Microscopic solutions of the Boltzmann-Enskog equation in the series representation

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.


Introduction
The present paper is devoted to the Boltzmann-Enskog equation, which describes the kinetics of a hard sphere gas: where the unknown f = f (x, v, t) ≥ 0 denotes the density function of the system, x ∈ R 3 , v ∈ R 3 and t ∈ R denote position, velocity and time respectively. Moreover a > 0 is the diameter of a hard sphere. If the function is normalized, i.e., collision configuration and (v , v 1 ) is the corresponding pair of outgoing velocities defined by the elastic reflection rules (with ω being the unit vector directed from the center of the first sphere to the center of the second one) Finally, λ > 0 is a parameter modulating the collision rate. The integral in (1) is referred to as the collision integral. This equation differs from the Boltzmann kinetic equation for hard spheres by the terms ±aω in the arguments of f in the collision integral. Namely the Boltzmann equation assumes the size of spheres to be negligibly small (in comparison to the scale of spatial variation of f ), while the Boltzmann-Enskog equation takes the size of spheres into account. Formally, the Boltzmann equation for hard spheres is obtained from the Boltzmann-Enskog equation in the limit a → 0. The convergence of solutions of the Boltzmann-Enskog equation to solutions of the Boltzmann equation is proved in [2,3,4]. Thus Eq. (1) is often regarded as a correction to the Boltzmann equation which is both a simpler mathematical model and a kinetic description of a "dense gas".
An important problem in the theory of the Boltzmann equation is its derivation from the microscopic dynamics (Hamiltonian particle system), see [8,22,12,16] and, for more recent research, [7,15,14,20,18,13]. Up to now, the Boltzmann equation for hard spheres and other short-range potentials has been rigorously derived for short times only. In contrast to the Boltzmann equation, it is not clear whether an Enskog type equation can be rigorously derived from deterministic dynamics.
It is interesting that, in case of hard spheres, Bogolyubov's microscopic derivation in [8] leads not to the Boltzmann equation but to the Boltzmann-Enskog equation (1). However, the latter equation does not provide a better approximation to the hard sphere dynamics, but just a natural intermediate description [20].
Moreover, the Boltzmann-Enskog equation has an interesting property, which relates it to the microscopic dynamics in a different way. Namely, Bogolyubov discovered [9] (see also [10]) that, for any finite N , the Boltzmann-Enskog equation with λ = N a 2 has solutions of the form of time-evolving empirical measures of N hard spheres: Here z = (x, v) ∈ R 6 is a point in the phase space and z i (t) = (x i (t), v i (t)) is the phase point of the i-th hard sphere determined by free motion between collisions plus elastic pairwise reflections at distance a. Setting z(t) = (z i (t)) N i=1 = T N (t)z 0 , and z 0 = (z 0 i ) N i=1 the time evolved and initial configuration respectively, the flow z 0 → z(t) is defined for almost all z 0 with respect to the Lebesgue measure. In fact configurations leading to collisions of more than two hard spheres simultaneously, to grazing collisions or to infinite collisions in a finite time have zero measure [1,11,12].
Solutions of type (3) are referred to as microscopic solutions. It may be surprising that the Boltzmann-Enskog equation "contains" the N -particle dynamics in itself, while the kinetic equation is valid in the limit of large N only.
Such solutions are even more surprising if we recall that the Boltzmann-Enskog equation describes the irreversible dynamics of the gas. Namely, consider the functional where B(x, a) is the ball around x and radius a, and ρ(x) = R 3 f (x, v) dv is the spatial density. This functional does not increase if f is a solution of the Boltzmann-Enskog equation [3]. In contrast, microscopic solutions are reversible in time (i.e., a transformation t → −t and v → −v maps a measure of type (3) to another measure of this type, which is also a solution of (1)). Of course, functional (4) is not defined on solutions of type (3), hence, formally, there is no contradiction. However, this shows that the reversibility or irreversibility of the Boltzmann-Enskog equation depends on the considered class of solutions: we have irreversible behaviour if we consider regular solutions, and reversible behaviour in the case of suitable singular measures (3).
A difficulty with the microscopic solutions (3) is that their formal substitution into (1) yields products of delta functions and, hence, it is ill-defined. A rigorous sense to these solutions was given in [25] by means of regularizations for delta functions and the collision integral (see also [23,26] for other variants). On the other hand it was proven in [19] that the empirical distributions (3) (more precisely the family of the empirical marginals) solve the BBGKY hierarchy for hard spheres. Actually, Bogolyubov's remark in [9] was that the Boltzmann-Enskog equation coincides formally with the first equation of the BBGKY hierarchy for hard spheres.
As discussed in [19], the approach therein developed, based on the standard notion of Duhamel series solution, has no simple adaptation to (1). In the present paper, we introduce a modified notion of series solution for which (3) does solve the Boltzmann-Enskog equation. Such a notion is based on tree expansions with partially ordered trees, in contrast with the standard expansion on totally ordered trees. This, together with a regularization based on separation of collision times, allows to formulate our main result (Theorem 4.1 in Section 4 below).
The paper is organized as follows. In Section 2 we introduce the trees and partially ordered trees together with the standard series solution. In Section 3 the concept of series solution for measures of type (3) is precisely formulated. We establish also a semigroup property which will be crucial, in Section 4, for the proof of Theorem 4.1. In the latter section we finally compare the strategy of [19] with the one of the present paper.
2 Series solution and tree expansion

