The Boltzmann–Grad limit of a hard sphere system: analysis of the correlation error

Abstract

We present a quantitative analysis of the Boltzmann–Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions, the correlation errors, measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k particles are connected by a chain of interactions preventing the factorization. We show that, provided $k < \varepsilon ^{-\alpha }$, such an error flows to zero with the average density $\varepsilon$, for short times, as $\varepsilon ^{\gamma k}$, for some positive $\alpha ,\gamma \in (0,1)$. This provides an information on the size of chaos, namely j different particles behave as dictated by the Boltzmann equation even when j diverges as a negative power of $\varepsilon$. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.

Appendices

Appendices

1.1 Appendix A: Chaotic states of hard spheres

We consider here the most natural construction of hard sphere states which factorize in the Boltzmann–Grad limit, and show that they satisfy the hypotheses stated in Sect. 2.3.

Let $\mathbf{W }^{\varepsilon }_{0}$ be the grand canonical state over $\mathcal{M}$ with system of densities

$$\begin{aligned} \frac{1}{n!}W^{\varepsilon }_{0,n}(\mathbf{z }_n)=\frac{1}{\mathcal{Z}_\varepsilon } \frac{e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n}{n!} f^{\otimes n}_0(\mathbf{z }_n) , \end{aligned}$$

(A.1)

where $\varepsilon ^2{\mu }_\varepsilon = 1$,

$$\begin{aligned} \mathcal{Z}_\varepsilon =\sum _{n\ge 0} \frac{e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n}{n!}\mathcal{Z}_{n}^{can}, \end{aligned}$$

(A.2)

and where the “canonical” normalization constant is

$$\begin{aligned} \mathcal{Z}_{n}^{can} = \int _{\mathcal{M}_n}d\mathbf{z }_n \,f^{\otimes n}_0(\mathbf{z }_n) = \int _{{\mathbb R}^{6n}} d\mathbf{z }_n \,f^{\otimes n}_0(\mathbf{z }_n)\prod _{1\le i < k \le n}\bar{\chi }^0_{i,k}, \end{aligned}$$

(A.3)

($\mathcal{Z}_0^{can}=1$) with $\bar{\chi }^0_{i,k}$ the indicator function of the set $\{ |x_i -x_k| > \varepsilon \}$. The function $f_0$ can be any probability density over ${\mathbb R}^3 \times {\mathbb R}^3$ satisfying $f_0(x,v) \le (h(x)/2) e^{-(\beta /2)v^2}$, for some $h \in L^1({\mathbb R}^3;{\mathbb R}^+)$ with ${{\mathrm{ess\,sup}}}_x h(x) = z$, and $z,\beta >0$.

Remark

• The state introduced is a “maximally factorized state” in the sense that the only correlations are due to the hard sphere exclusion. A Gibbs state in equilibrium statistical mechanics is of this form.

• The probability of finding n particles is $p_n = \mathcal{Z}^{can}_n\mathcal{Z}_{\varepsilon }^{-1} (1/n!)e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n$ and the distribution of the n particles $(\mathcal{Z}^{can}_n)^{-1}f_0^{\otimes n}$.

• The asymptotic behaviour of the normalization constants can be proved to be $\mathcal{Z}_{n}^{can} \sim e^{-Cn^2\varepsilon ^3}$ ($n \gg \varepsilon ^{-2}$, $C>0$) and $\mathcal{Z}_{\varepsilon } \sim e^{-C\varepsilon ^{-1}}$ (see e.g. [30]).

Proposition A.1

The state of the system defined by (A.1) admits r.c.f. satisfying Hypotheses 2.1, 2.2 and 2.3.

Proof

For this particular state, it is convenient to check (2.16)–(2.17) first. Since the only correlations are due to the exclusion, no combinatorial tools are required and a simple expansion of the non-overlap constraint suffices to reconstruct (2.16).

By definitions (2.7) and (2.11), the rescaled correlation functions are

$$\begin{aligned} f_{0,j}^{\varepsilon }(\mathbf{z }_j) = \frac{F^\varepsilon (\mathbf{z }_j)}{\mathcal{Z}_\varepsilon } f^{\otimes j}_0(\mathbf{z }_j), \end{aligned}$$

(A.4)

where

$$\begin{aligned} F^\varepsilon (\mathbf{z }_j)=\sum _{n\ge 0}\, \frac{e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n}{n!}\, F^{j+n}_{can}(\mathbf{z }_j) \end{aligned}$$

(A.5)

and

$$\begin{aligned} F^{j+n}_{can}(\mathbf{z }_j) = \int _{{\mathbb R}^{6n}} d\mathbf{z }_{j,n} f^{\otimes n}_0(\mathbf{z }_{j,n})\left( \prod _{i=1}^j\prod _{k=j+1}^{j+n}\ \bar{\chi }^0_{i,k}\right) \left( \,\prod _{j+1\le i < k \le {j+n}}\bar{\chi }^0_{i,k}\right) \end{aligned}$$

(A.6)

($F^{j}_{can}(\mathbf{z }_j)=1$).

For any $j,n \ge 1$, we rewrite $F^{j+n}_{can}(\mathbf{z }_j)$ by using

$$\begin{aligned} \prod _{i=1}^j\prod _{k=j+1}^{j+n}\ \bar{\chi }^0 _{i,k}=\prod _{i=1}^j (1- \chi ^0_{i,J^c}) \end{aligned}$$

(A.7)

where $J^c = \{j+1,\ldots ,j+n\}$ and

Expanding the product in (A.7), we find

$$\begin{aligned} \prod _{i=1}^j\prod _{k=j+1}^{j+n}\ \bar{\chi }^0_{i,k} = \sum _{K \subset J}(-1)^k \chi ^0_{K,J^c}, \end{aligned}$$

(A.8)

with

$$\begin{aligned} \chi ^0_{K,J^c}=\prod _{i\in K} \chi ^0_{i,J^c}. \end{aligned}$$

Inserting (A.8) into (A.6), we arrive to

$$\begin{aligned} f_{0,j}^{\varepsilon }(\mathbf{z }_j) = \sum _{L \subset J}f^{\otimes L}_0(\mathbf{z }_L) {E}^{\mathcal{B},0}_{J {\setminus } L}(\mathbf{z }_{J {\setminus } L}), \end{aligned}$$

(A.9)

where ${E}^{\mathcal{B},0}_\emptyset = 1$ and, for $k\ge 1$,

$$\begin{aligned} {E}^{\mathcal{B},0}_K(\mathbf{z }_k)= & {} (-f_0)^{\otimes k}(\mathbf{z }_k) \frac{1}{\mathcal{Z}_\varepsilon } \sum _{n\ge 1} \frac{e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n}{n!} \int d\mathbf{z }_{k,n}\,f_0^{\otimes n}(\mathbf{z }_{k,n}) \ \chi ^0_{K,K^c}\nonumber \\&\quad \prod _{k+1\le i < h \le {k+n}}\bar{\chi }^0_{i,h}. \end{aligned}$$

