Derivation of a Non-autonomous Linear Boltzmann Equation from a Heterogeneous Rayleigh Gas

A linear Boltzmann equation with nonautonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution semigroups for Kolmogorov equations of associated probability measures on collision histories.


Introduction
Particles undergoing hard sphere collisions are a classical topic in dynamical systems theory, e.g. the long term behaviour are of high interest in ergodic theory. Kinetic equations provide a different kind of limit description. The times remain finite, but the number N of discrete particles undergoing collisions tends to infinity as the diameter ε of individual spheres tends to zero. We are interested in a global continuum description of the particle gas in the Boltzmann-Grad scaling of N and ε, where the overall volume of the N particles tends to zero, but expected number of collisions per particle remains finite in a time of order one. The primary example is the Boltzmann equation given by, where f = f t (x, v) represents the distribution of the gas at position x and velocity v at time t, the operator Q represents the effect of self-interaction amongst the particles and f 0 is some given initial distribution. Instead of considering a system of a large number of 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. With such a model one then considers the linear Boltzmann equation, where the operator Q now encodes the effect of the tagged particle interacting with the scatterers, rather than self interactions amongst the particles. We will deal with a variant of the hard sphere flow and show the validity of a linear Boltzmann equation, i.e. we show that the distribution of a particle of interest in the many particle flow is well approximated by solutions to the appropriate Boltzmann equation. This is what we mean by the derivation of a continuum description. The first major work for the full Boltzmann equation was by Lanford [21] giving convergence of the one particle distribution function of a hard-sphere particle model to solutions of the Boltzmann equation for short times by using the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the evolution of k-particle distribution functions for all k ∈ N, see e.g. [10,13,37]. The convergence was valid for short times, a fraction of the average free flight time. This proof was simplified in [38] by employing Cauchy-Kowalevski arguments. Global in time convergence results were proved in [20,33,19] with the assumption of sufficiently large mean free paths. Gallagher, Saint-Raymond and Texier [16] continued this development by giving detailed convergence results for short times. Major work on short-range potentials includes [34]. The main challenge remains to provide convergence for arbitrarily long times. The first derivation of a linear Boltzmann equation for a Rayleigh gas was given in [39]. In [7] Bodineau, Gallagher and Saint-Raymond were able to utilise the tools from [16] 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 with an explicit rate of convergence. They used the linear Boltzmann equation as an intermediary step to prove convergence 1 to Brownian motion of a tagged particle. In [8], a variant gave the derivation of the Stokes-Fourier equation again using linear Boltzmann equations as an intermediary step. The books [12,13,36] give an introduction to the BBGKY hierarchy and its link to the Boltzmann equation. The derivation of the Boltzmann equation from a system of particles interacting with long range potentials, where each particle effects every other particle regardless of their distance, has proved more difficult. A first result was given in [14]. One recent result was proved in the linear case via the BBGKY hierarchy with strong decay assumptions on the potential and for arbitrarily long times in [3]. A different variant to show the validity of Boltzmann-type equations has been developed in a series of papers [29,30,28]. There we employ infinite dimensional dynamical system and semigroup techniques to study the evolution of the probability to see collision trees. This relates the distribution of the history of the particles up to a certain time to the distribution of the particles at a specific time.
While terms in Duhamel formulas providing solutions to the BBGKY hierarchy can be interpreted as pseudo histories, this approach uses other solution techniques and has a clearer connection to typical particle behaviour. So far we have been able to prove convergence for arbitrary times for various simplified particle models based on a many particle hard-sphere flows, e.g. kinetic annihilation and the dynamics for a tagged particle interacting with moving background particles, which do not change. The aims of this paper are to develop the semigroup approach and provide an example for the rigorous derivation of a non-autonomous linear Boltzmann equation. A tagged particle is undergoing collisions with n background particles. If the background particles are fixed and of infinite mass then this is the Lorentz gas first introduced in [25]. An autonomous linear Boltzmann equation can be derived as a scaling limit from a Lorentz gas with randomly placed scatterers, see for example [9,17,35] and a large number of references found in [36,Chap. 8].
