The Derivation of the Linear Boltzmann Equation from a Rayleigh Gas Particle Model

A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.


1.
Introduction. The derivation of continuum equations from atomistic particle models is currently a major problem in mathematical physics with origins in Hilbert's Sixth Problem. A particular interest in this area is the derivation of the Boltzmann equation from atomistic particle dynamics. The first major work in this area was by Lanford [17] which showed convergence from a hard-sphere particle model for short times by using the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, see e.g. [8,11,33]. A recent major work by Gallagher, Saint-Raymond and Texier [14] continued the BBGKY development and proved convergence to the Boltzmann equation for short times for both hard-sphere and short-range potentials. The latter of which was improved by Pulvirenti [29]. A particular difficulty has been proving convergence for arbitrarily long times, since the aforementioned results hold only for times up to a fraction of the mean free flight time. In [5] Bodineau, Gallagher and Saint-Raymond were able to utilise the tools from [14] to prove convergence from a hard-sphere particle model to the linear Boltzmann equation for arbitrary long times in the case that the initial distribution of the background is near equilibrium. They were further able to use the linear Boltzmann equation as an intermediary step to prove convergence to Brownian motion. Further in a following paper [6] the authors were able to consider weaker assumptions on the initial data and prove convergence to the Stokes-Fourier equations. For a general overview of the Boltzmann equation and the BBGKY hierarchy we refer to the books [10,11,32].
A new method to tackle this problem has recently been developed in a series of papers [24,25,26]. This method employs semigroup techniques to study the evolution of collision trees rather than the BBGKY hierarchy. This comes from studying the distribution of the history of the particles up to a certain time rather than the distribution of the particles at a specific time. These papers have been able to prove convergence for arbitrary times but only for a simplified particle interaction system. This is the approach that we continue to develop in this paper.
1.1. The BBGKY hierarchy. The standard approach to tackling the derivation of the Boltzmann equation is via the BBGKY hierarchy. We refer specifically to [5] for hard spheres. If f N (t) represents the N particle distribution resulting from hard sphere dynamics at time t then, away from collisions, f N satisfies the Liouville equation, ∂ t f N (t) + v · ∂ x f N (t) = 0. Considering this equation in weak form and integrating we have, away from collisions, ∂ t f (s) for s = 1, . . . , N where f (s) N denotes the s particle marginal and where C s,s+1 is the effect on the distribution of the first s particles by a collision with another particle given explicitly in (8) below. The system of N equations (1) is known as the BBGKY hierarchy. If the initial distribution of the N particles is given by, where 1 no conditions on no initial overlap and Z N is a normalising constant, then the initial distribution of f (s) N is given by, After successive time integration of (1) one obtains a representation in the form of a finite sum f (s) . . . T s+k (t k )f where T s is the flow map of s hard spheres.
A popular method to establish the convergence of solutions of (1) is to first demonstrate that the mild form of the BBGKY hierarchy is given by a contracting operator if t is sufficiently small. This step ensures that the sum (2) converges absolutely, and in a second step one checks the convergence of the individual terms. It is noteworthy that the necessity to establish the contraction property of the mild BBGKY operator (and thereby restricting the analysis to small values of t) is due to the fact that the individual terms in (2) are unsigned.
The key achievement of this paper is to demonstrate that in a simpler setting a representation formula similar to (2) for f (s) N can be found so that the individual terms are non-negative and can be interpreted as probabilities. This representation offers two significant advantages: 1. It is possible to allow irregular initial data which may not be exponentially tight. 2. The need to establish a contraction property is replaced by a tightness bound, which is related to the properties of physically relevant objects such as collision histories. In particular, convergence can be established for all times.
1.2. The Lorentz and Rayleigh gas. Instead of considering a system of N identical hard spheres evolving via elastic collisions one can consider a single tagged or tracer particle evolving among a system of fluid scatterers or background particles. If the background particles are fixed and of infinite relative mass to the tracer particle then one has a model known as the Lorentz gas first introduced by Lorentz in [21] to study the motion of electrons in a metal.