(A.10)

Let a be the maximum number of three-dimensional hard spheres that can be simultaneously overlapped by a single one, and $q = q(\mathbf{x }_k)$ the minimum number of different spheres in $K^c$ necessary to satisfy the condition $\chi ^0_{K,K^c}=1$ (any sphere in K is overlapped by at least one sphere in $K^c$). Then $k/a \le q \le k$ and

$$\begin{aligned} \chi ^0_{K,K^c} \le \sum _{\begin{array}{c} Q \subset K^c \\ |Q| = q \end{array}} \chi ^0_{Q,K}. \end{aligned}$$

(A.11)

It follows that

$$\begin{aligned} |{E}^{\mathcal{B},0}_K(\mathbf{z }_k)|\le & {} (z/2)^{k} e^{-(\beta /2)\sum _{i \in K} v_i^2}\, \frac{1}{\mathcal{Z}_\varepsilon } \, \sum _{n\ge q}\, \frac{e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n}{n!} \nonumber \\&\times \sum _{\begin{array}{c} Q \subset K^c \\ |Q| = q \end{array}}\,\int d\mathbf{z }_{k,n}\,f_0^{\otimes n}(\mathbf{z }_{k,n}) \,\chi ^0_{Q,K}\, \prod _{k+1\le i < h \le {k+n}}\bar{\chi }^0_{i,h}.\quad \quad \quad \quad \end{aligned}$$

(A.12)

Note now that $\chi ^0_{Q,K} = \prod _{i\in Q}\chi ^0_{i,K}$ and, for all $i\in Q$,

$$\begin{aligned} \int dz_{i}\,f_0(z_{i})\, \chi ^0_{i,K} \le (z/2) (2{\pi }/\beta )^{3/2} \,k\, B\,\varepsilon ^{3} \end{aligned}$$

(A.13)

where B is the volume of the unit ball. The remaining $n-q$ integration variables reconstruct $\mathcal{Z}^{can}_{n-q}$, so that we get

$$\begin{aligned} |{E}^{\mathcal{B},0}_K(\mathbf{z }_k)| \le \ \left( z/2\right) ^{k} e^{-(\beta /2)\sum _{i \in K} v_i^2} \frac{1}{\mathcal{Z}_\varepsilon } \ \sum _{n\ge q} \, \frac{e^{-{\mu }_{\varepsilon }}{\mu }_{\varepsilon }^n}{n!}\, \left( {\begin{array}{c}n\\ q\end{array}}\right) \,k^q\,(C \varepsilon ^3)^q \, \mathcal{Z}^{can}_{n-q}. \end{aligned}$$

(A.14)

Here and below we indicate by C a positive constant, possibly changing from line to line and depending on $z,\beta ,a,B$.

Using $\frac{1}{n!}\left( {\begin{array}{c}n\\ q\end{array}}\right) k^q \le \frac{(ke)^q}{q^q(n-q)! }\le C^q / (n-q)!$ and reminding (A.2), we deduce

$$\begin{aligned} |{E}^{\mathcal{B},0}_K(\mathbf{z }_k)|\le & {} (z/2)^{k} e^{-(\beta /2)\sum _{i \in K} v_i^2} (C \varepsilon )^q \nonumber \\\le & {} z^{k} e^{-(\beta /2)\sum _{i \in K} v_i^2}\, C^k\, \varepsilon ^{k/a}. \end{aligned}$$

(A.15)

This implies the estimate (2.17) by choosing $\gamma '_0 < 1/a$ and $\varepsilon$ small enough.⁹

Hypotheses 2.1 and 2.3 follow immediately.

Finally, observe that Hypothesis 2.2 and (2.16)–(2.17) are equivalent. Indeed, starting from (2.16), setting $f_0^{\otimes H} = (f^\varepsilon _{0,1}-E_1^{\mathcal{B},0})^{\otimes H}$ and expanding, one finds formula (2.13) with

$$\begin{aligned} E^0_{K} = \sum _{Q \subset K} (-1)^q (E_1^{\mathcal{B},0})^{\otimes Q} E_{K {\setminus } Q}^{\mathcal{B},0}, \end{aligned}$$

(A.16)

hence (2.17) implies $|E_K^0| \le 2^k \varepsilon ^{\gamma '_0 k}z^k\ e^{-(\beta /2)\sum _{i \in K} v_i^2} < \varepsilon ^{\gamma _0 k}z^k\ e^{-(\beta /2)\sum _{i \in K} v_i^2}$ for any $\gamma _0<\gamma '_0$ (and $\varepsilon$ small). The proof of the inverse statement is similar (one finds $\gamma '_0<\gamma _0$). $\square$

We conclude this appendix with the proof of the properties presented in Sect. 2.4.2.

Proof of Property 1

Hypothesis 2.2 is obtained from Property 1 in the case $S = {\mathcal J}= J$.

Let us show that Hypothesis 2.2 implies (2.29) for a generic partition of the set S. In this case, (2.28) is a rougher truncation and $E^0_{\mathcal{K}}$ takes into account only correlations among particles of different clusters.

Inverting (2.28) we find

$$\begin{aligned} E^0_{\mathcal{K}} =\sum _{\mathcal{Q}\subset \mathcal{K}} (-1)^{|\mathcal{Q}|} \left( \prod _{S \in \mathcal{Q}}f^{\varepsilon }_{0,S} \right) f^{\varepsilon }_{0,K{\setminus } Q}. \end{aligned}$$

(A.17)

We use the notation $K = \cup _{i\in \mathcal{K}}S_i$, $Q = \cup _{i\in \mathcal{Q}}S_i$ etc. By using (2.13), it follows that

$$\begin{aligned} E^0_{\mathcal{K}} =\sum _{\mathcal{Q}\subset \mathcal{K}} (-1)^{|\mathcal{Q}|} \sum _ {\begin{array}{c} L_1, \ldots , L_{|\mathcal{Q}|} \\ L_r \subset S_{i_r} \end{array}} \prod _{r=1}^{|\mathcal{Q}|} E_{L_r}^0 \sum _{L_0 \subset K {\setminus } Q} E^0_{L_0} \left( f^{\varepsilon }_{0,1}\right) ^{\otimes L^c}, \end{aligned}$$

(A.18)

where $i_1, \ldots , i_{|\mathcal{Q}|}$ are the indices of the clusters in $\mathcal{Q}$, and $L^c = K {\setminus } \cup _{r=0}^{|\mathcal{Q}|}L_r$. Note that the first sum is over subsets of clusters, while the other sums run over subsets of indices of particles. Setting $L = \cup _{r=0}^{|\mathcal{Q}|}L_r$ we notice that, for given $\mathcal{Q}$ and L, one has $L_r = L \cap S_{i_r}$ and $L_0 = L \cap (K {\setminus } Q)$. Therefore we rewrite (A.18) as

$$\begin{aligned} E^0_{\mathcal{K}} =\sum _{L \subset K} \left( f^{\varepsilon }_{0,1}\right) ^{\otimes K {\setminus } L} \sum _{\mathcal{Q}\subset \mathcal{K}} (-1)^{|\mathcal{Q}|} E^0_{L \cap (K {\setminus } Q)} \prod _{r=1}^{|\mathcal{Q}|} E_{L \cap S_{i_r}}^0. \end{aligned}$$