The linear Boltzmann equation can however fail as a valid approximation if there are (non-random) periodic scatterers, see e.g. [18,26]. A different limiting stochastic process for the periodic Lorentz gas was derived in [27] from the Boltzmann-Grad limit. We consider the closely related Rayleigh gas, where the background particles move and are no longer of infinite mass. In [24] Lebowitz and Spohn proved the convergence of the distribution of the tagged particle to the linear Boltzmann equation for background data at equilibrium for arbitrarily long times, see also [22,23,39]. In their case the distribution of the background particles does not change in time, the equilibrium distribution remains invariant under the pure transport. In this paper we derive a non-autonomous linear Boltzmann equation via the Boltzmann-Grad limit of a Rayleigh gas particle model, where one tagged particle evolves amongst a large number of background particles, which do not interact with each other. In contrast to [28] the initial distribution of the background particles is now spatially heterogeneous and away from the equilibrium distribution. The background g then satisfies a transport equation with explicit solution g(x, v, t) = g 0 (x − vt, v), this g will introduce the non-autonomous background in the linear Boltzmann equation. We assume that at a collision between the tagged particle and a background particle there is a full hard sphere collision in which both particles change direction, we will show that this change in the background is not relevant for the limit. The main result is theorem 2.4, which states that the distribution of the tagged particle evolving among N background particles converges as N tends to infinity to the solution of the non-autonomous linear Boltzmann equation. The convergence holds for arbitrarily large times and with moderate moment assumptions on the initial data. We extend the methods used in [30,28]. The idealised equation on collision trees is stated and semigroup methods are used to show that there exists a solution. Then it is shown that the distribution on collision trees induced by the many particle dynamics solves an empirical equation, which is of similar form as the equation for the idealised distribution. This leads in section 5 to the convergence of solutions of the empirical and idealised equations, which then implies the main theorem. A major technical difference to our paper [28] is in section 3 on the idealised equation. The introduction of a spatial dependence on the initial distribution of the background creates non-autonomous equations, which require us to study evolution system results. There is no specific evolution semigroup result for us to refer to for positive solutions of the non-autonomous equations, so our problem is viewed in the framework of general evolution system theory, which creates a number of more technical requirements. As the question of honesty of the semigroup solution of the non-autonomous linear Boltzmann equation is also more difficult than in the autonomous case, we were unable to directly verify honesty from existing results. Instead honesty of the solution is proven indirectly via the connection to the idealised equation. The change in collisions, where a collision between the tagged particle and a background particle is now a full hard sphere collision, makes only a minimal difference on our proof.

Model and Main Result
We now give our Rayleigh gas particle model in detail. The model differs from the model in [28] in two ways: i) we no longer assume that the initial distribution of the background particles is spatially homogeneous and ii) now when the tagged particle collides with a background particle the collision is treated as a full hard sphere collision and so the background particle changes velocity rather than continuing with the same pre-collision velocity. Let U = [0, 1] 3 ⊂ R 3 , with periodic boundary conditions. Let N ∈ N. One tagged particle evolves amongst N background particles. The tagged particle has random initial position and velocity given by f 0 ∈ L 1 (U × R 3 ) and the N background particles have random and independent initial position and velocity given by g 0 ∈ L 1 (U × R 3 ). The tagged particle and background particles are modelled as spheres with unit mass and diameter ε > 0 given by the Boltzmann-Grad scaling, N ε 2 = 1.
The tagged particle travels with constant velocity while it remains at least ε away from all background particles. Each background particles travels with constant velocity while it remains at least ε away from the tagged particle. Background particles do not effect each other and freely pass through each other. When the position of the tagged particle comes within ε of the position of a background particle both particles instantaneously change velocity as described by Newtonian hard-sphere collisions. We describe this process explicitly. Let the position and velocity of the tagged particle at time t ≥ 0 be denoted (x(t), v(t)) and for 1 ≤ j ≤ N , let the position and velocity of background particle j at time t be given by ( If there exists a 1 ≤ j ≤ N such that |x(0) − x j (0)| ≤ ε then we assume that the two particles pass through each other unaffected (indeed this is well defined since the velocities are only equal with probability zero). That is, any initial overlap is ignored and not treated as a collision. Now let t > 0. If for all 1 ≤ j ≤ N , |x(t) − x j (t)| > ε then dv(t) dt = 0 and dv j (t) dt = 0.
Else there exists a 1 ≤ j ≤ N such that |x(t) − x j (t)| = ε and both particles experience an instantaneous collision at time t. We denote by v(t − ) and v j (t − ) the velocity of the tagged particle and background particle j instantaneously before the collision and define v(t) and v j (t) to the velocity of the tagged particle and background particle j instantaneously after the collision. Define the collision parameter ν ∈ S 2 by Figure 1. The parameters of a collision between the tagged particle and particle j.
Then v(t) and v j (t) are given by Proposition 2.1. 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 zero measure.
Proof. The proof of this is unchanged from [28, prop.1], which is based upon [16, prop. 4 Definition 2.2. For t ≥ 0 and N ∈ N letf N t denote the distribution of the tagged particle at time t evolving via the Rayleigh gas dynamics described above amongst N background particles.
We are interested in the behaviour off N t as N increases to infinity, or equivalently as ε converges to zero. In the main theorem 2.4 we show that for any fixed T > 0 and under some assumptions on f 0 and g 0 ,f N t converges to f 0 t , the solution of the non-autonomous linear Boltzmann equation, in L 1 as N tends to infinity uniformly for any t ∈ [0, T ].
Then g 0 is background-admissible if all of the following hold R 3ḡ and there exists a M > 0 and an 0 < α ≤ 1 such that for almost all v ∈ R 3 and for any x, y ∈ U We now state the relevant non-autonomous linear Boltzmann equation. Firstly for t ≥ 0 define the operators Q 0,+ t and Q 0,− t : where we use the notation g t (x, v) := g 0 (x − tv, v) and where the pre-collision velocities, v ′ andv ′ , are given by The non-autonomous linear Boltzmann equation is given by We now state the main theorem.
Theorem 2.4. Let 0 < T < ∞ and suppose that f 0 and g 0 are tagged and background admissible probability densities respectively. Then, uniformly for t ∈ [0, T ],f N t , the distribution of the tagged particle at time t among N background particles under the above particle dynamics, converges in L 1 (U × R 3 ) as N tends to infinity to f 0 t , a solution of the non-autonomous linear Boltzmann equation (2.9). 4

Remarks.