Much research has been done deriving the linear Boltzmann equation from a Lorentz gas with randomly placed scatterers, for example [7,15,30] and a large number of references found in Part I Chapter 8 of [32]. The linear Boltzmann equation can however fail to hold if we consider non-random periodic scatterers, as shown for example by Golse [16] and Marklof [22]. The existence of a limiting stochastic process for the periodic Lorentz gas from the Boltzmann-Grad limit was shown by Marklof and Strmbergsson in [23].
When a force field is present the convergence of the distribution of the tracer particle in an absorbing Lorentz gas to the solution of a gainless linear Boltzmann equation was proved in [13]. The authors also proved that if the scatterers move with a constant random initial velocity then the convergence can be proven with significantly weaker assumptions on the force field.
Closely related to the Lorentz gas is the Rayleigh gas, where the background particles are no longer of infinite mass. Convergence to Brownian motion is discussed in Part I Chapter 8 of [32]. In [20] Lebowitz and Spohn proved the convergence of the momentum process for a test particle to a jump process associated to the linear Boltzmann equation. This was proved for arbitrarily long times, via the BBGKY hierarchy, when the initial distribution of the velocities is at equilibrium. This builds on their previous work [18,19,34].
In this paper we consider a Rayleigh gas where the background particles are of equal mass to the tagged particle and have no self interaction. The particles evolve via a simplified form of hard-sphere dynamics whereby the background particles do not change velocity. We consider non-equilibrium initial data but require that the background particles are spatially homogeneous. Convergence is proved for arbitrary times.
2. Model and main result. Our Rayleigh gas model in three dimensional space is now detailed. Define U := [0, 1] 3 with periodic boundary conditions. Here a tagged particle evolves via the hard sphere flow and the remaining N particles do not interact, i.e. move along straight lines. The initial distribution of the tagged particle is f 0 ∈ L 1 (U × R 3 ). The N background particles are independently distributed according to the law g 0 ∈ L 1 (R 3 ) in velocity space and uniform in U .
The tagged particle and the background particles are spheres with diameter ε > 0 which is related to N via the Boltzmann-Grad scaling, The background particles always travel in free flow with their velocities never changing from the initial value. The tagged particle travels in free flow whilst its centre remains at least ε away from the centre of all the background particles. When the centre of the tagged particle comes within ε of the centre of a background particle the tagged particle collides as if it was a Newtonian hard-sphere collision and changes velocity.
Explicitly this is described as follows. Denote the position and velocity of background particle 1 ≤ j ≤ N at time t by (x j (t), v j (t)). Then for all t ≥ 0, Further denote the position and velocity of the tagged particle at time t by (x(t), v(t)). Then for all t ≥ 0, If at time t for all 1 ≤ j ≤ N , |x(t) − x j (t)| > ε then dv(t)/dt = 0. Otherwise there exists a 1 ≤ j ≤ N such that |x(t) − x j (t)| = ε and the tagged particle experiences an instantaneous change of velocity. Define the collision parameter ν ∈ S 2 by, εν Figure 1. The collision parameter ν Then the velocity of the tagged particle instantaneously after the collision, v(t), is given by It is noted that in this model we do not have conservation of momentum. The background particles do not change velocity and the root particle does. Proposition 1. For N ∈ N and T > 0 fixed these dynamics are well defined up to time T for all initial configurations apart from a set of measure zero.
The proof is given in section 6, which establishes that almost surely all collisions involve only pairs. We are interested in studying the distribution of a tagged particle among N background particles,f N t , under the above particle dynamics as N tends to infinity or equivalently as ε tends to zero.
The distribution of the tagged particle is shown to converge to the solutions of a linear Boltzmann equation up to a finite arbitrary time T . We now state the main theorem of this paper.