Series expansion and trees
We will use the following notations: Also, for shortness, we will write f (t) and f ⊗j (t) for f (z 1 , t) and f ⊗j (z j , t) respectively. Let us introduce the operators S i (t) and C ± i,j+1 : · ω ≥ 0} and S 2 is the unit sphere in R 3 . Then, the Boltzmann-Enskog equation (1) can be rewritten as or, in the integrated form, as where f 0 ≡ f 0 (z 1 ). If we iterate equation (5) (i.e., iteratively substitute (5) into the right-hand side of itself), we obtain a series solution in the form and S 1...j (s) = S 1 (s) . . . S j (s). The substitution of (7) into (6) yields . Expansion (8) leads to a natural tree representation, where r i is a "parent" of the particle j + i. (Again, if we are interested only on the single-particle distribution function, we should set j = 1. Here we are introducing the 'hierarchy' of equations for the functions f j , which will be useful in the sequel.) The collection of integers r n = {r 1 , · · · , r n } is called 'tree'. The name is justified by the fact that a tree is conveniently represented graphically. For instance the tree r 5 = {1, 1, 2, 3, 2} for j = 1 is given by Fig. 1 where the i-th branch is generated by particle r i ; see also    Remark 1 For the tree representation we refer to [22] ("Collision Histories"). Later on this notion has been used frequently by several authors [17,14,18,19,20,21,5,6].
The series expansion (8) can be further specified by splitting any operator C j into its positive and negative part. The result is: where σ n = {σ 1 , · · · , σ n } and σ i = ±.