(1) We prove the result in dimension 3. The result should also hold in the case d = 2 or d ≥ 4 up to a change in moment assumptions. (2) With stronger moment assumptions on the initial distributions f 0 and g 0 it may be possible to calculate explicit convergence rates and convergence in L 1 -spaces involving moments. In particular to show (3.8) we use the dominated convergence theorem which proves convergence without any explicit rate. With further assumptions on our initial data it may be possible to prove this with a more quantitative method. Even under our mild assumptions we can quantify corrections terms in Theorem 4.10 explicitly. We will not express the explicit dependence on T in the estimates, but all error estimates will grow in T , e.g. linearly in Lemma 5.4. (3) Our main extensions to [28] are that methods developed here can deal with an evolving background. Hence the methods will be relevant for more involved particle models where the background particles evolve, such as the addition of an external force acting on the particles. In such a situation the relevant linear Boltzmann equation would include the additional force term. Also the distribution of the background particles at time t would include the effects of this force. This would add additional complications to the various bounds computed throughout. (4) We hope that evolving backgrounds can be a route to approximate the behaviour of a full many particle flow, where the background is regularised by introducing appropriate counters. A collision happens between background particle i and j if both particles have experienced less than k collisions. It remains open if the methods will be stable under letting k tend to infinity and if eventually this leads to improvements of the time interval of validity of the nonlinear Boltzmann equation compared to [21]. (5) The conditions on f 0 and g 0 in (2.1) and (2.4) considerably relax the moment conditions compared to the exponential moments in the function spaces in [7], while the transport effects due the spatial heterogeneity can be well controlled using (2.3). This allows us to use e.g. stretched exponentials or other non-Maxwellian distributions in the background, which should be helpful, when attempting extensions as in the previous remark.

2.2.
Method of Proof and Propagation of Chaos. We consider two Kolmogorov equations on the set of all possible collision histories or 'trees'. Section 3 is mostly devoted to proving theorem 3.1, where we prove that there exists a solution to the idealised equation by an iterative construction process and then prove that a number of properties hold, including the connection to the solution of the linear Boltzmann equation. In this section we introduce an ε dependence in both the idealised equation and the linear Boltzmann equation to enable convergence proofs that follow later. In section 4 we prove that the distribution of all possible collision histories from our particle dynamics solves the empirical equation, at least for well controlled situations, which resembles the idealised equation. We do this by explicitly calculating the rate of change of the distribution on all possible collision histories. Finally in section 5 we prove the main theorem 2.4, by proving the convergence between the solutions of the idealised and empirical equations. A detailed comparison with the classical BBGKY approach of [21] adapted by [16] has been given in [28]. A major challenge in that approach is proving the propagation of chaos. The tree history approach allows us to avoid the issue of proving the propagation of chaos explicitly. This approach was developed in [35] to circumvent the issues around the propagation of chaos by focusing on good histories or trees. The idealised distribution P ε t considers that the particles are chaotic so the probability of seeing a background particle at (x, v) at time t is given exactly by g 0 (x − tv, v). On the other hand for the empirical distribution no assumption of chaos is made and the particles evolve as described by the particle dynamics. Therefore the probability of seeing a background particle at (x, v) at time t is more involved than just g 0 (x − tv, v) since we need to consider the effect of a background particle colliding, changing velocity and then arriving at (x, v) at time t. This issue is resolved by introducing good collision trees. Good trees, defined precisely in definition 4.7 below, require, among other properties, that each background particle that the tagged particle collides with will not re-collide with the tagged particle up to time T . This means that if we restrict our attention to good trees then we know that there cannot be any re-collisions and so the distribution of the background particles is much clearer. For this reason we only investigate the properties of the empirical distributionP ε t on this set of good histories. It is then shown in proposition 5.5 below that good histories have full measure, in the sense that the contribution of histories that are not good is vanishing as ε tends to zero. Therefore to prove convergence between the idealised distribution and the empirical distribution, which is the key step to proving the main theorem, we only need to compare the idealised and empirical distributions on good histories and remark that the effect of histories that are not good is vanishing in the limit. Hence the propagation of chaos is proved implicitly with this collision history method. The idealised distribution assumes chaos whereas the empirical distribution does not. By proving the convergence from the empirical distribution to the idealised distribution we prove the propagation of chaos implicitly. We emphasise that good histories, due to their lack of re-collisions, mean that the propagation of chaos holds for the particles relevant for the tagged particle.
2.3. Tree Set Up. Trees are defined in the same way as in [30,28], where more detailed explanations can be found. We consider non-cyclic rooted trees of height at most one. A tree represents a specific history of collisions. The nodes of the tree represent particles and are marked with information about that particle, while the edges of the trees represent collisions. The root of the tree represents the tagged particle and is marked with the tagged particle's initial position (x 0 , v 0 ) ∈ U × R 3 . The child nodes of the tree represent background particles that the tagged particle collides with and are marked with (t j , ν j , v j ) ∈ [0, T ] × S 2 × R 3 the time of collision, the collision parameter and the incoming velocity of the background particle before the collision respectively. The graph structure of the tree is of little significance in the current paper. 1 1 1 Figure 2. A tree with two collisions, the time of the final collision is τ .