Theorem 2.2. Let 0 < T < ∞ and f 0 , g 0 be admissible. Thenf N t converges to a time-dependent density f t in the TV sense. Moreover, the limit f t satisfies the linear Boltzmann equation where the collision operator Q is defined by Q := Q + − Q − and Q + and Q − known respectively as the gain and loss term are given as follows, where the pre-collision velocities, v andv , are given by The reader is reminded that solutions of (7) only conserve mass, but not energy. 2. The analysis of the Rayleigh gas can also be done using traditional BBGKY approach. Here one uses the collision operator in R d : The hard sphere collision operator is given by The only difference between C Rayl s,s+1 and C hs s,s+1 is the fact that in the hard sphere case one sums over all indices i = 1, . . . , s and in the Rayleigh case only over i = 1. This gives estimates on C Rayl s,s+1 , which are independent of s in the contraction proof for the mild form of the associated BBGKY hierarchy. Using the function spaces X ε,β,µ with norm . ε,β,µ as in [14,Def 5.1.4] for measurable functions G : t ∈ [0, T ] → (t) = (g s (t)) s≥1 ∈ X ε,β,µ one can introduce another time-dependent variant compared to [14,Def 5.1.4] |G| ε,β,µ,λ := sup This will lead to a contraction for arbitrary large times T . For a slightly different approach assuming only finite moments see [31, Section II.B]. 3. Our method can be used to derive quantitative error estimates at the expense of more complex notation and additional regularity requirements for f 0 and g 0 . In particular, see lemmas 4.13 and 4.16 for some quantitative expressions. 4. The result should also hold in the case d = 2 or d ≥ 4 up to a change in moment assumptions on the initial data and minor changes in estimates and calculations throughout the paper. 5. One could consider a spatially inhomogeneous initial distribution for the background particles g 0 = g 0 (x, v). This adds a complication to the equations since for example the operator Q in (7) becomes time-dependent, i.e.
and Q + t analogous. Since the operator now depends on the time t this would require evolution semigroup results to echo the semigroup results in [3]. 6. One could also attempt to adapt these results to more complex and involved models. For example a model where each particle has an associated counter and a collision occurs between particle i and j if and only if both counters are less than k, in the hope of letting k tend to infinity. The main difficulty here will be that one will need to keep track of the current distribution of the background g t in contrast to our model where the background has constant with time distribution g 0 .

2.2.
Method of proof. We closely follow the method of [26]. That is we study the probability distribution of finding a given history of collisions, a given tree, at time t. Firstly in section 3 we prove the main result, theorem 3.1, which shows that there exists a solution P t to a Kolmogorov differential equation (11) and relate this solution to the solution of the linear Boltzmann equation. We show existence by explicitly building a solution on the most simple trees and using this to iteratively build a full solution.
In section 4 we consider the distributionP t of finding a given history of collisions from our particle dynamics and show by direct calculation that this solves a similar differential equation in theorem 4.6 for sufficiently well controlled (good) trees.
Finally in section 5 we prove the main theorem of the paper, theorem 2.2, by proving the convergence between P t andP t in theorem 5.6 and then relating this to f t andf N t . 2.3. Tree set up. We construct trees in a similar way to [26].
A tree represents a specific history of collisions. The nodes of the tree are denoted by m and represent particles while the edges, denoted E, represent collisions. The root of the tree represents the tagged particle and is marked with the initial position of the tagged particle (x 0 , v 0 ) ∈ U × R 3 . The child nodes of the root represent background particles that the root collides with and are denoted (t j , ν j , v j ) ∈ (0, T ]× S 2 × R 3 , where t j represents the collision time, ν j the collision parameter and v j the incoming velocity of the background particle. Since the background particles only collide with the root particle and not each other we only consider trees of height at most 1, so the trees simplify to the initial position of the tagged particle and a list of its n ≥ 0 collisions. The graph structure is mainly suppressed. Definition 2.3. The set of collision trees MT is defined by, For a tree Φ ∈ MT , n(Φ) is the number of collisions.