(A.19)

Observe that, in the above sum, L must be such that $|L\cap S_{i} |> 0$ for all $i \in \mathcal{K}$. Otherwise if $L\cap S_{i} =\emptyset$ for some i, setting $S^*=S_{i}$, since $E_{L \cap S_{i}}^0=1$,

$$\begin{aligned}&\sum _{\begin{array}{c} \mathcal{Q}\subset \mathcal{K}\\ S^* \in \mathcal{Q} \end{array}} (-1)^{|\mathcal{Q}|} E^0_{L \cap (K {\setminus } Q)} \prod _{r=1}^{|\mathcal{Q}|} E_{L \cap S_{i_r}}^0 +\sum _{\begin{array}{c} \mathcal{Q}\subset \mathcal{K}\\ S^* \notin \mathcal{Q} \end{array}} (-1)^{|\mathcal{Q}|} E^0_{L \cap (K {\setminus } Q)} \prod _{r=1}^{|\mathcal{Q}|} E_{L \cap S_{i_r}}^0 \nonumber \\&\quad = - \sum _{\mathcal{Q}\subset \mathcal{K}{\setminus } \{S^*\}} (-1)^{|\mathcal{Q}|} E^0_{L \cap (K {\setminus } Q)} \prod _{r=1}^{|\mathcal{Q}|} E_{L \cap S_{i_r}}^0 + \sum _{\mathcal{Q}\subset \mathcal{K}{\setminus } \{S^*\}} (-1)^{|\mathcal{Q}|} E^0_{L \cap (K {\setminus } Q)} \prod _{r=1}^{|\mathcal{Q}|} E_{L \cap S_{i_r}}^0 \nonumber \\&\quad = 0. \end{aligned}$$

(A.20)

As a consequence, using (2.12) and (2.14) in (A.19), we deduce

$$\begin{aligned} |E^0_{\mathcal{K}}|\le & {} 2^{|\mathcal{K}|}\,z^k\ e^{-(\beta /2)\sum _{i \in K} v_i^2} \sum _{\begin{array}{c} L \subset K \\ L \cap S_i \ne \emptyset \ \forall i \end{array}}\varepsilon ^{\gamma _0 |L|}\nonumber \\\le & {} 4^k \varepsilon ^{\gamma _0 |\mathcal{K}|}\, z^k\ e^{-(\beta /2)\sum _{i \in K} v_i^2}, \end{aligned}$$

(A.21)

so that (2.29) follows by reducing the values of $\gamma _0,\alpha _0$. $\square$

Proof of Property 2

We rewrite (2.30) as

$$\begin{aligned} f^{\varepsilon }_{0,S} = \sum _{\mathcal{H}\subset {\mathcal J}} \bar{\chi }_{\mathcal{H},{\mathcal J}}^0\left( \prod _{i \in \mathcal{H}}\bar{\chi }_{S_i}^0 f^{\varepsilon }_{0,S_i}\right) \bar{\chi }_{J {\setminus } H}^0 E^0_{{\mathcal J}{\setminus } \mathcal{H}} \end{aligned}$$

(A.22)

where $\bar{\chi }^0_{\mathcal{H},{\mathcal J}}=1$ if and only if all the particles in $S_i$ do not overlap with any other particle in $S_k$ for any choice of $i\in \mathcal{H}, k \in {\mathcal J}$, $k\ne i$.

We expand now the exclusion constraint (using the ideas explained in Sect. 4.2—Step 1 in the context of dynamical correlations). By virtue of Lemma 4.4,

$$\begin{aligned} \bar{\chi }_{\mathcal{H},{\mathcal J}}^0 =\sum _{ \mathcal{Q}\subset \mathcal{H}} R(\mathcal{Q},{\mathcal J}{\setminus } \mathcal{H}) \end{aligned}$$

(A.23)

and then we get

$$\begin{aligned} |R(\mathcal{Q},{\mathcal J}{\setminus } \mathcal{H})| \le C^{|\mathcal{Q}|}\, |\mathcal{Q}|! \,\chi ^0_{\mathcal{Q},\mathcal{Q}\cup ({\mathcal J}{\setminus } \mathcal{H})}. \end{aligned}$$

(A.24)

Inserting (A.23) in (A.22) we obtain (2.31) with

$$\begin{aligned} \bar{E}^0_{\mathcal{K}} = \sum _{\begin{array}{c} \mathcal{H}_1,\mathcal{H}_2 \\ \mathcal{H}_1 \cup \mathcal{H}_2=\mathcal{K}\\ \mathcal{H}_1 \cap \mathcal{H}_2=\emptyset \end{array}} R(\mathcal{H}_1,\mathcal{H}_2) \left( \prod _{i \in \mathcal{H}_1} \,\bar{\chi }_{S_i}^0 \, f^{\varepsilon }_{0,S_i}\right) \left( \bar{\chi }^0_{H_2} E^0_{\mathcal{H}_2}\right) . \end{aligned}$$

(A.25)

The bound (2.32) follows from (A.24). $\square$

1.2 Appendix B: Graph expansion

We prove in this section the graph expansion Lemma. The strategy is explained informally in Sect. 4.2—Step 1, where $\bar{\chi } _{L,L\cup L_0}$ is the non-recollision condition of the trees in the set L (and $\chi$ is the overlap constraint).

Proof of Lemma 4.4

By addition/subtraction we find

$$\begin{aligned} \bar{\chi } _{L,L\cup L_0}= & {} 1-\sum _{\begin{array}{c} L_1,L_2 \\ L_1 \cup L_2=L\\ L_1 \cap L_2=\emptyset \\ l_1 \ge 1 \end{array}} \chi _{L_1,L \cup L_0} \bar{\chi } _{L_2,L \cup L_0 }\nonumber \\= & {} 1-\sum _{\begin{array}{c} L_1,L_2 \\ L_1 \cup L_2=L\\ L_1 \cap L_2=\emptyset \\ l_1 \ge 1 \end{array}} \chi _{L_1,L_1 \cup L_0} \bar{\chi } _{L_2,L_0 \cup L_1 \cup L_2}. \end{aligned}$$

(B.1)

