1.1 Previous work and the goal of this paper

Motivated by the fact that the Boltzmann equation is valid only for very dilute gases and by the observations of [Reference Russ and von Grünberg29] in [Reference Ampatzoglou and Pavlović5], we suggested a kinetic model which goes beyond binary interactions incorporating sums of higher-order interaction terms. In particular, we introduced a generalized equation, which could serve as a toy model for incorporating higher-order interactions among particles and is of the form

(1.1) $$ \begin{align} \begin{cases} \partial_t f+v\cdot\nabla_xf=\displaystyle\sum_{k=2}^m Q_k(\underbrace{f,f,\cdots,f\,}_{k\text{-times}}),\quad (t,x,v)\in(0,\infty)\times\mathbb{R}^{d}\times\mathbb{R}^d,\\ f(0,x,v)=f_0(x,v),\quad (x,v)\in\mathbb{R}^{d}\times\mathbb{R}^d, \end{cases} \end{align} $$

where, for $k=1,...,m$ , the expression $Q_k(f,...,f)$ is the k-th order collisional operator and $m\in \mathbb {N}$ is the highest order collisions allowed. Notice that for $m=2$ , equation (1.1) reduces to the classical Boltzmann equation. We note that equations similar to (1.1) were studied for Maxwell molecules in the works of Bobylev, Gamba and Cercignani [Reference Bobylev, Gamba, Cercignani, Capriz, Giovine and Mariano8, Reference Bobylev, Gamba and Cercignani7] using Fourier transform methods.

The task of rigorously deriving an equation of the form (1.1) from a classical many particle system, even for the case $m=2$ (i.e., the Boltzmann equation), is a challenging problem that has been first settled for short times and hard sphere interactions in the pioneering work of Lanford [Reference Lanford26], and for short range potentials by King [Reference King25]. This program was revisited by Gallagher, Saint-Raymond, Texier in [Reference Gallagher, Saint-Raymond and Texier18], where important quantitative information on the convergence was provided. See also [Reference Cercignani12, Reference Pulvirenti, Saffirio and Simonella28, Reference Spohn30, Reference Uchiyama31, Reference Gerasimenko and Gapyak19] and the references mentioned in these papers. More recent works related to derivation of the Boltzmann equation itself have been carried out using the notion of fluctuations in, for example, [Reference Bodineau, Gallagher, Saint-Raymond and Simonella9, Reference Bodineau, Gallagher, Saint-Raymond and Simonella10, Reference Gerasimenko and Gapyak20]. Regarding longer times, the equation was derived for hard spheres for long times originally only for initial data near vacuum in [Reference Illner and Pulvirenti24]. However, recently, a different derivation has been carried out by Deng, Hani and Ma [Reference Deng, Hani and Ma15] as long as the Boltzmann equation itself is well-posed.

A relevant step towards rigorously deriving (1.1) for $m=3$ has been recently obtained in [Reference Ampatzoglou and Pavlović5], where we considered a certain type of three-particle interactions that lead us to derive a purely ternary kinetic equation, which we called a ternary Boltzmann equation. However, the derivation of (1.1) for $m=3$ has not been addressed yet, and that is exactly what we do in this paper. Furthermore, we expect that this paper can serve as a guideline for rigorously deriving generalized Boltzmann equation.

We start by describing challenges that we faced when introducing a framework that allows detection of binary and ternary interactions, while also accommodating a decoupling of such interactions so that it is clear which one is responsible for a creation of a binary or ternary collision terms in the nonlinear equation (1.1).

1.2 Challenges of detecting both binary and ternary interactions

The first challenge we face in deriving (1.1) for $m=3$ is to provide a mathematical framework allowing us to detect both binary and ternary interactions among particles. We achieve that by assuming the following:

• • Binary interactions are modeled as elastic collisions of hard spheres of diameter $\epsilon $ (i.e., two particles interact when the distance of their centers defined as

  $$ \begin{align*}d_2(x_i,x_j):=|x_i-x_j|\end{align*} $$
  becomes equal to the diameter $\epsilon $ ). We call this an $(i,j)$ interaction. As known, the relevant scaling to observe binary interactions is the Boltzmann-Grad scaling [Reference Grad21, Reference Grad22]
  (1.2) $$ \begin{align} N\epsilon^{d-1}\simeq 1, \end{align} $$
  as the number of particles $N\to \infty $ and their diameter $\epsilon \to 0^+$ .