Definition 2.5. The set of collision trees is defined by, For a tree Φ ∈ MT , n denotes the number of collisions. The final collision plays an important role in this theory. We define τ = τ (Φ) and for n ≥ 1 we use the notation (τ, ν, v ′ ) = (t n , ν n , v n ). Finally, for n ≥ 1, we defineΦ as the pruned tree of Φ with the final collision removed.
We define a metric, d, on MT as follows. For any Φ, Ψ ∈ MT , For Φ ∈ MT and h > 0 we define We obtain the Borel σ-algebra from the metric. All measures of interest will be absolutely continuous in each component of MT with respect the corresponding Lebesgue measure for 6 We note that for a given ε ≥ 0, the realisation of Φ at a time t ∈ [0, T ] uniquely determines (x(t), v(t)), the position and velocity of the tagged particle, and (x j (t), v j (t)), the position and velocity of the j background particles involved in the tree. We note that (x(t), v(t)) is independent of ε (since regardless of ε the tagged particle has given velocities and collision times), but each (x j (t), v j (t)) is ε dependent (since the relevant background particle must be ε from the tagged particle at the collision). Background particles might collide several times with the tagged particle, in such case less than n different particles are involved in a tree with n(Φ) collisions. Our parametrisation of the trees does not immediately identify such trees, we will later show that the resulting trees are rare. Furthermore the realisation of Φ gives information on the remaining (at least) N − n background particles, since we know that they have not interfered with the tree.

The Idealised Distribution
The idealised equation is the first of two Kolmogorov equations in this paper. In this section we show that there exists a solution to the idealised equation and relate it to the solution of the linear Boltzmann equation. We construct a solution by first considering the probability of finding the tagged particle at a certain position and velocity such that it has not yet had any collisions. From this we iteratively define a function and check that it solves the idealised equation and that the required connection to the linear Boltzmann equation holds. A significant problem in this section is showing that we have the required evolution system to solve the non-autonomous equation that describes the probability of finding the tagged particle such that it has not yet experience any collisions. In the autonomous case we were able to quote specific semigroup results for the Boltzmann equation from [4]. However in this non-autonomous case we have to resort to more general evolution system theory. This leads to a number of technical results to check the various assumptions of the general theory.
Then the ε dependent non-autonomous linear Boltzmann equation is given by We can now state the idealised equation. For ε ≥ 0 consider where the gain term depends on the pruned treeΦ as given in definition 2.5 For a tree Φ ∈ MT , t ≥ 0 and ε ≥ 0 we introduce the notation and note that this implies . Moreover for any t ∈ [0, T ] and for any Ω ⊂ U × R 3 define, Theorem 3.1. Suppose that f 0 and g 0 are tagged and background admissible respectively in the sense of definition 2.3. Then for all ε ≥ 0 there exists a solution P ε : And for any ε ≥ 0, t ∈ [0, T ] and any Ω ⊂ U × R 3 measurable, From now we assume that f 0 and g 0 are tagged and background admissible respectively. The rest of this section is devoted to proving theorem 3.1. We split this into a number of subsections.
In the first subsection, 3.1, we prove that there exists a solution P ε,(0) t to the gainless linear Boltzmann equation and that this solution has a particular form given by an evolution system U ε . This subsection takes a number of technical lemmas in order to prove various semigroup properties. Then in subsection 3.2 we show that the ε dependent non-autonomous linear Boltzmann equation has a solution, in the evolution system sense. Then in section 3.3 we construct P ε t and show that it indeed satisfies the properties of theorem 3.1. We finish this section by using theorem 3.1 to prove that the solution of the ε dependent non-autonomous linear Boltzmann equation is a probability measure.
3.1. The Evolution Semigroup. In this subsection we prove that there exists a solution to the ε dependent gainless linear Boltzmann equation (3.11) by following standard evolution system theory as in [32]. This requires a number of technical results.
Moreover the solution is given by P can be thought of as the probability of finding the tagged particle at (x, v) such that it has not yet experienced any collisions.
To prove this proposition we aim to apply [32, Theorem 5.3.1], which gives that there exists a evolution system defining the solution to (3.11). First we present lemmas checking that conditions (H1), (H2) and (H3) hold. This tells us that there exists a unique evolution system satisfying (E1), (E2) and (E3). Next we show that U ε (t, s) is a strongly continuous evolution system and that it satisfies (E1), (E2) and (E3), so is indeed the evolution system described by [32,Theorem 5.3.1]. This tells us that a solution to (3.11) is given by U ε (t, 0)f 0 . Proof. By [1, Theorem 10.4] we see that for t ≥ 0, A ε (t) generates the C 0 semigroup S ε t given by Since each S ε t is a contraction semigroup this is a stable family, which proves condition (H1) of [32, The following two lemmas, lemma 3.6 and lemma 3.7, are used to help prove that condition (H2) holds, which is shown in lemma 3.8.