Note that $l_1 = |L_1| >0$ and $\chi _{L_1,L \cup L_0}=\chi _{L_1,L_1 \cup L_0}$, because any vertex in $L_2$ is not connected. Iterating once,

$$\begin{aligned} \bar{\chi } _{L,L\cup L_0}= & {} 1-\sum _{\begin{array}{c} L_1 \subset L \\ l_1 \ge 1 \end{array}} \chi _{L_1,L_1 \cup L_0}\nonumber \\&+ \sum _{\begin{array}{c} L_1,L_2,L_3 \\ L_1 \cup L_2\cup L_3 =L\\ L_i \cap L_j=\emptyset , i\ne j \\ l_1 \ge 1, l_2 \ge 1 \end{array}} \chi _{L_1,L_0\cup L_1} \chi _{L_2,L_0\cup L_1 \cup L_2} \bar{\chi } _{L_3,L_0 \cup L_1\cup L_2 \cup L_3} .\nonumber \\ \end{aligned}$$

(B.2)

Then, successive iterations yield the following expansion:

$$\begin{aligned} \bar{\chi } _{L,L\cup L_0} =\sum _{r=0}^{|L|} (-1)^r \sum _{\begin{array}{c} L_1, \ldots , L_r \\ \cup _i L_i \subset L \\ l_i \ge 1\\ L_i \cap L_j= \emptyset , i\ne j \end{array}} \chi _{L_1,L_0\cup L_1 } \cdots \chi _{L_r, L_0 \cup L_1 \cdots \cup L_r }, \end{aligned}$$

(B.3)

where the $r=0$ term has to be interpreted as 1. I.e.

$$\begin{aligned} \bar{\chi } _{L,L\cup L_0} =\sum _{Q\subset L} R(Q,L_0), \end{aligned}$$

(B.4)

with

$$\begin{aligned} R(Q,L_0):= \sum _{r=1}^{q} (-1)^r \sum _{\begin{array}{c} L_1, \ldots , L_r \\ \cup _i L_i=Q \\ l_i \ge 1\\ L_i \cap L_j= \emptyset , i\ne j \end{array}} \chi _{L_1,L_0\cup L_1 } \cdots \chi _{L_r, L_0 \cup L_1 \cdots \cup L_r }, \end{aligned}$$

(B.5)

and $R(\emptyset ,L_0)=1$.

From this expression it follows that

$$\begin{aligned} |R(Q, L_0) |\le & {} \sum _{r=1}^{q} \sum _{\begin{array}{c} L_1, \ldots , L_r \\ \cup _i L_i=Q \\ l_i \ge 1\\ L_i \cap L_j= \emptyset , i\ne j \end{array}} \chi _{Q, Q\cup L_0 }\nonumber \\\le & {} \chi _{Q, Q\cup L_0 } \sum _{r=1}^{q} \sum _{\begin{array}{c} l_1, \ldots , l_r \\ l_i \ge 1 \end{array}} \frac{q !}{l_1! \cdots l_r!} \nonumber \\\le & {} \chi _{Q, Q\cup L_0 }\,q!\,C^q . \end{aligned}$$

(B.6)

$\square$

1.3 Appendix C: Reduction to energy functionals

In this appendix we prove the technical result stated in Sect. 4.3.3. We divide the proof in three parts where we truncate respectively number of particles, energy, and cross-sections. The truncation errors are controlled by slight variants of Lanford’s short time estimate.

Proof of Lemma 4.5

(a) We first use the bound (4.23) and the assumptions on the initial state (see (4.28)) to estimate $E_K$ as given by (4.27). Notice that (4.28) can be applied for $k + n < \varepsilon ^{-\alpha _0}$, which is ensured by $k < \varepsilon ^{-\alpha }$, $n \le \log \varepsilon ^{-\theta _2k}$ for arbitrary positive $\theta _2$ and $\alpha < \alpha _0$, as soon as $\varepsilon$ is small enough. We deduce:

$$\begin{aligned}&\int d\mathbf{v }_K |E_K(t)| \le z^k\,C^k\sum _{\begin{array}{c} L_0,Q, B \\ \subset \ K \\ \hbox {{disjoint}} \end{array}} q!\,b! \sum _{n=0}^{\log \varepsilon ^{-\theta _2 k}} z^n \sum _{\Gamma (k,n)} \int d\mathbf{v }_k d{\Lambda }\nonumber \\&\quad \quad \times \prod |B^\varepsilon | \chi _{L_0 }^{rec}\, \chi ^{ov}_{Q, K} \, \chi ^{0}_{B, K}\, \varepsilon ^{\gamma _0(k-q-l_0-b)} e^{-(\beta /2) \mathcal{H}_K} \nonumber \\&\quad \quad +z^k\,C^k \sum _{\begin{array}{c} L_0,Q \\ \subset \ K \\ \hbox {{disjoint}} \end{array}} q!\,(k-q-l_0)! \sum _{n > \log \varepsilon ^{-\theta _2 k}}\ z^n \ \sum _{\Gamma (k,n)} \int d\mathbf{v }_k d{\Lambda }\prod |B^\varepsilon | \, e^{-(\beta /2) \mathcal{H}_K}.\nonumber \\ \end{aligned}$$

(C.1)

The symbol C is always used for pure positive constants. Note that, in the error produced by the truncation on n, the last line of (4.27) has been estimated simply by $z^{k+n}C^k e^{-(\beta /2) \mathcal{H}_K} (k-q-l_0)!$, as follows from (2.32), (A.17) and Hypothesis 2.1.

Proceeding exactly as in the proof of Lemma 4.2 (case $a=1$), the last term in (C.1) is bounded, for $t < \bar{t}$ (see (4.8)), by

$$\begin{aligned}&C^k \,k!\, (4{\pi }/\beta )^{\frac{3}{2}k}\,(C(z,\beta )e)^{k} \sum _{n > \log \varepsilon ^{-\theta _2 k}}(\bar{t} \, C(z,\beta )e)^n \nonumber \\&\quad \le (C')^k \,k^k\,\varepsilon ^{\theta _2 \log (\bar{t} C(z,\beta )e)^{-1} k }\nonumber \\&\quad \le (C')^k \,\varepsilon ^{\theta _2 \log (\bar{t} C(z,\beta )e)^{-1} k -\alpha k} \le \varepsilon ^{\gamma k}/4, \end{aligned}$$

(C.2)

for a suitable $C'= C'(z,\beta )>0$. In the last line we used $k < \varepsilon ^{-\alpha }$,

$$\begin{aligned} \gamma < \theta _2 \log (\bar{t} C(z,\beta )e)^{-1}-\alpha \end{aligned}$$

(C.3)

and $\varepsilon$ small enough.