• • Ternary interactions that we consider in this paper are going to be of an interaction zone type as in [Reference Ampatzoglou and Pavlović5], by which we mean a particle i interacts with the pair of uncorrelated particles $(j,k)$ when the non-symmetric ternary distance

  $$ \begin{align*}d_3(x_i;x_j,x_k):=\sqrt{|x_i-x_j|^2+|x_i-x_k|^2 }\end{align*} $$
  becomes $\sqrt {2}\epsilon $ . We call this an $(i;j,k)$ interaction. The particle i is called the central particle of the interaction, and the particles $j,k$ are called adjacent particles.In terms of scaling, one could interpret an $(i;j,k)$ interaction of interaction zone $\epsilon $ as a special hard sphere interaction of radius $\sqrt {2}\epsilon $ in $\mathbb {R}^{2d}$ since the collisional condition $d_3(x_i;x_j,x_k)=\sqrt {2}\epsilon $ can be equivalently written as
  $$ \begin{align*}|\boldsymbol{x_{i,i}}-\boldsymbol{x_{j,k}}|_{2d}=\sqrt{2}\epsilon,\end{align*} $$
  where $\boldsymbol{x_{i,i}}=\displaystyle \binom {x_i}{x_i}$ and $\boldsymbol{x_{j,k}}=\displaystyle \binom {x_j}{x_k}$ . Then a $2d$ -particle with position $\boldsymbol{x_{i,i}}$ would span a volume of order $\epsilon ^{2d-1}$ in a unit of time. Assuming there are N-particles in the system, in order to observe $O(1)$ interaction per unit of time, there are $N^2-1$ options for the $2d$ -particle positioned at $\boldsymbol{x_{j,k}}$ . We obtain that $N^2\epsilon ^{2d-1}= O(1)$ , or equivalently,
  (1.3) $$ \begin{align} N\epsilon^{d-1/2}\simeq 1, \end{align} $$
  as the number of particles $N\to \infty $ and the interaction zone $\epsilon \to 0^+$ , which is the scaling used in [Reference Ampatzoglou and Pavlović5] to control ternary interactions.

Simultaneous consideration of both binary and ternary interactions brings the first crucial obstacle which is of conceptual nature; the apparent incompatibility of the Boltzmann-Grad scaling (1.2) dictated by binary interactions and the scaling (1.3) of ternary interactions, if both of them are of order $\epsilon $ . This incompatibility creates major difficulties even at the formal level. We overcome this scaling obstacle by assuming that, at the N-particle level, hard spheres of diameter $\epsilon _2$ can participate in binary interactions as well as in ternary interactions via an interaction zone $\epsilon _3$ . Imposing scalings (1.2) with $\epsilon :=\epsilon _2$ and (1.3) with $\epsilon :=\epsilon _3$ , we obtain the common scaling

(1.4) $$ \begin{align} N\epsilon_2^{d-1}\simeq N\epsilon_3^{d-1/2}\simeq 1, \end{align} $$

as $N\to \infty $ and $\epsilon _2,\epsilon _3\to 0^+$ . Notice that the scaling (1.4) implies that for sufficiently large N, we have

(1.5) $$ \begin{align} \epsilon_2<<\epsilon_3, \end{align} $$

which will have a prominent role in this paper.

Figure 1

Both binary and ternary interactions at the same time.

The next challenge we address is the need to decouple binary and ternary interactions for a system of finitely many particles. More precisely, our framework a-priori allows that particles i and j interact as hard spheres:

$$ \begin{align*}d_2(x_i,x_j)=\epsilon_2,\end{align*} $$

while at the same time there is another particle k such that the particle i interacts with the particles j and k:

$$ \begin{align*}d_3(x_i;x_j,x_k)=\sqrt{2}\epsilon_3.\end{align*} $$

Such a configuration is illustrated in Figure 1. Pathological configurations, including the one we just described, are going to be shown to be negligible. This is far from trivial, and for more details on the microscopic dynamics, see Subsection 1.3 and Section 3. In particular, we shall show that as long as $0<\epsilon _2<\epsilon _3<1$ , only the following two interaction scenarios are possible with nontrivial probability under time evolution:

1. 1. Two particles interact as hard spheres, while all other particles are not involved in any binary or ternary interactions at the same time. This type of configurations generates the binary collisional operator. It is illustrated in Figure 2.

   Figure 2

   Binary interaction: $\epsilon _2^2+\lambda _2^2>2\epsilon _3^2, \quad \lambda _2>\epsilon _2$ .

   
   Figure 3

   Ternary interaction: $\lambda _1^2+\lambda _2^2=2\epsilon _3^2,\quad \lambda _1,\lambda _2>\epsilon _2$ .

   

2. 2. Three particles interact via an interaction zone, while none of them is involved in a binary interaction with either of the other two particles of the interaction zone at the same time. The rest of the particles are not involved in any binary or ternary interactions. This type of configurations is responsible for generating the ternary collisional operator. It is illustrated in Figure 3.

Finally, since we will eventually let the number of particles $N\to \infty $ , the main challenge we need to address is the stability of a good configurationFootnote ¹ under the adjunction of one or two collisional particles. Assume, for a moment, that we have a good configuration of m-particles and we add $\sigma $ particles to the system, where $\sigma \in \{1,2\}$ , such that a binary or ternary interaction is formed among one of the existing particles and the $\sigma $ new particles. In general, under backwards time evolution, the system could run into another binary or ternary interaction; see, for example, Figure 4, which illustrates the mathematically most difficult case where the newly formed $(m+2)$ -configuration runs into a binary interaction. To the best of our knowledge, this is the first time there was the need to address the possibility of a newly formed interacting configuration running into an interaction of a different type (binary to ternary or ternary to binary) backwards in time. However, in Section 8 and Section 9, we develop novel algebraic and geometric tools which help us eliminate pathological scenarios, including the one described in Figure 4, by showing that outside of a small measure set, negligible in the limit, the newly formed configuration does not run into any additional interactions backwards in time. For more details on the technical difficulties faced, see Subsection 1.6.