The final collision in a tree Φ plays a significant role. Define the maximum Further for n(Φ) ≥ 1 the marker for the final collision is denoted by, The realisation of a tree Φ at a time t ∈ [0, T ] for a particle diameter ε > 0 uniquely defines the position and velocity of the tagged particle for all times up to t since the initial position and the collisions the root experiences are known. Further it determines the initial positions of the n(Φ) background particles involved in the tree since we can work backwards from the collision and we know that their velocity does not change. Finally it also includes information about the other N − n background particles, because it is known that they do not interfere with the root up to time t.
If the root collides at the instant t denote the pre-collisional velocity by v(t − ) and the post-collisional velocity by v(t). Throughout this paper the dependence on Φ is often dropped from these and other variables when the context is clear.
For n ≥ 1, defineΦ as being the pruned tree of Φ with the node representing the final collision, which occurs at time τ , removed.  Define a metric d on MT as follows. For Φ, Ψ ∈ MT with components Φ j and Ψ j respectively.
The standard Lebesgue measure on MT is denoted by dλ.
3. The idealized distribution. In this section we show that there exists a solution, denoted P t , to the idealized equation, equation (11), and relate this solution to the solution of the linear Boltzmann equation. We prove existence by constructing a solution iteratively on different sized trees. In section 5 P t is compared to the solution of a similar evolution equation defined by the particle dynamics in order to show the required convergence. The idealized system plays the same role as the Boltzmann hierarchy in [14]. For a given tree Φ ∈ MT , P 0 (Φ) is zero unless Φ involves no collisions, in which case P 0 (Φ) is given in terms of the initial distribution f 0 . P t (Φ) remains zero until t = τ when there is an instantaneous increase to a positive value depending on P τ (Φ) and the final collision in Φ. For t > τ , P t (Φ) decreases at a rate that is obtained by considering all possible collisions.
The idealized equation is given by, where, Then where f t is the unique mild solution of the linear Boltzmann equation given in proposition 2.
From now on assume that f 0 and g 0 are admissible with the provision that either (6) or (14) holds. We prove the existence by construction, taking several steps to build a solution by solving on the most simple trees first and using this solution to iteratively build a full solution. We begin by solving the linear Boltzmann equation. We establish existence, uniqueness and regularity of solutions of (7) by adapting methods from semigroup theory. The difficulty here is that after writing the linear Boltzmann equation as the sum of two unbounded operators we need to ensure that a honest semigroup is generated in order to prove existence and uniqueness. Next we adapt these semigroup techniques to define functions P (j) t that describe the distribution of finding the tagged particle such that it has experienced j collisions. This is key to connecting P t to the solution of the linear Boltzmann.
The following notion of mild solution suitable for transport equations is used (c.f.
Definition 3.2. Consider the following system, Where L : We split the right hand side of (7) into two operators, A and B. These will appear in the construction of P t .
Proposition 2. Suppose that the assumptions in theorem 3.1 hold. Then there exists a unique mild solution f : (7). Furthermore f t remains non-negative and of mass 1, and hold for all t ∈ [0, T ].
Proposition 3. There exists a unique mild solution, to the following evolution equation, Where A is as in (16).

KARSTEN MATTHIES, GEORGE STONE AND FLORIAN THEIL
The distribution P (0) t (x, v) can be thought of as the probability of finding the tagged particle at (x, v) at time t such that it has not yet experienced any collisions.