Since $\chi ^{ov}_{Q, K} \, \chi ^{0}_{B, K} \le \chi ^{ov}_{Q\cup B, K}$ (overlap at time zero implies overlap in [0, t]), renaming $Q \cup B \rightarrow Q$, (C.1) yields

$$\begin{aligned}&\int d\mathbf{v }_K |E_K(t)| \nonumber \\&\quad \le z^kC^kk!\,\sum _{\begin{array}{c} L_0,Q\\ \subset \ K \\ {\mathrm{disjoint}} \end{array}} \sum _{n=0}^{\log \varepsilon ^{-\theta _2 k}} z^n \sum _{\Gamma (k,n)} \int d\mathbf{v }_k d{\Lambda }\nonumber \\&\quad \quad \times \prod |B^\varepsilon | \chi _{L_0 }^{rec} \chi ^{ov}_{Q, K} \varepsilon ^{\gamma _0(k-q-l_0)} e^{-(\beta /2) \mathcal{H}_K} + \frac{\varepsilon ^{\gamma k}}{4}. \end{aligned}$$

(C.4)

(b) Next we truncate the integration domain to the sphere of energy smaller than $2\varepsilon ^{-\theta _3}$, for arbitrary $\theta _3 > 0$. The corresponding error is bounded by

(C.5)

for a suitable $C''= C''(z,\beta )>1$. From second to third line we repeated the proof of Lemma 4.2 with $a=1$ and $\beta \rightarrow \beta /2$. Note that (C.5) is in turn bounded, for $k< \varepsilon ^{-\alpha }$, by $(C'')^k\, e^{-(\beta /4) \varepsilon ^{-\theta _3}+\varepsilon ^{-\alpha }\log \varepsilon ^{-\alpha }+\varepsilon ^{-\alpha }\log \varepsilon ^{-\theta _2}\log C''}$, which is smaller than $\varepsilon ^{\gamma k}/4$ if $\theta _3>\alpha$ and $\varepsilon$ is small enough.

Remembering (4.29), it follows that

$$\begin{aligned}&\int d\mathbf{v }_K |E_K(t)| \nonumber \\&\quad \le z^kC^kk!\,\sum _{\begin{array}{c} L_0,Q\\ \subset \ K \\ \hbox {{disjoint}} \end{array}} \sum _{n=0}^{\log \varepsilon ^{-\theta _2 k}} z^n \sum _{\Gamma (k,n)} \int d\mathbf{v }_k d{\Lambda }\nonumber \\&\quad \quad \times \prod |B^\varepsilon | \chi _{L_0 }^{rec} \chi ^{ov}_{Q, K} \varepsilon ^{\gamma _0(k-q-l_0)} F_{\theta _3}(K) + \frac{2\varepsilon ^{\gamma k}}{4}. \end{aligned}$$

(C.6)

(c) Finally, we introduce a truncation of the cross-section factors $\prod |B^\varepsilon |$. We want actually to eliminate these factors from (C.6). Such a simplification of formulas will be very useful for the recollision estimates. (This procedure was already applied in [30].)

To this purpose we apply the following corollary of Lanford’s short time estimate, Lemma 4.2, of which we adopt here the notation.

Lemma C.1

Let $F \le 1$ be any positive measurable function of the variables $\mathbf{z }_j$, $\mathbf{t }_n,$ ${\varvec{\omega }}_n,$ $\mathbf{v }_{j,n}$. Let $N>0$ and $\theta _1>0$. There exists $\bar{C}'>0$ such that, for any $t<\bar{t}$,

(C.7)

The result holds also when $B^\varepsilon , {\varvec{\zeta }}^\varepsilon$ are replaced by $B^{{\mathscr {E}}},{\varvec{\zeta }}^{{\mathscr {E}}}$ (Enskog flow) or $B,{\varvec{\zeta }}$ (Boltzmann flow).

Note that the last integral does not contain any more cross-section factors, the only residual being the characteristic function that prohibits overlaps at creation times.

To deduce the corollary, it is enough to observe that the integral on the l.h.s., when restricted to the set such that $\prod |B^\varepsilon | > \varepsilon ^{-\theta _1 j}$, is bounded by $\varepsilon ^{\theta _1 j}$ times the integral with respect to $d\mathbf{v }_j$ of the left hand side in (4.1) with $a=2$. Applying Lemma 4.2, we obtain the result by taking $\bar{C}' = \bar{C} (4{\pi }/\beta )^{3/2}$.

Computing the l.h.s. in Lemma C.1 via the mixed flow (4.18) instead of the IBF causes, of course, no modification, except for the expression of the characteristic function in (C.7). Therefore we may apply the result to (C.6), which produces an error $C^k k! (\bar{C}')^k \varepsilon ^{\theta _1 k} \le (C''')^k k^k \varepsilon ^{\theta _1 k}$ for a suitable $C'''= C'''(z,\beta )>0$ and arbitrary $\theta _1>0$. This is, in turn, smaller than $\varepsilon ^{\gamma k}/4$ for $k < \varepsilon ^{-\alpha }$, $\gamma < \theta _1-\alpha$ and $\varepsilon$ small enough.

We conclude that, for any $t<\bar{t}$,

(C.8)

where the characteristic functions are those defined after (4.19). $\square$

Remark (Choice of parameters) If we choose the parameters as in (4.32), then (4.34) ensures that all the conditions in the proof above are satisfied. In fact, in part (a) of the proof we just need to check (C.3) which reads $\gamma < (1/2) - \alpha$ and follows from $\gamma< a(\gamma _0) - 3 \alpha < 1/4 - 3\alpha$. In part (b), the condition $\alpha < \theta _3 = 1/5$ follows from $\alpha< (1/3) a(\gamma _0) < 1/12$. Finally in part (c) the condition $\gamma < \theta _1 - \alpha = a(\gamma _0)-\alpha$ is guaranteed by $\gamma < a(\gamma _0) - 3\alpha$.

1.4 Appendix D: Estimate of internal recollisions

Proof of Lemma 4.12

It is convenient to use the Enskog backwards flow ${\varvec{\zeta }}^{\mathscr {E}}$ introduced in Sect. 3.5.2. For any given value of the variables $(x_1,\Gamma (1,n_1),\mathbf{v }_{n_1+1},{\varvec{\omega }}_{n_1},\mathbf{t }_{n_1})$, if the IBF ${\varvec{\zeta }}^\varepsilon$ delivers an internal recollision, then the EBF ${\varvec{\zeta }}^{\mathscr {E}}$ delivers an internal overlap (two particles of the flow having a distance smaller than $\varepsilon$). That is,

$$\begin{aligned} \chi ^{int} \le \chi ^{i.o.}({\varvec{\zeta }}^{\mathscr {E}}), \end{aligned}$$

(D.1)

where

$$\begin{aligned} \chi ^{i.o.}=\chi ^{i.o.}({\varvec{\zeta }}^{\mathscr {E}})=1 \end{aligned}$$

(D.2)

if and only if the EBF associated to the 1-particle tree exhibits at least one overlap between two particles. Therefore in what follows we shall focus on the proof of the estimate

$$\begin{aligned} \sum _{\Gamma (1,n_1)} \int dv_{1} d{\Lambda }\,\chi ^{i.o.}\, e^{-(\beta /2)\sum _{i\in S(1)}v_i^2} \le \frac{\varepsilon ^{\gamma _1}}{2}(Dt)^{n_1}. \end{aligned}$$

(D.3)

Remind that $d{\Lambda }= d{\Lambda }(\mathbf{t }_{n_1},{\varvec{\omega }}_{n_1},\mathbf{v }_{1,1+n_1})$ and $\Gamma (1,n_1)=(k_1,\ldots ,k_{n_1})$.