Since P ∈ Y and using (2.5) we can integrate each of these terms over U . Hence for almost all By bounding the exponential term in (3.13) by 1 we have Further we note that by (2.5) for some C > 0, Also, and Y is invariant under S ε t by lemma 3.6 so the only remaining property to check is that for any P ∈ Y be a test function and let δ > 0. We show for s > 0 sufficiently small, Hence for |v| < R, and this converges to zero as s converges to zero. Therefore Since η is continuous on U ×R 3 , is it uniformly continuous on U ×B R (0) so we can make s sufficiently small so that this is less than δ/3. Now by (2.5) we have for almost all x Hence for s sufficiently small, again by the uniform continuity of η on U × B R (0), Also, by a similar process to (3.21) we see that for s sufficiently small Together (3.23) and (3.24) give that for s sufficiently small Therefore with (3.21), we see that for s sufficiently small, (3.20) holds Using this and (3.20) finally (3.19) can be proved.
By the calculations in the proof of lemma 3.6 we have that for any s, t, ε ≥ 0 there exists a C ≥ 1 such that for any P ∈ Y , . Hence to apply [32, Theorem 5.2.2] we need to show that there exists an M ≥ 1 and ω ≥ 0 such that for any k ∈ N, any sequence 0 ≤ t 1 ≤ t 2 ≤ · · · ≤ t k ≤ T , any list 0 ≤ s 1 , . . . , s k and any P ∈ Y , To that aim fix k ∈ N, 0 ≤ t 1 ≤ t 2 ≤ · · · ≤ t k ≤ T and 0 ≤ s 1 , . . . , s k and P ∈ Y . Define ΠP := k j=1 S ε t j (s j )P . By repeatedly applying (3.13) we see that, Denoting the expression inside the exponential by −W we have, by the same calculation as in (3.22), for almost all x ∈ U , Hence, by bounding exp(−W ) ≤ 1 and using that for any s ≥ 0, s ≤ exp(s) we have Hence we see that for M = 4π(M 1 + 1) and ω = 1 (3.25) holds. Thus we can apply [32, Theorem 5.2.2] which proves that A ε (t)| Y is a stable family in Y , which completes the proof of the lemma.
Having proved condition (H2) in the previous lemma we now move on to proving that condition (H3) holds.
for some C > 0. Hence Y ⊂ D(A ε (t)) and A ε (t) is bounded as a map Y → X. It remains to prove that t → A ε (t) is continuous in the B(Y, X) norm. Let P ∈ Y , t ≥ 0 and δ > 0. We seek an η > 0 such that for all s ≥ 0 with |t − s| < η, we have

Now by the definition of
Now take R ≥ 1 sufficiently large such that R > 2C/δ, where C is as in lemma 6.1 in section 6 below. Further take η > 0 sufficiently small so that, CR 5 η α < δ/2. Then lemma 6.1 gives that for any s such that |t − s| α < η and for almost all x ∈ U , Hence substituting this into (3.26), Taking the supremum over all The above lemmas have proved that conditions (H1), (H2) and (H3) hold. We now prove that the evolution system that results from [32, Theorem 5.3.1] is indeed U ε as defined in (3.12). We first show in the following lemma that U ε is indeed an evolution system. Lemma 3.10. Let U ε be as in (3.12). U ε is an exponentially bounded evolution family on L 1 (U ×R 3 ).
Proof. We use [31, definition 3.1]. It is clear to see that U ε (s, s) is the identity operator. Further, for 0 ≤ s ≤ r ≤ t and f ∈ L 1 (U × R 3 ) we have by (3.12), Exponential boundedness follows with M = 1, ω = 0 by bounding the exponential term in U ε by 1.
In the following proposition we now prove that U ε is indeed strongly continuous.
Proposition 3.11. The evolution family U ε is strongly continuous.
To prove this proposition we use part 2 of [31, Proposition 3.2]. In the following lemmas we prove that iii) holds, that is, uniformly for 0 ≤ s ≤ t in compact subsets, s) is bounded. The proposition gives that this is equivalent to i), strong continuity. We note that c) has been proved in lemma 3.10. We prove a) and b) separately in the following two lemmas.
To simplify notation here define: , t ≥ 0 and δ > 0. We show that for 0 ≤ s ≤ t sufficiently close to t, This implies for t − s sufficiently small By the uniform continuity of η on U × B R (0) Hence by (3.30) and (3.31) for t − s sufficiently small, The required result follows by (3.32) and comparing f and U ε (t, s)f with η and U ε (t, s)η respectively.
Proof. Fix f ∈ L 1 (U × R 3 ) and δ > 0. Let h > 0. By lemma 3.10 U ε is an evolution family so, we can follow the proof of lemma 3.12 to prove that for h sufficiently small, It remains to prove that Fix δ > 0. Let h > 0. Then using (3.27), Now since E ε (t, s, x, v) ≤ 1 we have that We can make this less than δ/2 by approximating f with a test function η ∈ C ∞ c (U × R 3 ) as in the above lemma. We now look to I 2 . Firstly since f ∈ L 1 (U × R 3 ) there exists an R > 0 such that, 14 Hence, By the mean value theorem for any α, β ≤ 0 there exists an θ ∈ (α, β) ∪ (β, α) such that By lemma 6.1, for any R 2 ≥ 1 and for almost all x ∈ U , Hence, by a similar calculation to (3.29), . 15 This gives that Now take R 2 ≥ 1 sufficiently large such that and h > 0 sufficiently small so that both Substituting this into (3.34) gives I 2 < δ/2. Returning to (3.33) this gives for h > 0 sufficiently small, which completes the proof of the lemma.