Backward and forward Boltzmann-Enskog flows
The operators C σi ri,j+i are integrals over dω i and dv j+i which, together with t i , will be called the 'node variables' (see Fig. 1).
Given n, r n , σ n and values of the node variables, we now define the so called Boltzmann-Enskog backward flow for the configuration of particles ζ(s), s ∈ [0, t], with ζ i (s) = (ξ i (s), η i (s)), respectively position and velocity of particle i. Let us define the flow. Note that the number of particles is increasing backward in time.
We are considering the case j = 1 for simplicity. Firstly, (recall that (x 1 , v 1 ) are the arguments of the function f 1 in (9)). At the instant t, only the first particle is under consideration. It moves freely, back in time, up to the instant of the first creation t 1 , i.e.
The time instant t 1 corresponds to a contact with the particle 2 with the configuration (x 1 − σ 1 aω 1 , v 1 ) ("contact" means that the distance between the centres of the hard spheres is exactly a). We add the configuration of this particle to ζ. If σ 1 = −, then the configuration of the particles is precollisional and both particles continue to move freely (backward in time) with their velocities (v 1 , v 2 ), so that If σ 1 = +, then the configuration of the particles is postcollisional, and the particles continue to move freely (backward in time) with the precollisional velocities (v 1 , v 2 ): Both particles continue to move freely (backward in time) up to their next collision, and so on. Since r i is the "parent particle" of the particle i, i = 2, . . . , n + 1, the general formula reads for i = 2, . . . , n + 1. The above prescription defines the backward Boltzmann-Enskog flow. Thus, given t > 0, we have a (r n , σ n )-dependent map where t n = (t 1 , . . . , t n ), ω n = (ω 1 , . . . , ω n ), v n = (v 2 , . . . , v n+1 ). This map is important since it appears in formula (9), by writing explicitly the transport and collision operators. The transformation (13) is a Borel map and we will denote its image asÃ rnσn (t). Obviously, the image for n = 0 is R 6 . The Jacobian determinant of the transformation is in modulus a 2n n i=1 |ω i · (v i+1 − η ri (t i + 0))|, so that the map induces the equivalence of measures We will use also the inverse of map (13). To construct it, we introduce the Boltzmann-Enskog forward flow, dependent on (k n , σ n ): where ζ(0) ∈Ã rnσn (t). A tree or, algebraically, the n-tuple r n prescribes the sequences of creations (i.e., collisions) of the n particles. The Boltzmann-Enskog forward flow is defined as follows. A particle i moves freely up to a contact with some other particle j. If the particle j is not the next collision partner of the particle i or the particle i is not the next collision partner of the particle k, then they ignore each other, i.e., they go through each other reaching a mutual distance ≤ a. Otherwise, if the two particles are the next collision partners as specified by r n , then the particle with the larger number disappears from ζ F . For definiteness, let i > k, hence k = r i . If σ i−1 = −, then the particle k does not change its velocity. If σ i−1 = +, then the particle k changes its velocity according to law (2) (with v and v 1 substituted by the precollisional velocities of the particles k and i respectively, and ω being the unit vector directed from the center of the hard sphere i to the center of the hard sphere k). If, according to the tree, the particles i and k are the next collision partners, then the existence of a moment of their contact is guaranteed by the condition ζ(0) ∈Ã rnσn (t).
By construction, if we apply the backward flow (13) to (x 1 , v 1 , t n , ω n , v n ) with s = 0 to obtain ζ(0) and then apply the forward flow to ζ(0), we will obtain Remark 2 It may be worth to underline that forward and backward flows are not directly connected with the real dynamics of a hard-sphere system. In these flows particles can overlap and the tree r n specifies which pair of particles must collide once at contact and which particles overlap freely. This formalism is just a way to represent the solution of a partial differential equation. A connection with the hard-sphere motion will be discussed later on.

Partially ordered trees
Preliminary to the construction of measure valued solutions discussed in the next section, we introduce here a rearrangement of the previous series expansion, based on a different notion of tree which will be called 'partially ordered tree' in contrast with the fully ordered tree -or simply 'tree'-defined above. Here we disregard the mutual ordering of collision times, except for those particles having the same parent. Equivalently, we ignore the time ordering of the birthdays of descendants of different parents.
For instance consider the tree r 5 = {1, 1, 2, 3, 2} in Fig. 1, and the other two trees in Fig. 2, in which the time ordering t 5 < t 4 and t 3 < t 2 is ignored. All these trees are equivalent to the first one as unlabelled graphs. We call then 'partially ordered tree' a tree in which only the ordering of branches generated by a given one is fixed. In Fig. 3 a partially ordered tree is the collection of a fully ordered trees with different ordering of the times t 2 , t 3 and t 4 , t 5 and so on.
Each partially ordered tree is fully specified by the sequence of integers where k i is the number of collisions of the particle i on the way to its final point (after its creation). In particular, k 1 is the number of particles generated by the first particle. The name assigned to each particle is specified as follows.