We start with

$$\begin{aligned} \chi ^{i.o.} \le \sum _{s=2}^{n_1} \,\sum _{h=k_s,s+1}\,\sum _{\begin{array}{c} i=1,\ldots ,s\\ i\ne k_s,s+1 \end{array}} \, \chi ^{i.o.}_{(i,h), s}, \end{aligned}$$

(D.4)

where $\chi ^{i.o.}_{(i,h), s} = \chi ^{i.o.}_{(i,h), s}({\varvec{\zeta }}^{\mathscr {E}}) = 1$ if and only if:

• (i) going backwards in time, the first overlap between particles i and h takes place at a time $\tau \in (0, t_s]$;

• (ii) particles i and h move freely in $(\tau ,t_s)$;

• (iii) at time $t_s$

  $$\begin{aligned} \eta ^{{\mathscr {E}}}_h(t^-_s)\ne \eta ^{{\mathscr {E}}}_{k_s}(t^+_s). \end{aligned}$$

  (D.5)

  Notice that particle h is involved in the creation process at time $t_s$. See Fig. 8 below for a scheme of the possible situations and observe that, by virtue of (iii), we are excluding case 2 for incoming collision configurations at the creation time $t_s$.

From (D.3) to (D.4) one gets

$$\begin{aligned}&\sum _{\Gamma (1,n_1)} \int dv_{1} d{\Lambda }\,\chi ^{i.o.}\, e^{-(\beta /2)\sum _{i\in S(1)}v_i^2} \nonumber \\&\quad \le \sum _{\Gamma (1,n_1)}\,\sum _{s=2}^{n_1} \,\sum _{h=k_s,s+1}\,\sum _{\begin{array}{c} i=1,\ldots ,s\\ i\ne k_s \end{array}} \, \int dv_{1} d{\Lambda }\, \chi ^{i.o.}_{(i,h), s}\, e^{-(\beta /2)\sum _{i\in S(1)}v_i^2}.\nonumber \\ \end{aligned}$$

(D.6)

Note that $\chi ^{i.o.}_{(i,h), s}$ depends actually only on ${\varvec{\zeta }}^{\mathscr {E}}_{1+s}$, hence we can immediately integrate out the node variables

$$\begin{aligned} t_{s+1},\ldots ,t_{n_1},\omega _{s+1}\ldots ,\omega _{n_1},v_{s+2},\ldots ,v_{1+n_1} \end{aligned}$$

and sum over the tree variables $k_{s+1},\ldots ,k_{n_1}$. Applying (4.7),

$$\begin{aligned} \sum _{k_{s+1},\ldots ,k_{n_1}}\int d\mathbf{t }_{s,n_1-s} = (s+1)(s+2)\cdots (n_1) t^{n_1-s}/(n_1-s)! \le e^{n_1}t^{n_1-s}, \end{aligned}$$

thus we infer that

$$\begin{aligned}&\sum _{\Gamma (1,n_1)} \int dv_{1} d{\Lambda }\,\chi ^{i.o.}\, e^{-(\beta /2)\sum _{i\in S(1)}v_i^2} \le e^{n_1}\,\sum _{s=2}^{n_1} \,(D't)^{n_1-s}\, \sum _{\Gamma (1,s)}\,\sum _{h=k_s,s+1}\nonumber \\&\quad \times \sum _{\begin{array}{c} i=1,\ldots ,s\\ i\ne k_s \end{array}} \, \cdot \int dv_{1} d{\Lambda }(\mathbf{t }_s,{\varvec{\omega }}_s,\mathbf{v }_{1,1+s})\,\chi ^{i.o.}_{(i,h), s}\, e^{-(\beta /2)\sum _{i = 1}^{1+s}v_i^2}, \end{aligned}$$

(D.7)

where $D' = 4\pi \, (2\pi /\beta )^{3/2}$ and, in the last line, we are left with integrals associated to 1-particle, s-collision trees.

If $\chi ^{i.o.}_{(i,h), s} = 1$, then there are two possibilities: either $h = s+1$ (h is created at $t_s$) or $k_s = h$ (h is the progenitor of $s+1$), see Fig. 8. Let us resort to the notation of virtual trajectories, to deal with both cases simultaneously (Definition 4.11, applied to $\bar{{\varvec{\zeta }}}={\varvec{\zeta }}^{\mathscr {E}}$). We set

$$\begin{aligned} W=\eta ^{\mathscr {E}}_h (t_s^-)-\eta ^{\mathscr {E}}_i(t_s), \quad W_0= \eta ^{{\mathscr {E}},h}(t_s^+) - \eta ^{\mathscr {E}}_i(t_s) \end{aligned}$$

and

$$\begin{aligned} Y=\xi ^{\mathscr {E}}_h (t_s^-)-\xi ^{\mathscr {E}}_i(t_s), \quad Y_0=\xi ^{{\mathscr {E}},h} (t_{s-1}^-)-\xi ^{\mathscr {E}}_i(t_{s-1}). \end{aligned}$$

We remind that $t^+, t^-$ denote the limit from the future (post-collision) or from the past (pre-collision) respectively. Note that (D.5) is, in this notation,

$$\begin{aligned} \eta ^{{\mathscr {E}},h}(t^-_s)\ne \eta ^{{\mathscr {E}},h}(t^+_s), \end{aligned}$$

(D.8)

namely the virtual trajectory of particle h changes velocity at time $t_s$.

The overlap-condition implies

$$\begin{aligned} \inf _{\tau \in (0,t_s)} | Y-W\tau | \le \varepsilon . \end{aligned}$$

(D.9)

Put $\hat{W}=\frac{W}{|W|}$ if $W\ne 0$ and $W = (1,0,0)$ otherwise. Eq. (D.9) implies in turn

$$\begin{aligned} |Y\wedge \hat{W}| \le \varepsilon , \end{aligned}$$

i.e.

$$\begin{aligned} |(Y_0-W_0 t_{s-1}) \wedge \hat{W} + (W_0 \wedge \hat{W}) t_s | \le 2\,\varepsilon , \end{aligned}$$

(D.10)

where the factor 2 takes into account the jump in position in the virtual trajectory of particle h at time $t_s$, case 1. Therefore, we may bound the last line in (D.7) by replacing $\chi ^{i.o.}_{(i,h), s}$ with the indicator function of the events (D.10) and $W\ne W_0$ (which takes into account (D.8)).

