+2}(X_{s+2,N}) - \big ( 1 - {\hat{\chi }}^-_{s+2}(X_N) \big ) {\hat{\chi }}^+_{s+2}(X_{s+2,N}) \end{aligned}$$

we distinguish two more contributions $I_2 (Z_s) \le I_{2,1} (Z_s) + I_{2,2} (Z_s)$ with

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle I_{2,1} (Z_s) := {\mathcal Z}_N^{-1} N \Vert g_{\alpha ,0}\Vert _{L^\infty } \, \int \big ( 1 - \chi ^+_{s+1}(X_{s+1,N}) \big ) \; \big ( 1 - {\hat{\chi }}^-_{s+2}(X_N) \big ) {\hat{\chi }}^+_{s+2}(X_{s+2,N}) \, dX_{s+1,N} \, ,\\ \displaystyle I_{2,2} (Z_s): = {\mathcal Z}_N^{-1} N \Big | \int M_\beta (v_{s+1}) g_{\alpha , 0}(z_{s+1}) \big ( 1 - \chi ^+_{s+1}(X_{s+1,N}) \big ) \; {\hat{\chi }}^+_{s+2}(X_{s+2,N}) \, dz_{s+1} dX_{s+2,N} \Big |\,. \\ \end{array}\right. } \end{aligned}$$

The term $I_{2,1}$ takes into account two constraints : $s+1$ is close to a particle in $X_{s+2,N}$ and one particle in $X_{s+2,N}$ is close to $X_s$. Since $N {\varepsilon }= \alpha$, we deduce that

$$\begin{aligned} I_{2,1} (Z_s) \le N s {\varepsilon }^2 \, (N-s-1) ^2 {\varepsilon }^2 \, \frac{{\mathcal Z}_{N-s-3} }{ {\mathcal Z}_N} \Vert g_{\alpha ,0}\Vert _{L^\infty } \le s \alpha ^3 {\varepsilon }\exp ( C s) \Vert g_{\alpha ,0}\Vert _{L^\infty }\,. \end{aligned}$$

The term $I_{2,2}$ does not depend on $X_s$, thus one can integrate over $z_{s+1}$ and use again the assumption $\displaystyle \int _\mathbb {D}M_\beta g_{\alpha ,0} (z) dz = 0$. To see this, it is enough to note that the function

$$\begin{aligned} x_{s+1} \mapsto \int \big ( 1 - \chi ^+_{s+1}(X_{s+1,N}) \big ) \; {\hat{\chi }}^+_{s+2}(X_{s+2,N}) dX_{s+2,N} \end{aligned}$$

is independent of $x_{s+1}$ thanks to the periodic structure of $\mathbb {D}^{N- s-2}$. Thus $I_{2,2} (Z_s) = 0$.

Combining the previous estimates, we conclude Proposition 2.6. $\square$