Figure 4

In the next subsection, we investigate more precisely what happens when a binary or a ternary interactions occurs and describe the time evolution of such a system.

1.3 Dynamics of finitely many particles

Let us describe the evolution in $\mathbb {R}^d$ , $d\geq 2$ , of a system of N hard spheres of diameter $\epsilon _2$ and interaction zone $\epsilon _3$ , where $0<\epsilon _2<\epsilon _3<1$ . The assumption $\epsilon _2<\epsilon _3$ is necessary for ternary interactions to be of non trivial probability; see Remark 3.1 for more details.

1.3.1 Interactions considered

We first define the interactions considered in this paper.

Definition 1.1. Let $N\in \mathbb {N}$ , with $N\geq 3$ , and $0<\epsilon _2<\epsilon _3<1$ . We define binary and ternary interactions, also referred to as collisions, as follows:

• • Consider two particles $i,j\in \{1,...,N\}$ with positions $x_i,x_j\in \mathbb {R}^{d}$ . We say that the particles $i,j$ are in an $(i,j)$ binary interaction if the following geometric condition holds:

  (1.6) $$ \begin{align} d_2(x_i,x_j):=|x_i-x_j|=\epsilon_2. \end{align} $$

• • Consider three particles $i,j,k\in \{1,...,N\}$ , with positions $x_i,x_j,x_k\in \mathbb {R}^{d}$ . We say that the particles $i,j,k$ are in an $(i;j,k)$ interactionFootnote ² if the following geometric condition holds:

  (1.7) $$ \begin{align} d_3(x_i;x_j,x_k):=\sqrt{|x_i-x_j|^2+|x_i-x_k|^2}=\sqrt{2}\epsilon_3. \end{align} $$

When an $(i,j)$ interaction occurs, the velocities $v_i,v_j$ of the i-th and j-th particles instantaneously transform according to the binary collisional law:

(1.8) $$ \begin{align} \begin{aligned} v_i^{\prime} &=v_i+\langle\omega_1,v_j-v_i\rangle\omega_1,\\ v_j^{\prime}&=v_j-\langle\omega_1,v_j-v_i\rangle\omega_1, \end{aligned} \end{align} $$

where

(1.9) $$ \begin{align} \omega_1:=\frac{x_j-x_i}{\epsilon_2}. \end{align} $$

Thanks to (1.6), we have $\omega _1\in \mathbb {S}_1^{d-1}$ . The vector $\omega _1$ is called binary impact direction and it represents the scaled relative position of the colliding particles. Moreover, one can see that the binary momentum-energy system

(1.10) $$ \begin{align} \begin{aligned} v'+v_1^{\prime}&=v+v_1,\\ |v'|^2+|v_1^{\prime}|^2&=|v|^2+|v_1|^2, \end{aligned} \end{align} $$

is satisfied.

When an $(i;j,k)$ interaction happens, the velocities $v_i,v_j,v_k$ of the i-th, j-th and k-th particles instantaneously transform according to the ternary collisional law derived in [Reference Ampatzoglou and Pavlović5]

(1.11) $$ \begin{align} \begin{aligned} v_i^{*}&=v_i+\frac{\langle\omega_1,v_j-v_i\rangle+\langle\omega_2,v_k-v_i\rangle}{1+\langle\omega_1,\omega_2\rangle}(\omega_{1}+\omega_{2}),\\ v_j^{*}&=v_j-\frac{\langle\omega_1,v_j-v_i\rangle+\langle\omega_2,v_k-v_i\rangle}{1+\langle\omega_1,\omega_2\rangle}\omega_{1},\\ v_k^{*}&=v_k-\frac{\langle\omega_1,v_j-v_i\rangle+\langle\omega_2,v_k-v_i\rangle}{1+\langle\omega_1,\omega_2\rangle}\omega_2, \end{aligned} \end{align} $$

where

(1.12) $$ \begin{align} (\omega_1,\omega_2):=\left(\frac{x_j-x_i}{\sqrt{2}\epsilon_3},\frac{x_k-x_i}{\sqrt{2}\epsilon_3}\right). \end{align} $$

Thanks to (1.7), we have $(\omega _1,\omega _2)\in \mathbb {S}_1^{2d-1}$ . The vectors $(\omega _1,\omega _2)$ are called ternary impact directions, and they represent the scaled relative positions of the interacting particles. Moreover, it has been shown that the ternary momentum-energy system

(1.13) $$ \begin{align} \begin{aligned} v^{*}+v_1^{*}+v_2^{*}&=v+v_1+v_2,\\ |v^{*}|^2+|v_1^{*}|^2+|v_2^{*}|^2&=|v|^2+|v_1|^2+|v_2|^2, \end{aligned} \end{align} $$

is satisfied. In particular, expression (1.11) provides the unique solution to (1.13) equipped with the extra condition

$$ \begin{align*} \begin{aligned} v_2^{*}=v_2+c\omega_1,\quad v_3^{*}=v_3+c\omega_2,\quad c\in\mathbb{R}. \end{aligned} \end{align*} $$

We note that we had a choice in selecting the additional condition to uniquely solve (1.13). However, the one we chose in this work expresses the uncorrelation of the adjacent particles since their velocities are transformed uniformly with respect to the impact directions.