By definition of the Enskog flow, $Y_0$ and $W_0$ do not depend on $t_s$ (since they concern the previous history). Moreover, the velocities in $(0,t_{s})$, which we denote

$$\begin{aligned} (\eta ^-_1,\ldots ,\eta ^-_{s+1}) = (\eta ^{\mathscr {E}}_1(t^-_s),\ldots ,\eta ^{\mathscr {E}}_{s+1}(t^-_{s})), \end{aligned}$$

(D.11)

are also independent of the times $t_1,\ldots ,t_s$: they depend only on previous velocities and impact vectors. In particular, W does not depend on $t_s$, so that in (D.10) a linear relation in $t_s$ appears. On the other hand, the integral in $t_s$ over the condition (D.10) is bounded by $\min (t,4\varepsilon |W_0\wedge \hat{W}|^{-1})$. Hence, for an arbitrary $\gamma _1\in (0,1)$,

$$\begin{aligned}&\sum _{\Gamma (1,s)}\,\sum _{h=k_s,s+1}\,\sum _{\begin{array}{c} i=1,\ldots ,s\\ i\ne k_s \end{array}} \, \int dv_{1} d{\Lambda }(\mathbf{t }_{s},{\varvec{\omega }}_s,\mathbf{v }_{1,1+s})\, \,\chi ^{i.o.}_{(i,h), s}\,e^{-(\beta /2)\sum _{i = 1}^{1+s}v_i^2}\nonumber \\&\quad \le (4/t)^{\gamma _1} t \,\varepsilon ^{\gamma _1} \sum _{\Gamma (1,s)}\,\sum _{h=k_s,s+1}\,\sum _{\begin{array}{c} i=1,\ldots ,s\\ i\ne k_s \end{array}} \nonumber \\&\quad \quad \times \int dv_{1} d{\Lambda }'(\mathbf{t }_{s-1},{\varvec{\omega }}_s,\mathbf{v }_{1,1+s})\, \frac{1}{|W_0\wedge \hat{W}|^{\gamma _1}} \, e^{-(\beta /2)\sum _{i = 1}^{1+s}v_i^2}, \end{aligned}$$

(D.12)

where $d{\Lambda }'(\mathbf{t }_{s-1},{\varvec{\omega }}_s,\mathbf{v }_{1,1+s})$ is the measure $d{\Lambda }(\mathbf{t }_{s},{\varvec{\omega }}_s,\mathbf{v }_{1,1+s})$ deprived of $dt_s$ and multiplied, in case 2 of Fig. 8, by the characteristic function of $\omega _s \cdot (v_{1+s}-\eta ^{{\mathscr {E}},h}(t^+_s))>0$ (coming from the condition $W \ne W_0$).

It remains to prove that the integral of the singular function $|W_0\wedge \hat{W}|^{-\gamma _1}$ converges. To do so, let us first express $W_0$ in terms of the pre-collisional variables (D.11). Applying the elastic collision rule (2.3), one finds

$$\begin{aligned} W_0= & {} \left( \eta ^{{\mathscr {E}},h}(t_s^+)- \eta ^{{\mathscr {E}},h}(t_s^-)\right) +W \\= & {} P_s W_\ell + W, \end{aligned}$$

where

$$\begin{aligned} W_\ell = \eta ^-_\ell - \eta ^-_h \end{aligned}$$

and

$$\begin{aligned} P_s X := {\left\{ \begin{array}{ll} \displaystyle P_{\omega _s}^{\perp } X := X - \omega _s (\omega _s\cdot X)&{} \mathrm{case 1, outgoing collision}\\ \displaystyle X &{} \mathrm{case 1, incoming collision}\\ \displaystyle P_{\omega _s}^{\parallel } X := \omega _s (\omega _s\cdot X)&{} \mathrm{case 2, outgoing collision}\\ \displaystyle 0 &{} \mathrm{case 2, incoming collision} \end{array}\right. }. \end{aligned}$$

(D.13)

Cases 1, 2 are those in Fig. 8, while we remind that the incoming/outgoing collisions are depicted in Fig. 2 (here corresponding respectively to the negative / positive sign of the scalar product $\omega _s \cdot (v_{1+s}-\eta ^{{\mathscr {E}},h}(t^+_s))$). Moreover, the “case” depends only on the structure of the chosen tree $\Gamma (1,s)$. It follows that

$$\begin{aligned} \frac{1}{|W_0\wedge \hat{W}|} = \frac{1}{|P_s W_\ell \wedge \hat{W}|} \end{aligned}$$

(D.14)

which we may insert into (D.12).

Next, we change variables according to $v_1,v_2,\ldots ,v_{s+1} \rightarrow \eta ^-_1,\ldots ,\eta ^-_{s+1}$. This is an invertible and measure-preserving transformation, for any fixed value of $\omega _1,\ldots ,\omega _s$, (since the single hard-sphere collision (2.3) is so). Moreover, by the conservation of energy at collisions, $e^{-(\beta /2)\sum _{i = 1}^{1+s}v_i^2}= e^{-(\beta /2)\sum _{i = 1}^{1+s}(\eta ^-_i)^2}$. From (D.7), (D.12) and (D.14), we thus obtain

$$\begin{aligned}&\sum _{\Gamma (1,n_1)} \int dv_{1} d{\Lambda }\,\chi ^{i.o.}\, e^{-(\beta /2)\sum _{i\in S(1)}v_i^2}\le e^{n_1}\,\sum _{s=2}^{n_1} \,(D't)^{n_1-s}\, (4/t)^{\gamma _1} t \,\varepsilon ^{\gamma _1} \,s! \, 2s \nonumber \\&\quad \quad \times \frac{t^{s-1}}{(s-1)!}\, \int d{\varvec{\omega }}_s \int d{\varvec{\eta }}^-_{s+1} \nonumber \\&\quad \quad \times \left( \frac{e^{-(\beta /2)\sum _{i = 1}^{1+s}(\eta ^-_i)^2}}{|P_{\omega _s}^{\perp } W_1\wedge \hat{W}|^{\gamma _1}} + \frac{e^{-(\beta /2)\sum _{i = 1}^{1+s}(\eta ^-_i)^2}}{|W_1\wedge \hat{W}|^{\gamma _1}} + \frac{e^{-(\beta /2)\sum _{i = 1}^{1+s}(\eta ^-_i)^2}}{|P_{\omega _s}^{\parallel } W_1\wedge \hat{W}|^{\gamma _1}}\right) ,\nonumber \\ \end{aligned}$$

(D.15)

where we renamed 1, 2, 3 particles $\ell ,h,i$ respectively (hence $W_1 = \eta ^-_1-\eta ^-_2, W = \eta ^-_2-\eta ^-_3$).