Proof. By lemma 6.1 A generates the substochastic C 0 semigroup T (t) given in (85). By the Hille-Yoshida theorem, [ Remark 2. This lemma is entirely expected. The probability of finding the tagged particle at (x, v) at time t is given by f t (x, v) and the probability of finding it at (x, v) at time t such that it has not experienced any collisions up to time t is given by P t . Then since f t and P (0) t are mild solutions of (7) and (20) By (19) and Now noting that f θ is non-negative it follows that Bf θ and hence T (t − θ)Bf θ are non-negative also. Hence F Definition 3.5. For j ∈ N ∪ {0} denote by T j the set of all trees with exactly j collisions. Explicitly, The required solution P t can now be defined iteratively on the space MT . For Else define, The right hand side of this equation depends on P τ (Φ) but sinceΦ has degree exactly one less than Φ the equation is well defined.
The proof that P t has the required properties of theorem 3.1 is given shortly. We first define the function P (j) t which is thought of, in parallel to P (0) t , as being the probability of finding the tagged particle at a certain position at time t such that it has experienced exactly j collisions up to time t. The P (j) t will be required to show the connection between P t and the solution of the linear Boltzmann equation. Definition 3.6. Let t ∈ [0, T ] and Ω ⊂ U × R 3 be measurable. Recall in theorem 3.1 we define the set, S t (Ω) = {Φ ∈ MT : (x(t), v(t)) ∈ Ω} -the set of all trees such that the tagged particle at time t is in Ω. Define for all j ∈ N ∪ {0}, is absolutely continuous with respect to the Lebesgue measure on U × R 3 .
Proof. Let j = 1. Then we have by (24), We define a coordinate transform (ν, This transformation has Jacobi matrix, where the blank entries are not required for the computation of the matrix's determinant. The 2x2 matrix in the bottom right has determinant −1 and hence the absolute value of the determinant of the entire matrix is 1. With this (25) becomes, Hence we see that if the Lebesgue measure of Ω equals zero then so does P (1) t (Ω). For j ≥ 2 we use a similar approach, using the iterative formula for P t (Φ) (24).
is an absolutely continuous measure on U × R 3 , the Radon-Nikodym theorem (see Theorem 4.2.2 [12]) implies that P (j) t has a density, which we denote by P (j) t also. This gives,

KARSTEN MATTHIES, GEORGE STONE AND FLORIAN THEIL
Hence for almost all (x, v) ∈ U × R 3 , Remark 4. A similar formula holds for P (0) t since the set S 0 t (x, v) contains exactly one tree: the tree Φ with initial root data (x − tv, v) and such that the root has no collisions, as defined above is almost everywhere the unique mild solution to the following differential equation, where A is given in (16) and B in (17).
The following lemma helps prove the proposition for the case j = 1 which allows the use of an inductive argument to prove the proposition in full.  Proof. We show that for any Ω ⊂ U × R 3 measurable we have, By the definition of T (t) in equation (85), the definition of B in (17), and the proof of lemma 3.7, for w Proof of proposition 4. Consider induction on j. First let j = 1. We seek to apply then the proposition holds so by the above lemma P (1) t is the unique mild solution.
To this aim note that since P This implies, as required. Now consider j ≥ 2 and assume the proposition is true for j − 1.
a similar argument to lemma 3.4 shows that P (j−1) t ≤ f t . By (19) and (92), P (j−1) t ∈ D(B) so the right hand side is well defined. A similar approach to lemma 3.8 shows that for any t ∈ [0, T ] and almost where T (t) is the semigroup given in (85). The rest follows by the same argument as in the j = 1 case.

Proposition 5. For all t ∈ [0, T ] and almost all
where f t is the unique mild solution of the linear Boltzmann equation given in proposition 2.

Recalling from the proof of proposition 4 that D(A)
Further summing (28) and (30) for j ≥ 1, is a mild solution of (90) and therefore since f t is the unique mild solution the proof is complete.
We now have all the results needed to prove that P t satisfies all the requirements of theorem 3.1.