Figure 4:
Defining we fix the following ordering: particles created by particle i obtain the numbers K i + 1, . . . , K i + k i .We also set k n+1 := 0, since the particle with the last number does not suffer collisions.
We denote the set of partially ordered trees by K n = {k n = (k 1 , . . . , k n )} ⊂ {0, . . . , n} n . We will use that it is given by n-tuples of integers such that The variables k i are related by the fact that the tree k n must be realizable as graph. For instance the sequence (1, 0, 2, 1) with n = 4 is not admissible, because it contains descendants without parents. First, we require k 1 ≥ 1 always. Then if k i > 0, particle i must be already created in the procedure described above. Note that we can represent the solution of the Boltzmann-Enskog equation in terms of partially ordered trees as where the operators Q are given by the following recursive formula: For each Q kn ( ; t −1 ) we repeat the procedure to arrive to Eq. (16).
Observe that the notions of backward and forward Boltzmann-Enskog flow, introduced in the previous section, can be extended easily to the present context. We define as before the (k n , σ n )-dependent map the only difference being that the times t n = (t 1 , . . . , t n ) are only partially ordered. Moreover we denote by A knσn (t) the image of this Borel map.
Notice that two fully ordered trees yield the same partially ordered tree if they are equivalent as topological (unlabelled) graphs. Therefore k n can be thought as an equivalence class of fully ordered trees and we write r n ∈ k n if r n belongs to the equivalence class specified by k n . We have The introduction of the backward Boltzmann-Enskog flow allows to write the series solution in a more explicit way, namely (20) where r i is the parent of particle i in the partially ordered tree, ζ(0) = ζ(0, k n , σ n ), χ kn (t n ) is the indicator function of the partial order dictated by the tree k n and ζ(s, k n , σ n ), s ∈ (0, t) is the backward flow. By using the change of variables induced by the map (19), we arrive to for any bounded continuous function φ : R 6 → R.
A fundamental property of the series expansion is the semigroup property. Denote T (t) the evolution operator sending f 0 to f (t), as given by the above formulas. Then f (t) = T (t)f 0 . If τ 1 , τ 2 ≥ 0, then by algebraic manipulations it follows that In Section 3, more attention will be paid to the proof of the semigroup property for measure valued solutions, which will be needed to construct the microscopic solutions of the Boltzmann-Enskog equation.

Convergence
The first relevant property of the expansion in terms of partially ordered trees is that their number is much smaller than the number of fully ordered trees. Indeed, while rn 1 = n!, This allows to prove Proposition 1 If λ < a 2 /8 f 0 (where · is the L 1 -norm), the series (21) is convergent (in L 1 (R 6 )) for all t > 0.
Proof. We have The generic term is smaller than We see that the series geometrically converges whenever λ < a 2 /8 f 0 . This slightly improves the result of [17] due to another algebraic way of description of the same graphical tree structure: we specify a tree just by an n-tuple k n .
Moreover it may be interesting to observe that one can prove the convergence for any λ, but small t . However, this is not a real restriction, since the free energy functional (4) provides a uniform (in time) estimate of absolute continuity of the distribution function which allows to iterate the procedure to reach arbitrary times, see [17].

Measure valued solutions
To prove the existence of microscopic solutions to the Enskog equation, we need to extend the series solution discussed so far to the case of initial singular measures. Proceeding formally we write, for any bounded continuous ϕ( where z F (t, z n+1 , k n , σ n )) is the forward Boltzmann-Enskog flow, µ 0 (dx 1 dv 1 ) on R 6 is any finite Borel measure and µ n 0 (dz n ) := µ 0 (dz 1 ) . . . µ 0 (dz n ). We rewrite here the integration variables ζ(0) as z n+1 , and ζ F 1 (t) as z F 1 (t) = z F 1 (t, z n+1 , k n , σ n ).