Let us now give a bound of the explicit integral $\int d{\varvec{\eta }}^-_{s+1} \frac{ e^{-(\beta /2)\sum _{i = 1}^{1+s}(\eta ^-_i)^2}}{|\tilde{P}_sW_1 \wedge \hat{W}|^{\gamma _1}}$, where $\tilde{P}_sW_1 = P_{\omega _s}^{\perp } W_1, W_1$ or $P_{\omega _s}^{\parallel } W_1$. Since $W^2+W_1^2 \le 2(\eta ^-_1)^2+4(\eta ^-_2)^2+2(\eta ^-_3)^2$, applying the translations $(\eta ^-_1,\eta ^-_2) \rightarrow (W_1 = \eta ^-_1-\eta ^-_2, W=\eta ^-_2-\eta ^-_3)$, we find

$$\begin{aligned}&\int d{\varvec{\eta }}^-_{s+1} \frac{ e^{-(\beta /2)\sum _{i = 1}^{1+s}(\eta ^-_i)^2}}{|\tilde{P}_sW_1\wedge \hat{W}|^{\gamma _1}} \le \int d{\varvec{\eta }}^-_{s+1} e^{-(\beta /2)\sum _{\begin{array}{c} i> 3 \end{array}}(\eta ^-_i)^2}\nonumber \\&\quad \quad \times \,e^{-(\beta /4)(\eta ^-_3)^2}\, \frac{e^{-(\beta /8)(W_1^2+W^2)} }{|\tilde{P}_sW_1\wedge \hat{W}|^{\gamma _1}}\nonumber \\&\quad = \int d{\varvec{\eta }}^-_{2,s-1} \,e^{-(\beta /2)\sum _{\begin{array}{c} i > 3 \end{array}}(\eta ^-_i)^2} \,e^{-(\beta /4)(\eta ^-_3)^2}\, \int dW_1 dW\frac{e^{-(\beta /8)(W_1^2+W^2)} }{|\tilde{P}_sW_1\wedge \hat{W}|^{\gamma _1}}\nonumber \\&\quad \le C^s_\beta \int dW_1 \frac{e^{-(\beta /8)W_1^2} }{|\tilde{P}_sW_1|^{\gamma _1}}\nonumber \\&\quad \le C^s_\beta \, C_{\beta ,\gamma _1}, \end{aligned}$$

(D.16)

for suitable constants $C_\beta ,C_{\beta ,\gamma _1}>0$ and for any $\gamma _1<1$ (with $C_{\beta ,\gamma _1}$ diverging in the case $\tilde{P}_sW_1=P_{\omega _s}^{\parallel } W_1$ as $\gamma _1 \rightarrow 1$).

Inserting (D.16) into (D.15) and performing the sums, we obtain the final result. $\square$

Fig. 8

Case 1: $h = s+1$, $k_s = \ell$. Case 2: $h = k_s$, $\ell = s+1$

1.5 Appendix E: Proof of Corollary 4.14

The result follows from minor modifications in the proof of Theorem 2.4.

First of all, by Property 2, case $S = {\mathcal J}= J$, applied to the state with r.c.f. $f^\varepsilon _j$ and correlation errors $\bar{E}_k \equiv E_k$,

$$\begin{aligned} |E_K|\le & {} \sum _{H \subset K} \left( C^{h}\, h!\, \chi ^0_{H,K}\, \left( f^{\varepsilon }_{1}\right) ^{\otimes H}\right) \, \left( \bar{\chi }^0_{K {\setminus } H} |E_{K {\setminus } H}|\right) \nonumber \\= & {} \sum _{H \subset K {\setminus } Q'} \left( C^{h}\, h!\, \chi ^0_{H,K}\, \left( f^{\varepsilon }_{1}\right) ^{\otimes H}\right) \, \left( \bar{\chi }^0_{K {\setminus } (Q' \cup H)} |E_{K {\setminus } H}|\right) . \end{aligned}$$

(E.1)

Remind that $\chi ^0_{H,K}=1$ if and only if any particle with index in H overlaps with a different particle in K, which implies $H \subset K {\setminus } Q'$. Moreover, $\bar{\chi }^0_{K {\setminus } H}=1$ if and only if all particles in $K {\setminus } H$ do not overlap among themselves, which implies $\bar{\chi }^0_{K {\setminus } H} = \bar{\chi }^0_{K {\setminus } (Q' \cup H)}$. In particular,

$$\begin{aligned} \int _{{\mathbb R}^{3q'}} d\mathbf{v }_{Q'} |E_K(t)|\le & {} \sum _{H \subset K {\setminus } Q'} \left( C^{h}\, h!\, \chi ^0_{H,K}\, \left( f^{\varepsilon }_{1}(t)\right) ^{\otimes H}\right) \, \bar{\chi }^0_{K {\setminus } (Q' \cup H)}\nonumber \\&\times \int _{{\mathbb R}^{3q'}} d\mathbf{v }_{Q'} |E_{K {\setminus } H}(t)| \end{aligned}$$

(E.2)

and we are allowed to insert expression (4.27) into (E.2). Note that this preparation is necessary, since (4.27) does apply only when the particles in $K {\setminus } H$ are sufficiently far from each other.

The estimate of the integral on the r.h.s. differs from that of the main theorem from the fact that we integrate only with respect to a subset of velocities $Q' \subset K$. Furthermore, we know that particles in $Q'$ are at distance larger than $\delta$ from any other particle in K, but we have no information on the relative distance of particles in $K {\setminus } Q'$. We shall not repeat here the proof of Sects. 4.3–4.5, which applies unchanged, except for the following modifications.

• 1. In Lemma 4.5, one integrates only over $d\mathbf{v }_{Q'}$. However the integral over velocities is never used in the reduction to energy functionals (see Appendix C). Therefore one gets the same result apart from an overall $(const.)^k$. This produces the first term in (4.128).

• 2. In Proposition 4.6, one integrates only over $d\mathbf{v }_{Q'}$ and $\varepsilon ^{\gamma _1 \frac{q + l_0}{2}}= \varepsilon ^{\gamma _1 \frac{|Q \cup L_0|}{2}}$ has to be replaced by $\varepsilon ^{\gamma _1 \frac{|(Q \cup L_0)\cap Q'|}{2}}$. Indeed in the proof of the proposition, Sect. 4.4.2.c, when the bullet $\alpha _i$ is outside $Q'$, Lemma 4.10 cannot be applied. Instead of estimate (4.51), one uses then the simple estimate

  $$\begin{aligned} \sum _{\Gamma _{\alpha _i}} \,\int d {\Lambda }_{\alpha _i} \, \chi ^{(\alpha _i,\beta _i)}\,F_{\theta _3}(\alpha _i) \le (D't)^{n_{\alpha _i}}. \end{aligned}$$

  (E.3)

• 3. In (4.36), one integrates only over $d\mathbf{v }_{Q'}$ and, by virtue of the previous two points, one gets $\varepsilon ^{\min [\gamma _0,\gamma _1/2] q'}$ instead of $\varepsilon ^{\min [\gamma _0,\gamma _1/2] k}$. This produces the second term in (4.128). $\square$