Remark 1.2. We note that both binary and ternary interactions are involutionary (i.e., reversible and measure-preserving). For more details, see Proposition 2.2 and Proposition 2.5 for binary and ternary interactions, respectively.

1.3.2 Phase space and description of the flow

Let $N\in \mathbb {N}$ , with $N\geq 3$ , and $0<\epsilon _2<\epsilon _3<1$ . The natural phase spaceFootnote ³ to capture both binary and ternary interactions is

(1.14) $$ \begin{align} \begin{aligned} \mathcal{D}_{N,\epsilon_2,\epsilon_3}=\big\{Z_N=(X_N,V_N)\in\mathbb{R}^{2dN}: d_2(x_i,x_j)\geq\epsilon_2,\text{ }\forall (i,j)\in\mathcal{I}_N^2, \\ \text{ and }d_3(x_i;x_j,x_k)\geq\sqrt{2}\epsilon_3,\text{ }\forall (i,j,k)\in\mathcal{I}_N^3\big\}, \end{aligned} \end{align} $$

where $X_N=(x_1,x_2,...,x_N)$ , $V_N=(v_1,v_2,...,v_N),$ represent the positions and velocities of the N-particles, and the index sets $\mathcal {I}_N^2,\mathcal {I}_N^3$ are given by

$$ \begin{align*} \mathcal{I}_N^2=\{(i,j)\in\{1,...,N\}^2:i<j\},\quad \mathcal{I}_N^3=\{(i,j)\in\{1,...,N\}^3:i<j<k\}. \end{align*} $$

Let us describe the evolution in time of such a system. Consider an initial configuration $Z_N\in \mathcal {D}_{N,\epsilon _2,\epsilon _3}$ . The motion is described as follows:

1. 1. Particles are assumed to perform rectilinear motion as long as there is no interaction

   $$ \begin{align*} \dot{x}_i=v_i,\quad \dot{v}_i=0,\quad\forall i\in\{1,...,N\}. \end{align*} $$

2. 2. Assume now that an initial configuration $Z_N=(X_N,V_N)$ has evolved until time $t>0$ , reaching $Z_N(t)=(X_N(t),V_N(t))$ , and that there is an interaction at time t. We have the following cases:

   • • The interaction is binary: Assuming there is an $(i,j)$ interaction, the velocities of the interacting particles instantaneously transform velocities according to the binary collisional law $(v_i(t),v_j(t))\to (v_i^{\prime }(t),v_j^{\prime }(t))$ given in (1.8).

   • • The interaction is ternary: Assuming there is an $(i;j,k)$ interaction, the velocities of the interacting particles instantaneously transform velocities according to the ternary collisional law

     $$ \begin{align*}(v_i(t),v_j(t),v_k(t))\to (v_i^{*}(t),v_j^{*}(t),v_k^{*}(t)),\end{align*} $$
     given in (1.11).

Let us note that (I)–(II) are not sufficient to generate a global in time flow for the particle system since the velocity transformations are not smooth. In general, pathologies might arise as time evolves, meaning more than one type of interactions happening at the same time, grazing interaction, or infinitely many interactions in finite time. Although well-defined dynamics were shown to exist in [Reference Alexander1] for hard spheres and in [Reference Ampatzoglou and Pavlović5] for the purely ternary case, those results do not imply well-posedness of the flow for the mixed case, where both binary and ternary interactions are taken into account. The reason for that is that a binary interaction can be succeeded by a ternary interaction and vice versa, a situation which was not addressed in [Reference Alexander1] or [Reference Ampatzoglou and Pavlović5]. However, we are showing that a non-grazing interaction cannot be succeeded by the same interaction. In other words, when two particles $(i,j)$ interact, the next interaction could be anything, binary or ternary, except a binary recollision of the particles $(i,j)$ . Similarly, when three particles run into an $(i;j,k)$ interaction, the next interaction can be anything except a ternary $(i;j,k)$ Footnote ⁴ interaction. This observation allows us to define the flow locally a.e. and then run some combinatorial covering arguments to geometrically exclude a zero Lebesgue measure set such that the flow is globally in time defined on the complement.

Let us informally state this result. For a detailed statement, see Theorem 3.23.

Existence of a global flow: Let $N\in \mathbb {N}$ and $0<\epsilon _2<\epsilon _3<1$ . There is a global in time measure-preserving flow $(\Psi _m^t)_{t\in \mathbb {R}}:\mathcal {D}_{N,\epsilon _2,\epsilon _3}\to \mathcal {D}_{N,\epsilon _2,\epsilon _3}$ described a.e. by (I)-(II) which preserves kinetic energy and and is time reversible. This flow is called the N-particle $(\epsilon _2,\epsilon _3)$ -interaction flow.