Pathologies
It may arise an ambiguity in this definition if the µ n+1 0 -measure of the boundary ∂A knσn (t) of the region A knσn (t) is not vanishing. Recall that A knσn (t) is the set of the initial n + 1-particle configurations leading to the tree (k n , σ n ). So, ∂A knσn (t) is the set of the initial n + 1-particle configurations that, depending on small perturbation, may realize or not the tree (k n , σ n ). A configuration z n+1 belongs to ∂A knσn (t) in several cases.
The first one corresponds to an initial configuration leading to a collision of the particles 1 and 2 exactly at the final instant t.
If a configuration leading to a collision at the instant t has a positive µ 0 -measure, then we need to consider either t−η or t+η instead. For this reason, we may demand (23) to be satisfied not for all t but for almost all t.
The second case when z n+1 ∈ ∂A knσn (t) corresponds to initial configurations leading, in the forward Boltzmann-Enskog flow, to so called grazing collisions (collisions with ω · (v − v 1 ) = 0 in (2)). However, since we are interested in initial data of the form (3) for t = 0 also such pathology can be avoided by simply using that the initial configuration does not deliver grazing collisions in a finite time. In fact the set of such configurations is of full Lebesgue measure.
The previous pathologies depend on the hard-sphere dynamics, but there is also a pathology depending on the structure of Eq. (23). We illustrate it by means of an example. Consider the case N = 2 with initial measure µ 0 (dζ 1 dζ 2 ) = 1 2 (δ(ζ 1 − z 1 ) + δ(ζ 2 − z 2 ))dζ 1 dζ 2 and with initial state (z 1 , z 2 ) leading to a collision in the time (0, t). If we compute the terms in the expansion relative to n = 0, 1, we easily recover the time evolved measure µ t . This means that particle 2 is created in some definite instant t 1 , to fit with the initial configuration. However, as pointed out in [19], there are other non vanishing contributions. For instance particle 3 can be created by particle 2 with σ 2 = − at time t 1 − 0. The initial contribution is Furthermore, an arbitrary number of branches with σ i = −, i > 1 can be created accumulating at the first node, see Fig. 5. In order to prevent such an unphysical event, we introduce next a suitable notion of measure valued solution.

Regularization via time separation
In order to overcome the pathologies discussed above, we modify definition (23) by setting The difference between (23) and (24) consists in the introduction of the parameter ε and in the replacement of A knσn (t) by A ε knσn (t) defined by where the set S ε knσn (t) consists of all the elements z n+1 delivering, in the Boltzmann-Enskog forward flow, the event in which two particles, born from the same progenitor, are created at times t i , t j , such that |t i − t j | ≤ ε. In other words, creations of particles from the same branch are time separated by ε > 0. Clearly, this avoids the main pathology envisaged in the previous section.
We take Equation (24) as definition of 'regularized series solution' for a given initial measure µ 0 . We will apply this notion only to initial measures which are empirical distributions, namely of the form (3) at time 0. Notice, however, that the regularized series solution makes sense for any initial µ 0 .
Remark 3 One could have defined the time separation on all the nodes of a fully ordered tree. This would not be good for our purpose, since simultaneous creations from different branches of a tree play a crucial role in the reconsruction of the hard-sphere dynamics (see Section 4.2 below).
A crucial property of the weak series solution is the semigroup property which we are going to illustrate.

Semigroup property
Proposition 2 (Semigroup property) Let µ 0 be such that configurations leading to grazing collisions have zero µ n+1 0 -measure for all n, k n , and σ n . Let also t > 0 and τ ∈ (0, t) be such that initial configurations leading to collisions at the instants τ or t have zero µ n+1 0 -measure for all n, k n , and σ n . If the following equalities are satisfied: then equality (24) is also satisfied. Conversely, if (24) is satisfied for t and τ , then (25) is also satisfied.