Proof of proposition 3.11. This proposition follows from lemma 3.12 and lemma 3.13.
Finally to prove proposition 3.3 it remains to prove that U ε satisfies the properties (E1), (E2) and (E3). Proof. By bounding the exponential term by 1 it is clear that (E1) holds with M = 1, ω = 0. Now let P ∈ Y and Ω ⊂ U × R 3 be measurable. Then And using (3.27) we have 16 Hence This proves (E2). Further Hence This proves (E3) which completes the proof of the lemma.
We can finally now combine all the results in this subsection to prove proposition 3.3.
Proof of proposition 3.3. Let ε ≥ 0. By lemmas 3.5, 3.8 and 3.9 we can apply [32, Theorem 5.3.1]. This gives that there exists a unique evolution system satisfying (E1), (E2), (E3). By lemma 3.10, U ε is an exponentially bounded evolution family and by proposition 3.11 it is strongly continuous. By proposition 3.14, U ε satisfies these conditions and hence the solution is given by P ε,(0) t = U ε (t, 0)f 0 as required.

Existence of Non-Autonomous Linear Boltzmann Solution.
In this subsection we prove that there exists a solution to the ε dependent non-autonomous linear Boltzmann equation (3.1). We prove the result by adapting the method of [2].
Remark 3.16. Later, in proposition 3.35, we are able to show that for any ε ≥ 0 and any t ∈ [0, T ], f ε t is a probability measure on U × R 3 and that f ε t converges in L 1 to f 0 t uniformly for t ∈ [0, T ]. We first introduce some notation. For ε, By the use of Carleman's representation (see [11] and [6, Section 3]) we have Lemma 3.17. For any f ∈ L 1 + (U × R 3 ) and any s, Then for any r > s we have, using the substitutionx = x − tv and Fubini's theorem, Hence Q(t) is finite for almost all t ≥ s which proves the lemma.
Lemma 3. 18. For any f ∈ L 1 + (U × R 3 ) and any s ≥ 0, U ε (t, s)f ∈ D(B ε (t)) for almost every t ≥ s and the mapping [s, ∞) ∋ t → B ε (t)U (t, s)f is measurable. Moreover, for any r ≥ s, Proof. By changing from pre to post collisional variables, see for example [15, Chapter 2, section 1.4.5], we have, for any x ∈ U , The required results now follow from the statement and proof of the previous lemma.
Proof of proposition 3.15. For this proof we use [2]. The above two lemmas give that the modification of [2, lemma 5.11, corollary 5.12 and assumptions 5.1] to our situation hold. Hence, as in [2, section 5.2], we see that [2, theorem 2.1] holds, which gives that there exists an evolution family V ε (t, s).
So by (3.37), Hence, Remark 3.19. We were unable to adapt the honesty results of [2] to our situation, so we cannot yet deduce that V ε is an honest semigroup and that the solution f ε t = V ε (t, 0)f 0 conserves mass in the expect way. Honestly is proved later in proposition 3.35 by exploiting the connection to the idealised equation.

Building the Solution.
In this subsection we construct the function P ε t (Φ) iteratively and prove that is satisfies the properties in theorem 3.1. After defining P ε t (Φ), we define P ε,(j) t , which similarly to P ε,(0) t , can be thought of as the probability that the tagged particle is at a certain position and has experienced exactly j collisions. Once a few properties of P ε,(j) t have been checked the majority of theorem 3.1 follows. Proving that P ε t is indeed a probability measure on MT requires a careful analysis of the time derivative of P ε t . This subsection differs from our previous work [28] in two ways. Firstly the ε dependence, which is important in the later proof to deal with spatial dependence for positive ε, but makes little technical difference. Secondly there are significant differences in proving that P ε t is a probability measure. Previously it followed from the honesty of the solution of the autonomous linear Boltzmann equation that the idealised distribution is a probability measure. However in this case we do not have the equivalent honesty result for the non-autonomous linear Boltzmann. Therefore we prove that P ε t is a probability measure by explicitly showing that the measure of the whole space has zero derivative with respect to time. This requires a significant number of calculations. That is, T j contains all trees with exactly j collisions. Let ε ≥ 0 and t ∈ [0, T ]. For Φ ∈ T 0 define , v(t)).

Else define
The right hand side of this equation depends on P ε τ (Φ) but sinceΦ has degree exactly one less than Φ and we have defined P ε t (Φ) for trees Φ with degree 0 the equation is well defined. Note that this definition implies that for any, ε ≥ 0, Φ ∈ MT and τ ≤ s ≤ t ≤ T .
is absolutely continuous with respect to the Lebesgue measure on U × R 3 .
Proof. Let j = 1 and Ω ⊂ U × R 3 be measurable. Then by (3.41) we have (3.44) We now introduce a change of coordinates (ν, Computing the Jacobian of this transformation, where the non-filled entries are not required to compute the determinant. We now see that the bottom right 2x2 matrix has determinant −1 and hence the absolute value of the determinant of the Jacobi matrix is 1. We note that under this transformation for t ≥ τ , (x(t), v(t)) = (x, v) and for Hence with this transformation (3.44) becomes Hence we see that if the Lebesgue measure of Ω is zero then P ε,(1) t (Ω) equals zero also. For j ≥ 1 we use a similar approach using the iterative formula for P ε t (Φ).