The global measure-preserving interaction flow yields the Liouville equationFootnote ⁵ for the evolution $f_N$ of an initial N-particle probability density $f_{N,0}$ .Footnote ⁶ Footnote ⁷

(1.15) $$ \begin{align} \begin{aligned} &\partial_tf_N+\sum_{i=1}^Nv_i\nabla_{x_i}f_N=0,\quad (t,Z_N)\in (0,\infty)\times\mathring{D}_{N,\epsilon_2,\epsilon_3},\\ &f_N(t,Z_N^{\prime})=f(t,Z_N),\quad t\in[0,\infty),\quad Z_N\text{ is a simple binary interaction},\\ &f_N(t,Z_N^{*})=f(t,Z_N),\quad t\in[0,\infty),\quad Z_N\text{ is a simple ternary interaction},\\ &f_N(0,Z_N)=f_{N,0}(Z_N),\quad Z_N\in\mathring{D}_{N,\epsilon_2,\epsilon_3}. \end{aligned} \end{align} $$

The Liouville equation provides a complete deterministic description of the system of N-particles. Although Liouville’s equation is a linear transport equation, efficiently solving it is almost impossible in the case where the particle number N is very large. This is why an accurate kinetic description is welcome, and to obtain it, one wants to understand the limiting behavior of it as $N\to \infty $ and $\epsilon _2,\epsilon _3\to 0^+$ , with the hope that qualitative properties will be revealed for a large but finite N.

1.4 The binary-ternary Botzmann equation

To obtain such a kinetic description, we let the number of particles $N\to \infty $ and the diameter and interaction zone of the particles $\epsilon _2,\epsilon _3\to 0^+$ in the common scaling (1.4):

$$ \begin{align*} N\epsilon_2^{d-1}\simeq N\epsilon_3^{d-\frac{1}{2}}\simeq 1, \end{align*} $$

which will lead the binary-ternary Boltzmann equation

(1.16) $$ \begin{align} \begin{cases} \partial_t f+v\cdot\nabla_x f=Q_2(f,f)+Q_3(f,f,f), \quad (t,x,v)\in (0,\infty)\times\mathbb{R}^d\times\mathbb{R}^d,\\ f(t=0)=f_0(x,v),\quad (x,v)\in\mathbb{R}^d\times\mathbb{R}^d. \end{cases} \end{align} $$

The operator $Q_2(f,f)$ (see, for example, [Reference Cercignani, Illner and Pulvirenti13]) is the classical hard sphere binary collisional operator given by

(1.17) $$ \begin{align} Q_2(f,f)=\int_{\mathbb{S}_1^{d-1}\times\mathbb{R}^d}b_2^+\left(f'f_1^{\prime}-ff_1\right)\,d\omega_1\,dv_1, \end{align} $$

where

$$ \begin{align*} \begin{aligned} &b_2=\langle\omega_1,v_1-v\rangle,\quad b_2^+=\max\{b_2,0\},\\ &f'=f(t,x,v'),\quad f=f(x,t,v),\quad f_1^{\prime}=f_1(t,x,v_1^{\prime}),&\quad f_1=f(t,x,v_1). \end{aligned} \end{align*} $$

The operator $Q_3(f,f,f)$ , introduced for the first time in [Reference Ampatzoglou and Pavlović5], is the ternary hard interaction zone operator given by

(1.18) $$ \begin{align} Q_3(f,f,f)=\int_{\mathbb{S}_1^{2d-1}\times\mathbb{R}^{2d}}b_3^+\left(f^{*}f_1^{*}f_2^{*}-ff_1f_2\right)\,d\omega_1\,d\omega_2\,dv_1\,dv_2, \end{align} $$

where

(1.19) $$ \begin{align} \begin{aligned} b_3(\omega_1,\omega_2,v_1-v,v_2-v):=\langle\omega_1,v_{1}-v\rangle+\langle\omega_2,v_{2}-v\rangle,\quad b_3^+=\max\{b_3,0\},&\\ f^{*}=f(t,x,v^{*}),\quad f=f(x,t,v),\quad f_i^{*}=f_i^{*}(t,x,v_i^{*}),\quad f_i=f(t,x,v_i), \,i\in\{1,2\}.& \end{aligned} \end{align} $$

We should mention that in [Reference Ampatzoglou, Gamba, Pavlović and Tascović3], global well-posedness near vacuum has been shown for (1.16) for potentials ranging from moderately soft to hard in spaces of functions bounded by Maxwellian. In fact, in [Reference Ampatzoglou, Gamba, Pavlović and Tascović3], it is seen that the ternary collisional operator allows consideration of softer potentials that the binary operator. In other words, the ternary correction to the Boltzmann equation does not behave worse than the classical Boltzmann equation.