Proof. Rewrite the integral in (26) as
, and χ A is a characteristic function of a set A.
Firstly, we give a heuristic proof of the proposition, then we give a formal proof. We apply formula (25) n + 1 times. Each n-tuple k n in expansion (26) corresponds to one tree for the interval [τ, t). Applications of (25) to (27) give n + 1 trees k 1 m1 , . . . , k n+1 mn+1 , which are, graphically, continuations of the final points of the tree k n (see Fig. 6). So, the composition of the tree k n with the trees k 1 m1 , . . . , k n+1 mn+1 yields a new tree K N , where N = n + m 1 + . . . + m n+1 . The Boltzmann-Enskog backward flows produced by the trees k 1 m1 , . . . , k n+1 mn+1 for the time interval [0, τ ] together constitute a continuation of the Boltzmann-Enskog backward flow produced by the tree k n . Their composition yields the Boltzmann-Enskog backward flow produced by the tree K N . Conversely, the Boltzmann-Enskog forward flow produced by the tree k n is a continuation of the n-tuple of the Boltzmann-Enskog forward flows produced by the trees k 1 m1 , . . . , k n+1 mn+1 . Their composition also yields the Boltzmann-Enskog forward flow produced by the tree K N .
From the graphical representation, one can understand that the summations over all k n and over all k 1 m1 , . . . , k n+1 mn+1 gives the summation over all joint trees K N for the whole interval [0, t].
Now we give a formal proof of the proposition. After the application of (25) to Figure 6: Sum of compositions of trees.
(27), a generic term is where Σ i = ± and Note that, in (28), we have replaced the product of limits lim εq→0 , q = 1, · · · , n+ 1 (running over the final points of the tree k n ), with a single limit lim ε→0 . The result is indeed independent on the precise way in which the regularization is removed in the different subtrees.
In the first factor in the right-hand side of (29), we see a composition of two Boltzmann-Enskog forward flows: for the interval [0, τ ] (determined by the "bottom trees" k 1 m1 , . . . , k n+1 mn+1 ) and for the interval [τ, t] (determined by the "top tree" k n ). We have The characteristic function in (29) means that, actually, the integration in (28) takes place over initial configurations in leading to configurations in A ε knσn (t − τ ) at the instant τ , i.e., over the set Consider another set: Since, for a fixed (k n , σ n ), the "bottom trees" and the joint tree (K N , Σ N ) are in a one-to-one correspondence, sets (30) and (31) coincide if ε = 0. If ε > 0, then these sets slightly differ. Namely, set (30) does not demand the time separation between subsequent collisions of the same particle in a neighbourhood of the instant τ . Consider, for example, the second composition of trees on Fig. 6, where t 1 ∈ [τ, t] and t 2 ∈ [0, τ ]. There is no demand of time separation between t 1 and t 2 in this composition: both t 1 and t 2 are allowed to be arbitrary close to τ . However, by assumption, the µ N +1 0 -measure of initial configurations leading to collisions in a ε-neighbourhood of τ tends to zero for all triples (N, K N , Σ N ). Hence, in the limit ε → 0, the integrals over set (30) and over set (31) with respect to the measure µ N +1 0 coincide. Thus, the right-hand side of (26) can be written as The summation over (m 1 , k 1 m1 , σ 1 m1 , . . . , m n+1 , k n+1 mn+1 , σ n+1 mn+1 ) can be replaced by the summation over (N, K N , Σ N ), because, as we said before, they are in one-toone correspondence for fixed (n, k n , σ n ). Hence, (32) can be rewritten as hence the summation over (n, k n , σ n ) in (33) gives the integral over A ε K N Σ N (t): which coincides with (24).

Microscopic solutions of the Boltzmann-Enskog equation 4.1 Main result
We are going to prove that a measure of form (3) is a weak series solution of the Boltzmann-Enskog equation, i.e., satisfies (24). We assume that the initial configuration z 0 = (z 0 1 , . . . , z 0 N ) is such that configurations z 0 i i∈In , for all nonempty I n ⊂ {1, . . . , N }, lead to a well-defined hard-sphere motion and do not lead to grazing collisions. Also, for simplicity, let us assume that the initial configuration z 0 does not lead to simultaneous collisions of different pairs of hard spheres. As previously recalled, the set of initial configurations satisfying these conditions has full measure. Proof. Denote by T n (s), s ∈ [0, t] the n-particle hard sphere flow. As before, denote z(s) = T N (s)z 0 . Let t be such that the configurations z 0 i i∈In do not lead to collisions at the instant t.
(ii) Let the particles l and m collide in the interval [θ j , θ j+1 ). Then both flows are free.
The left-hand side of (24) is, for all s ∈ [0, θ 1 ] Denote the terms on the right-hand side of (24) as ∞ n=0 Φ n . The n = 0 term is so that Φ 0 + Φ 1 is equal to (34). The higher-order terms vanish since A ε knσn (s) for n ≥ 2 does not intersect with supp µ 0 . Consider, for example, n = 2. Configurations like (z 0 i , z 0 j , z 0 r ) with i = j = r for σ 2 = (+, +) do not lead to two subsequent collisions, since the Boltzmann-Enskog forward flow coincides with the hard sphere dynamics for this choice of σ 2 . But, by property (i) of the intervals, hard spheres suffer at most one collision on the interval [0, θ 1 ). The same configurations for σ 2 = (+, −), (−, +), (−, −) do not lead to two collisions by property (ii): neither particle l ignoring particle m collides with another particle nor particle m ignoring particle l collides with another particle; other particles move freely. Configurations like (z 0 l , z 0 m , z 0 m ) ∈ supp µ 3 0 could be interpreted as configurations leading to two collisions: the first particle collides with the second and the third particles at the same instant τ . However, by definition of A ε knσn (s), the subsequent collisions of the same particle should be ε-separated in time.
Thus, (3) satisfies (24) in the interval [0, θ 1 ) in the case of a single collision in this interval. This concludes the proof of the theorem.
Remark 4 Note that Proposition 1, adapted to the present context, guarantees the convergence of series (24) only for small λ. However, in case of microscopic solutions, the series has only a finite number of non-vanishing terms, by construction, so that it converges for arbitrary large λ = N a 2 .