Proof of theorem 3.1. Using definition 3.6, proposition 5, and, since each P (j) t is positive, the monotone convergence theorem we have for any measurable Ω ⊂ U × R 3 , In particular, Hence P t ∈ L 1 (MT ). To show that P t is a solution of (11) first consider Φ ∈ T 0 . Since n(Φ) = 0, . Hence it solves the initial condition. Now for t > 0, since Φ ∈ T 0 , v(t) = v 0 and x(t) = x 0 + tv 0 . Hence by (85) and (21), The only dependence on t here is in the exponential term so we differentiate P t (Φ) with respect to t, Hence P t solves (11) on T 0 .
We now consider Φ ∈ T j for j ≥ 1. Since Φ ∈ T j we have n(Φ) = j and τ > 0.
Further for t > τ the only dependence on t is inside the exponential term and hence differentiating gives, Equations (32), (33) and (34) prove that P t solves (11) on T j . Since MT is the disjoint union of T j for j ≥ 0, P t is a solution of (11) on MT . Finally, the required connection between P t and the solution of the linear Boltzmann equation has been shown in (31). 4. The empirical distribution. We now consider the empirical distribution of treesP ε t defined by the dynamics of the particle system for particles with diameter ε. To ease notation we drop the dependence on ε and writeP t . The key result of this section is thatP t solves the differential equation (36) which is similar to the idealized equation (11). The similarity between the two equations is exploited in the next section to prove the required convergence as ε tends to zero.
We do this by restricting our attention to trees that are well controlled in various ways, calling these trees good trees.

Definition 4.2.
A tree Φ ∈ MT is called re-collision free at diameter ε if for all 0 ≤ ε ≤ ε, for all 1 ≤ j ≤ n(Φ) and for all t < t j , That is to say, if the tree involves a collision between the root particle and background particle j at time t j then the root particle has not previously collided with background particle j. So if a tree is re-collision free then it involves at most one collision per background particle. Further define R(ε) := {Φ ∈ MT : Φ is re-collision free at diameter ε}.
Definition 4.4. A tree is Φ ∈ MT is called free from initial overlap at diameter ε > 0 if initially the root is at least ε away from the centre of each background particle. Explicitly if, for j = 1, . . . , N , Define S(ε) ⊂ MT to be the set of all trees that are free from initial overlap at radius ε.

(ε) and Φ is non-grazing
Since M, V are decreasing for ε < ε we have G(ε) ⊂ G(ε ). Later some conditions on M and V are required to prove thatP t solves the relevant equation and to prove convergence.
Now define the operatorQ t which mirrors the idealized operator Q t in the empirical case. FixĈ 1 > 0, a constant depending only on Φ described later. Define the gain operator, Next for a given tree Φ, a time 0 < t < T and ε > 0, define the function 1 ε That is 1 ε is 1 if a background particle starting at the position (x,v) avoids colliding with the root particle of the tree Φ up to the time t. This allows us to define the loss operator, For someĈ 2 (ε) > 0 depending on t and Φ of o(1) as ε tends to zero. Finally define the operatorQ t as follows,Q Theorem 4.6. For ε sufficiently small and for Φ ∈ G(ε),P t solves the following The functions γ and ζ are given by and, Remark 5. When choosing the background particles according to some Poisson point process some of these terms simplify as in [25].
The proof is developed by a series of lemmas in which we prove the gain term, loss term and initial condition separately.
Definition 4.7. Define ω 0 := (u 0 , w 0 ) ∈ U × R 3 to be the random initial position of the test particle. By our model ω 0 has distribution f 0 .