Remark 3.23. By the Radon-Nikodym theorem it follows that P ε,(j) t has a density, which we also denote by P ε,(j) t . Hence for any Ω ⊂ U × R 3 we have that Proof. First consider j = 0. We prove that for any Ω ⊂ U × R 3 measurable, By the definition of B ε (3.10) and U ε (3.12) we have, for Hence by (3.45) we notice that this is equal to the right hand side of (3.47). Hence for j = 0 (3.46) holds for almost all (x, v) ∈ U × R 3 . For j ≥ 1 one takes a similar approach.
We can now prove the existence part of theorem 3.1. The next lemmas provide the additional properties.
Proof. Firstly let Ω ⊂ U × R 3 be measurable. By proposition 3.25, and since each P ε,(j) t is positive, the monotone convergence theorem and definition 3.21, Thus P ε t ∈ L 1 (MT ). Now we check that P ε t (Φ) indeed solves (3.2). For Φ ∈ T 0 noting that x(t) = x 0 + tv 0 and v(t) = v 0 , we have for t ≥ 0 We see that this gives the required initial value at t = 0, that it is differentiable with respect to t and differentiates to give the required term. Now consider Φ ∈ T j for j ≥ 1. By definition (3.41) we see that for t < τ , P ε t (Φ) = 0 and that P ε τ (Φ) has the required form. We also see that for t > τ we have We now prove (3.6). Let K > 0 be as in proposition 3.15. By a similar argument to (3.49), by using the same method as the proof of lemma 3.22 and by proposition 3.15 we have We see that (3.7) has been proved in (3.49).
The only remaining parts of theorem 3.1 are that P ε t is a probability measure on MT and that (3.8) holds. We remark here that for the corresponding result for P t in the autonomous case being a probability measure resulted from the fact that we were able to prove that the semigroup defining the solution of the autonomous linear Boltzmann equation was honest and hence conserved mass. However in this non-autonomous case we have not been able to find equivalent honesty results. Therefore we prove that P ε t is a probability measure explicitly by showing that MT P ε t (Φ) dΦ is differentiable with respect to t and has derivative zero. To that aim, the following lemmas calculate various limits that are required to show that MT P ε t (Φ) dΦ is differentiable, which is finally proved in lemma 3.32 and 3.33.
Lemma 3.27. Let ε ≥ 0 and Ψ ∈ MT . Then for t ≥ τ , Proof. We begin with (3.52). Let t ≥ τ . Firstly, Noting that for s, t ≥ τ , |x(s) − x(t)| = |t − s||v(τ )| it follows by lemma 6.1, with Now let δ > 0. By the proof of proposition 3.26 we know that P ε s (Φ) is differentiable with respect to s and hence continuous for s ≥ τ . So for h sufficiently small and any s ∈ [t, t + h] .
This, together with (3.54) and (3.55) proves (3.52). The proof of (3.53) is similar but we must exclude t = τ because in that case P ε s (Φ) in the integrand is always 0. For t > τ we take h sufficiently small so that t − h > τ and hence P ε s (Φ) is continuous with respect to s for s ∈ [t − h, t]. The result now follows by the same method as (3.52). Proof. Let t ∈ (0, T ]. Then MT t ∩ T 0 = ∅ since for any Φ ∈ T 0 , τ = 0. Now for any j ≥ 1, MT t ∩ T j is a set of co-dimension 1 in T j (since one component, the final collision time, is fixed) and hence has zero measure. Since, it follows that MT t is set of zero measure.
(3.58) 24 and for almost all Ψ ∈ MT Proof. Let t ∈ (0, T ]. We first prove (3.58). Let Ψ ∈ MT . If t < τ then the left hand side is zero and the right side is zero also, since for h sufficiently small P ε s (Ψ) = 0 for all s ∈ [t, t + h]. Suppose t ≥ τ . Note that, Hence to prove (3.58) we show that So by using (3.52) it remains to prove that (3.60) Recall Ψ ′ = Ψ ∪ (s,ν,v). Denote by w the velocity of the root particle of Ψ ′ after its final collision at s. Then, ≤ hπM g (1 + 2|v(τ )| + |v|).
It follows that Hence Let δ > 0. By (2.3) there exists an R > 0 such that, . Hence, Further for h sufficiently small, By substituting this and (3.62) into (3.61) we see that (3.60) holds, which concludes the proof of (3.58). We now prove (3.59), which we prove holds for all Ψ ∈ MT \ MT t . Indeed MT t is a set of zero measure by lemma 3.29. Let Ψ ∈ MT \ MT t . If τ > t then the left hand side of (3.59) is zero and the right hand side is also zero since for any s ∈ [t − h, t], P ε s (Φ) = 0. If t > τ we use the same method as we used for (3.58), using (3.53) instead of (3.52).