It is important to point out that, upon symmetrization of the ternary collisional operator (see [Reference Ampatzoglou2], [Reference Ampatzoglou, Gamba, Pavlović and Tasković4]), the corresponding binary-ternary Boltzmann equation enjoys similar statistical and entropy production properties, as well as conservation laws, as the classical Boltzmann equation. Inspired by this fact, in collaboration with Gamba, Tasković [Reference Ampatzoglou, Gamba, Pavlović and Tasković4], we studied the generation and propagation of polynomial and exponential moments, as well as the global well-posedness, of the space homogeneous binary-ternary Boltzmann equation. Interestingly, the results of [Reference Ampatzoglou, Gamba, Pavlović and Tasković4] show that the co-existence of binary and ternary collisions yields better generation properties and time decay than when only binary or ternary collisions are considered. This suggests that such a model could potentially serve as a correction of the classical Boltzmann equation.

Recently, in collaboration with Warner [Reference Ampatzoglou, Pavlović and Warner6], based on ideas introduced in the current paper, we were able to derive an equation of the type (1.1) for arbitrary order collisions based on a symmetric distance/collisional law among the particles. In other words, unlike the asymmetry present in the definition of the ternary distance (1.7), in [Reference Ampatzoglou, Pavlović and Warner6] particles are fully interchangeable.

1.5 Strategy of the derivation and statement of the main result

In order to pass from the N-particle system dynamics to the kinetic equation (1.16), we implement the program of constructing linear finite and infinite hierarchies of equations, pioneered by Lanford [Reference Lanford26] and refined by Gallagher, Saint-Raymond, Texier [Reference Gallagher, Saint-Raymond and Texier18], and connecting them to the new binary-ternary Boltzmann equation. In [Reference Ampatzoglou and Pavlović5], we extended this program to include ternary interactions, which led to the rigorous derivation of a purely ternary kinetic equation for particles with hard interaction zone in the scaling (1.3). However, rigorous derivation of (1.16) does not follow from [Reference Lanford26, Reference Gallagher, Saint-Raymond and Texier18] or the ternary work [Reference Ampatzoglou and Pavlović5]. As mentioned in Subsection 1.2, the first difficulty is the apparent incompatibility of scalings (1.2)-(1.3), which we overcome by introducing the common scaling (1.4). The most challenging task is to make the argument rigorous, though, is the analysis of all the possible recollisionsFootnote ⁸ of the backwards $(\epsilon _2,\epsilon _3)$ -flow. In contrast to the binary or the ternary case where each binary or ternary interaction is succeeded by a binary or ternary interaction, respectively, here we can have any possible interaction sequence of binary or ternary interactions. We keep track of this combinatorics using the set

(1.20) $$ \begin{align} S_k=\{\sigma=(\sigma_1,...,\sigma_k):\sigma_i\in\{1,2\},\quad\forall i=1,...,k\}. \end{align} $$

In addition to more involved combinatorics, careful analysis of all the possible interaction sequences requires development of novel geometric and algebraic tools, which we discuss in details in Subsection 1.6. For now, we continue to discuss the process of derivation.

More specifically, we first derive a finite, linear, coupled hierarchy of equations for the marginal densities

of the solution $f_N$ to the Liouville equation, which we call the BBGKY.Footnote ⁹ This hierarchy is given by

(1.21) $$ \begin{align} \partial_t f_N^{(s)}+\sum_{i=1}^sv_i\cdot\nabla_{x_i}f_N^{(s)}=\mathcal{C}_{s,s+1}^Nf_N^{(s+1)}+\mathcal{C}_{s,s+2}^Nf_N^{(s+2)},\quad s\in\{1,...,N-1\}. \end{align} $$

For the precise form of the operators $\mathcal {C}_{s,s+1}^N$ , $\mathcal {C}_{s,s+2}^N$ , see (4.15)–(4.16). Duhamel’s Formula yields that the BBGKY hierarchy can be written in mild form as follows:

(1.22) $$ \begin{align} \hspace{2cm} f_N^{(s)}(t,Z_s)=T_s^tf_{N,0}(Z_s)+\int_0^t T_s^{t-\tau}(\mathcal{C}_{s,s+1}^Nf_N^{(s+1)}+\mathcal{C}_{s,s+2}^Nf_N^{(s+2)})(\tau,Z_s)\,d\tau,\quad s\in\mathbb{N}, \end{align} $$

where for any continuous function $g_s:\mathcal {D}_{s,\epsilon _2,\epsilon _3}\to \mathbb {R}$ , we write $T_s^tg_s(Z_s):=g_s(\Psi _s^{-t}Z_s),$ and $\Psi _s^t$ is the $(\epsilon _2,\epsilon _3)$ -interaction zone flow of s-particles.

We then formally let $N\to \infty $ and $\epsilon _2,\epsilon _3\to 0^+$ in the scaling (1.4) to obtain an infinite, linear, coupled hierarchy of equations, which we call the Boltzmann hierarchy. This hierarchy is given by

(1.23) $$ \begin{align} \partial_t f^{(s)}+\sum_{i=1}^sv_i\cdot\nabla_{x_i}f^{(s)}=\mathcal{C}_{s,s+1}^\infty f^{(s+1)}+\mathcal{C}_{s,s+2}^\infty f^{(s+2)},\quad s\in\mathbb{N}. \end{align} $$

For the precise form of the operators $\mathcal {C}_{s,s+1}^\infty $ , $\mathcal {C}_{s,s+2}^\infty $ , see (4.28), (4.32), respectively. Duhamel’s Formula yields that the Boltzmann hierarchy can be written in mild form as follows:

(1.24) $$ \begin{align} f^{(s)}(t,Z_s)=S_s^tf_{0}(Z_s)+\int_0^t S_s^{t-\tau}(\mathcal{C}_{s,s+1}^\infty f^{(s+1)}+\mathcal{C}_{s,s+2}^\infty f^{(s+2)})(\tau,Z_s)\,d\tau,\quad s\in\mathbb{N}, \end{align} $$

where for any continuous function $g_s:\mathbb {R}^{2ds}\to \mathbb {R}$ , we write $S_s^tg_s(Z_s):=g_s(\Phi _s^{-t}Z_s),$ and $\Phi _s^t$ is the s-particle free flow of s-particles defined by $S_s^t Z_s=S_s^t(X_s,V_s)=(X_s-tV_s,V_s).$

It can be observed that for factorized initial data and assuming that the solution remains factorized in time,Footnote ¹⁰ the Boltzmann hierarchy reduces to the binary-ternary Boltzmann equation (1.16). This observation connects the Boltzmann hierarchy with the binary-ternary Boltzmann equation (1.16).

To make this argument rigorous, we first show that the BBGKY and Boltzmann hierarchy are well-posed in the scaling (1.4), at least for short times, and then that the convergence of the BBGKY hierarchy initial data to the Boltzmann hierarchy initial data propagates in the time interval of existence of the solutions. Showing convergence is a very challenging task, and is the heart of our contribution. We describe details in Subsection 1.6.

Now, we informally state our main result. For a rigorous statement, see Theorem 6.8 and Corollary 6.10.

Statement of the main result: Let $F_0$ be initial data for the Boltzmann hierarchy (1.23), and $F_{N,0}$ be some BBGKY hierarchy (1.23) initial data which ‘approximate’ Footnote ¹¹ $F_0$ as $N\to \infty $ , $\epsilon \to 0^+$ under the scaling (1.4). Let $\boldsymbol{F_N}$ be the mild solution to the BBGKY hierarchy (1.21) with initial data $F_{N,0}$ , and $\boldsymbol{F}$ the mild solution to the Boltzmann hierarchy (1.23), with initial data $F_0$ , up to short time $T>0$ . Then $\boldsymbol{F_N}$ converges in observables Footnote ¹² to $\boldsymbol{F}$ in $[0,T]$ as $N\to \infty $ , $\epsilon \to 0^+$ , under the scaling (1.4). In the case of Hölder continuous $C^{0,\gamma }$ , $\gamma \in (0,1]$ tensorized Boltzmann hierarchy initial data and approximation by conditioned BBGKY hierarchy initial data, we obtain convergence to the solution of the binary-ternary Boltzmann equation (1.16) with a rate $O(\epsilon ^r)$ for any $0<r<\min \{1/2,\gamma \}.$

1.6 Difficulties faced in the proof of the main result

The main idea to obtain convergence (Theorem 6.8) is to inductively use mild forms (1.22), (1.24) of the BBGKY hierarchy and Boltzmann hierarchy, respectively, to formally obtain series expansions with respect to the initial data:

(1.25) $$ \begin{align} \begin{aligned} &f_N^{(s)}(t,Z_s)=T_s^tf_{N,0}^{(s)}(Z_s)+\\ &\sum_{k=1}^\infty\sum_{\sigma\in S_k}\int_0^t\int_0^{t_1}...\int_0^{t_{k-1}}T_s^{t-t_1}\mathcal{C}_{s,s+\widetilde{\sigma}_1}^NT_{s+\widetilde{\sigma}_1}^{t_1-t_2}...\mathcal{C}_{s+\widetilde{\sigma}_{k-1},s+\widetilde{\sigma}_k}^NT_{s+\widetilde{\sigma}_k}^{t_k}f_{N,0}^{(s+\widetilde{\sigma}_k)}(Z_s)\,dt_k...\,dt_1, \end{aligned} \end{align} $$

(1.26) $$ \begin{align} \begin{aligned} &f^{(s)}(t,Z_s)=S_s^tf_{0}^{(s)}(Z_s)+\\ &\sum_{k=1}^\infty\sum_{\sigma\in S_k}\int_0^t\int_0^{t_1}...\int_0^{t_{k-1}}S_s^{t-t_1}\mathcal{C}_{s,s+\widetilde{\sigma}_1}^\infty S_{s+\widetilde{\sigma}_1}^{t_1-t_2}...\mathcal{C}_{s+\widetilde{\sigma}_{k-1},s+\widetilde{\sigma}_{k}}^\infty S_{s+\widetilde{\sigma}_k}^{t_k}f_{0}^{(s+\widetilde{\sigma}_k)}(Z_s)\,dt_k...\,dt_1, \end{aligned} \end{align} $$

where $S_k$ is defined in (1.20), and given $\sigma \in S_k$ , $\ell =1,...,k$ , we write $\widetilde {\sigma }_\ell :=\sum _{i=1}^\ell \sigma _i$ . We note that the summation over $S_k$ in (1.25)-(1.26) allows us to keep track of the possible interaction sequences occurring by ‘adding’ one or two particles in each time step. For more details, see Section 7.