Recollisions vs. contractions
There is another and more natural way to describe the hard-sphere dynamics for a microscopic state in terms of a series expansion.
The j-particle marginals associated to a time evolving particle configuration z N (t) = T N (t)z N (0) = {z 1 (t), · · · ,z N (t)} are defined by where z j = (z 1 · · · z j ). For j = 1 we recover the time-evolved empirical distribution (3). In [19] it has been proved that (35) satisfies the following series expansion (BBGKY hierarchy) where S int 1...j (t)F (z j ) = F (T j (−t)z j ) = F (z j (−t)) and T j is the j-particle interacting flow.
The validity of the above expansion, for j = 1 can be compared with the one of the Boltzmann-Enskog equation, which we rewrite here as where µ 0,j = µ ⊗j 0 . Eq. (37) has to be interpreted according to the notion of weak series solution discussed in this paper.
The right-hand sides of (36) for j = 1 and (37) must coincide, being identical the left hand sides. How is this possible?
First, observe that Note that the above identity holds on the physical phase space only, namely on the set of configurations for which |x k − x | > a for all k = . Furthermore we take into account that λ = a 2 N and that the the sum N −1 can be replaced by ∞ by adding vanishing terms. Then we conclude that the two series are identical but for the fact that S int (t) = S(t). More precisely, the backward flow S int (t) describes also the collisions among the particles already created. Such interactions are usually called 'recollisions'. In contrast, S(t) is just the free flow which implies that, once two particles are created, if they arrive at distance a (necessarily with outgoing velocities), they go ahead freely by backward overlapping. More precisely, [19] proves the validity of a representation of the form: where z F I 1 (t, z n+1 , r n , σ n )) is the interacting forward flow, taking into account the recollisions. Here there is no necessity of time separation and, as we have already illustrated, the sum over the fully ordered trees can be replaced by the sum over the partially ordered trees.

Remark 5
The above equation must be further clarified (see [19]). In fact, in contrast with the Boltzmann-Enskog flow, the corresponding (r n , σ n )-dependent map ('interacting backward flow') of whichĀ rnσn (t) is the image, is not globally invertible, but only locally. However (38) makes sense if we assume that ϕ has small enough support. We shall make this assumption in the discussion that follows.
The wavy line of the first tree denotes that particles 3 and 4 recollide at some time τ ∈ (0, t 3 ) according, say, to the hard-sphere dynamics in the figure.
Summarizing we see that the integration in This example shows also that a full time separation cannot work as regularization, because it cannot describe recollisions which, in the Boltzmann-Enskog expansion, are given by simultaneous creations of two particles, contracted with previously existing particles of the backward flow.

Remark 6
The previous argument shows that Eq. (38) could be derived by assuming (24) and vice versa.

Conclusions
The presented way of giving a rigorous sense to the microscopic solutions of the Boltzmann-Enskog equation appears to be the most natural one, in comparison to the previously proposed variants [23,24,25,26], since it does not involve regularizations of delta functions, but gives direct sense to weak solutions of the Boltzmann-Enskog equation by means of a notion of series solution. Here we have introduced a series expansion that involves only partial chronological ordering of collisions. The result is recovered by the time separation of collisions corresponding to this partial order. Thus, the Boltzmann-Enskog equation, which is known to describe irreversible dynamics and entropy production, contains also solutions corresponding to the reversible microscopic dynamics of hard spheres.
Formula (24) for weak series solutions is not very handable for practical purposes, but reveals a relation of solutions of the Boltzmann-Enskog equation with the hard sphere dynamics.