Further for j = 1, . . . , N define ω j := (u j , w j ) to be the random initial position and velocity of background particle j. Note that ω j has distribution Unif(U ) × g 0 . Finally define ω := (ω 1 , . . . , ω N ). Proof. Recall the definition of B h (Ψ) in (10). Since G(ε) is open there exists a h > 0 such that B h (Ψ) ⊂ G(ε). In the case n(Ψ) = 0 for all t ≥ 0,P t (Ψ) ≤ f 0 (x 0 , v 0 ) and hence absolute continuity follows. Suppose We view ϕ as having n(Φ) + 1 components, the first being the initial root position (x 0 , v 0 ), components j = 2 . . . , n being the marker (t j , ν j , ν j ) and the final component being (x(τ ) + εν − τ v , v ) -the initial position of the background particle that leads to the final collision with the root in Φ. We claim that, To prove this we first rotate our coordinate axis so that ν = e 1 . Then for k = 0, . . . , n define F 0,k := ∇ x0 ϕ k (Φ) and for j = 1, . . . , n define F j,k := ∇ tj ,νj ϕ k (Φ). We calculate, where the blank components are not needed. Also, F 0,n = ∇ x0 ϕ n (Φ) = Id(2). Further for j = 2, . . . , n + 1, F j,j = ∇ tj ,νj ϕ j (Φ) = Id (2). For any other j, k not already calculated, F j,k = 0. Hence det(∇ϕ)(Φ) is the product of the determinants of all F k,k for k = 1, . . . , n+ 1, proving the claim. Now define a second map,φ : This maps Φ to the initial position of each particle in Φ. By repeatedly applying (38), For h > 0 and j = 0, . . . , n define C h,j (Φ) = C h,j ⊂ U × R 3 to be the cube with side length h centred atφ j (Φ). Further for h > 0 define, By the fact that the probability of finding a tree is less than the probability that initially there is a particles at the required initial position, Recalling that λ denotes the Lebesgue measure on MT , by (39) it follows, Hence combining (40) and (41) and recalling (3), Since f 0 ∈ L 1 (U × R 3 ) and g 0 ∈ L 1 (R 3 ) let h tend to zero and the left hand side, which becomesP t (Φ)/λ(Φ) in the limit, remains bounded. This completes the proof.
We now prove the initial condition requirement onP t .
Lemma 4.9. Under the assumptions and set up of theorem 4.6 we havê Proof. In the case n(Φ) > 0, we haveP 0 (Φ) = 0. This is because the tree is free from initial overlap and the tree involves collisions happening at some positive time therefore the collisions cannot have occurred at time 0. Now consider n(Φ) = 0. In this situation the tree Φ contains only the root particle and the probability of finding the root at the given initial data (x 0 , v 0 ) is given by f 0 (x 0 , v 0 ). However this must be multiplied by a factor less than one because we must rule out situations that would give initial overlap of the root particle with a background particle. So we calculate the probability that there is no overlap. Firstly, Hence, as required.
Before we prove the loss term lemma we require a few technical estimates to calculate the rate at which the root particle experiences a collision.
That is W h (t) is the set of initial points in U × R 3 for the background particles that lead to a collision with the root particle of Φ between the time t and t + h. Further define, From now on assume that the functions V and M in definition 4.5 satisfy, for any 0 < ε < 1, and, Lemma 4.11. Recall definition 4.7. For ε sufficiently small, Φ ∈ G(ε) and t > τ , Proof. Note that by the inclusion exclusion principle the fact that the background particles are independent, Recalling (43),P Now we estimate the right hand side of (47) by estimating the numerator and denominator. Firstly by calculating the volume of the appropriate cylinder, for anȳ v ∈ R 3 , Define β := Note that by assumption (5), β < ∞. Since Φ ∈ G(ε) it follows that |v(t)| ≤ V(Φ) ≤ V (ε). Using these and (48) we estimate the numerator in (47),

KARSTEN MATTHIES, GEORGE STONE AND FLORIAN THEIL
Turning to the denominator of (43). Firstly note that, By using (48), t ≤ T and the same estimates from the numerator estimate, Hence for ε sufficiently small by (44), so by (51) and (52), Bounds for both the numerator and the denominator of (47) have been found in equations (50) and (53) respectively. Hence, Substituting this into (46) and recalling (3), This gives finally that, completing the proof of the lemma.
The previous lemma shows that the rate of seeing two collisions in a short time converges to zero. We now show that the rate of seeing one collision converges to the required loss term. Before we do this we first estimate the error caused by re-collisions.