Comparing expressions (1.25)-(1.26), we expect to obtain the required convergence under the scaling (1.4) as long as $f_{N,0}^{(s)}$ ‘approximates’ $f_0^{(s)}$ under the same scaling. However it is not possible to directly compare (1.25)-(1.26) because of the possible divergence of the backwards interaction flow from the free flow, which we call recollisions. Although recollisions were also faced in [Reference Gallagher, Saint-Raymond and Texier18] and [Reference Ampatzoglou and Pavlović5], the mixed case, where both binary and ternary interactions are considered, requires different conceptual treatment in many instances and is not implied by the results of these works. The reason for that is that a binary interaction can be succeeded by a ternary interaction and vice versa, a situation which was not addressed in [Reference Gallagher, Saint-Raymond and Texier18, Reference Ampatzoglou and Pavlović5]. The key to overcome these difficulties is that the diameter of the particles is much smaller than the interaction zone, as implied by the common scaling (1.4). This fact allows us to develop certain delicate algebraic and geometric arguments to extract a small measure set of pathological initial data which lead to recollisions. On the complement of this set, expansions (1.25)-(1.26) are comparable and the required convergence is obtained.

The main idea for eliminating recollisions is an inductive application in each time step of Proposition 9.2 and Proposition 9.4, which treat the binary adjunction, or Proposition 9.6 and Proposition 9.7, which treat the ternary adjunction. More precisely, we face the following different cases:

1. 1. Binary adjunction: One particle is added forming a binary interaction with one of the existing particles. The pathological situations that might arise under backwards time evolution are the following:

   • • The newly formed binary collisional configuration runs to a binary interaction under time evolution. This pathological situation is eliminated using arguments inspired by [Reference Gallagher, Saint-Raymond and Texier18]. This is actually the only case which is similar to the cases covered in [Reference Gallagher, Saint-Raymond and Texier18].

   • • The newly formed binary collisional configuration runs to a ternary interaction under time evolution. This pathological situation did not appear in any of the previous works since merely binary or ternary interactions were studied. However, due to the fact that $\epsilon _2<<\epsilon _3$ , which comes from the scaling (1.4), this pathological situation can be treated using techniques inspired by [Reference Ampatzoglou and Pavlović5] and adapting them to the binary case.

   Proposition 9.2 and Proposition 9.4 are the relevant results controlling recollisions after a binary adjunction.

2. 2. Ternary adjunction: Two particles are added forming a ternary interaction with one of the existing particles. The pathological situations that might arise under backwards time evolution are the following:

   • • The newly formed ternary collisional configuration runs to a ternary interaction under time evolution. This case was studied in depth in [Reference Ampatzoglou and Pavlović5]. We eliminate this pathological situation using Proposition 9.5. For its proof, we refer to [Reference Ampatzoglou and Pavlović5].

   • • The newly formed ternary collisional configuration runs to a binary interaction under time evolution. This is the most challenging case to treat and is the heart of the technical contribution because the scaling (1.4) does not directly help as in the case of the binary adjunction where one of the collisional particles enters an interaction zone. To treat this case, we need to use new algebraic tools (see Proposition 9.6) to exclude sets of initial data which lead to these pathological trajectories and develop elaborate geometric estimates to control its measure. The geometric estimates needed are thoroughly presented in Section 8. In particular, Subsection 8.2 is devoted to developing novel tools which rely on an appropriate representation of $(2d-1)$ -spheres (see (8.1)). More specifically, in 8.2.1, we perform some initial truncations to the impact directions, while in 8.2.2, we establish certain spherical cap and conic region estimates needed to control the precollisional case, while 8.2.3 focuses on developing the necessary annuli estimates enabling us to control the postcollisional case using precollisional arguments. After establishing the necessary geometric tools, we employ them in Proposition 9.7 to show that the corresponding set constructed in Proposition 9.6 is negligible.

1.7 Notation

For convenience, we introduce some basic notation which will be frequently used throughout the manuscript:

• • $d\in \mathbb {N}$ will be a fixed dimension with $d\geq 2$ .

• • Given $x,y\in \mathbb {R}$ , we write $x\lesssim y$ if there is a constant $C_d>0$ such that $x\leq C_d y$ . Similarly, we write $x\simeq y$ if there is a constant $C_d>0$ such that $x=C_d y$ .

• • Given $n\in \mathbb {N}$ , $\rho>0$ and $w\in \mathbb {R}^n$ , we write $B_\rho ^n(w)$ for the n-closed ball of radius $\rho>0$ , centered at $w\in \mathbb {R}^n$ . In particular, we write $B_\rho ^n:=B_\rho ^n(0)$ for the $\rho $ -ball centered at the origin.

• • Given $n\in \mathbb {N}$ and $\rho>0$ , we write $\mathbb {S}_\rho ^{n-1}$ for the $(n-1)$ -sphere of radius $\rho>0$ .

• • When we write $x<<y$ , we mean that there is a small enough constant $0<c<1$ such that $x<cy$ .