Notice that B h,t (Φ) is the set of all initial positions that a background particle can take such that it collides with the root once during (0, t) and once during (t, t + h).
Consider s ∈ (0, t − δ] and σ ∈ (t, t + h) fixed. The conditions defined by B h,t require that,v Hence, Thus the length of the curve is bounded by, Therefore for some C > 0, Noting that x(σ) = x(t) + (σ − t)v(t), forv given, (x,v) ∈ B h,t (Φ) requires that, Hence forv givenx is contained in cylinder of radius ε and length h|v(t)−v|. Denote this cylinder by C(v). By (6) or (14), for constants C that change on each line, It remains to show that (V (ε) + 1)|E| is o(1) as ε tends to zero. Recall (44), that δ = ε 1/3 and (55), as required. Now consider the second part of B h,t (Φ) for s ∈ (t − δ, t) denoted B 2 h,t (Φ). Since Φ is fixed, t > τ and δ = ε 1/3 let ε sufficiently small such that t − δ > τ . Hence for s ∈ (t−δ, t), v(s) = v(t). We change the velocity space coordinates so that v(t) = 0. If we require that a particle starting at (x,v) collides with the tagged particle in (t − δ, t) and again in (t, t + h) we require in the new coordinates that eitherv = 0 or that |v| is sufficiently large so that the background particle wraps round the torus having travelled at least distance 3/4 (for ε sufficiently small) within time (δ + h) to re-collide with the tagged particle. That is, .
For ε sufficiently small, Φ ∈ G(ε), t > τ andĈ 2 (ε) as in the above lemma, Proof. Since the initial data for each background particle is independent of the other background particles, By (54), Hence dividing (56) by h and taking h to zero we see that all terms in the sum for j ≥ 1 tend to zero, leaving only the contribution from the term j = 0. Hence, It remains to investigate, For B h,t (Φ) as defined in definition 4.12 we have, It then follows from lemma 4.13, Substituting this into (57), which proves the lemma.
With these estimates we can now prove that the loss term of (36) holds.
Noting that,P Using lemma 4.11 and lemma 4.14, , which proves the lemma.
We next move to proving the gain term in (36).
Note that U 0 := {Φ}. Then by lemma 4.8, For Ψ ∈ U h define V h (Ψ) ∈ U × R 3 to be the initial position of the background particle that leads to the final collision of Ψ and define V h ⊂ U × R 3 by, Note that V 0 = {(x(τ ) + εν − τ v , v )} that is V 0 contains only the initial point of the background particle that gives the final collision in Φ. Then by a change of coordinates, recalling (38), By absolute continuity ofP τ , Combining these into (59), Next consider the lower bound. By the inclusion-exclusion principle,

KARSTEN MATTHIES, GEORGE STONE AND FLORIAN THEIL
As in (60) it follows, Further, By the absolute continuity ofP t this implies, Hence by (62), Recalling that we need to prove (58) to prove the lemma, we see that with (61) and (63) we now need only to show that, That is, Since we are conditioning onΦ occurring there is a region of U that we must rule out for the initial position of the background particle if we know that is has velocity v . That is to say there is a region of U where we know the background particle cannot have started with initial velocity v because if it did it would have interfered withΦ. Denote this region of U by ∆. To calculate the volume of ∆ we imagine it as cylinders to get, for fixed incoming velocity v , where the first term calculates the volume of each cylinder as the root particle changes direction and the second term subtracts the over-estimate error caused counting certain parts twice as the particle changes direction. Therefore, recalling that |U | = 1, and this together with the fact that the velocity of the background particle has initial distribution g 0 gives the required (64).
, v(t)) U ∆ Figure 3. In the case v = 0 we are calculating the volume of ∆, since we know the background particle cannot start in ∆. For v = 0 the cylinders get shifted but the principle is the same. (Diagram not to scale)