Lemma 3.32. Let ε ≥ 0 and t ∈ (0, T ]. Then ∂ + t MT P ε t (Φ) dΦ exists and is equal to zero. Proof. Fix ε ≥ 0 and t ∈ (0, T ]. We want to show that For MT S as defined in definition 3.28 and h > 0 we have, We show that each of these terms converges and that their sum is zero. Firstly, Now note that for any h > 0, t+h t L ε s (Φ) ds ≤ hπM g (1 + |v(τ )|). Hence for any h > 0, By (3.6) it follows that Hence by the dominated convergence theorem and the fact that for any Φ with τ > t, P ε t (Φ) = 0, Hence by the dominated convergence theorem and (3.58), Combining (3.63),(3.64) and (3.65) we see that the limit indeed exists and is equal to zero, proving the lemma.
Proof. Fix ε ≥ 0 and t ∈ (0, T ]. We show that As in lemma 3.32 note that, We again show each limit exists and the sum is zero. Firstly By lemma 3.29, (3.59) and the dominated convergence theorem we have, (3.67) Now for the final term we first prove that for any Φ ∈ MT [0,t) To this aim fix Φ ∈ MT [0,t) . Then τ < t. Let h sufficiently small so that t − h > τ . Then since P ε s (Φ) is continuous for s ∈ [τ, T ] we have that P ε t−h (Φ) converges to P ε t (Φ) as h tends to zero. Further, This proves (3.68). Hence by the dominated convergence theorem and lemma 3.29 The following lemma is used to prove (3.8). Proof. Let ε be sufficiently small so that lemma 6.2 holds. We prove by induction on n, the number of collisions in Φ. Suppose n = 0. Then by definition 3.20, (3.35) and lemma 6.2, as required. Now suppose the result holds true for almost all Φ ∈ MT with n = j for some j ≥ 0 and let Ψ ∈ MT with n = j + 1 be such that the result holds forΨ and, Indeed by (2.4) and (2.6) this only excludes a set of zero measure. Let δ > 0. Then using (2.6) take ε sufficiently small so that, . 28 And using the inductive assumption take ε sufficiently small so that, Now by the inductive assumption for ε sufficiently small, So, as in the base case, take ε sufficiently small so that .

(3.72)
Hence by (3.70), (3.71) and (3.72) and bounding the exponential term by 1, for ε sufficiently small, This completes the inductive step and so proves the result.
We can now prove the remainder of theorem 3.1. Proof. The empirical distribution of the tagged particlef N t can be equivalently obtained by integrating over the background particles of the N + 1 particle distribution. The initial distribution of the N + 1 particles is given by which, under our assumptions, is absolutely continuous with respect to Lebesgue measure on (U × R 3 ) N +1 . As the N + 1 particle flow preserves Lebesgue measure, this implies that the empiric N + 1 particle distribution is absolutely continuous with respect to Lebesgue measure on (U × R 3 ) N +1 . Hence its marginalf N t is absolutely continuous with respect to the Lebesgue measure on U × R 3 . We now describe the set of 'good' trees that we will work with.
That is if the root collides with a background particle at time t j , it has not collided with that background particle before in the tree and up to time T it does not come into contact with that particle again. Define R(ε) := {Φ ∈ MT : Φ is re-collision free at diameter ε}.
Definition 4.6. A tree Φ ∈ MT is called free from initial overlap at diameter ε > 0 if initially the root is at least ε away from all the background particles. That is if for j = 1, · · · , N we define S(ε) := {Φ ∈ MT : Φ is free from initial overlap at diameter ε}.
and Φ is non-grazing .
Proof. The only non-trivial conditions are checking that S(ε) and R(ε) are increasing. To this aim suppose that ε ′ < ε and Φ ∈ S(ε). If n = 0 then it follows from the definition that Φ ∈ S(ε ′ ). Else n ≥ 1. For the background particles not involved in the tree it is clear that reducing ε to ε ′ will not cause initial overlap. For 1 ≤ j ≤ n the initial position of the background particle corresponding to collision j is and so Φ ∈ S(ε ′ ). Now suppose that ε ′ < ε and Φ / ∈ R(ε ′ ). Then in particular n ≥ 1 and there exists a 1 ≤ j ≤ n and t > t j such that, if we denote the velocity of the background particle j after its collision at time t j byv, Hence since the left in side is continuous with respect to t it must be that there exists a t ′ such that, x(t ′ ) − (x(t j ) + (t ′ − t j )v) ∈ −εν j + εS 2 , i.e. Φ / ∈ R(ε). Hence R(ε) ⊂ R(ε ′ ) and so G(ε) ⊂ G(ε ′ ).
The last lemma allows a simplified definition of G(ε) compared to [28], where the monotonicity was enforced by taking unions. We will later give restrictions on V and M in order to control bounds in order to prove required results. Lemma 4.9. Let ε > 0 and Φ ∈ G(ε) thenP ε t is absolutely continuous with respect to the Lebesgue measure λ on a neighbourhood of Φ.
Proof. The only difference to the proof of [28,Lemma 4.8] is that instead of C h,j g 0 (v) dx dv, we have C h,j g 0 (x, v) dx dv due to g 0 depending on x. As g 0 ∈ L 1 (U × R 3 ) by assumption, the argument can be concluded in the same way.
From now on we letP ε That is ½ ε t [Φ](x,v) 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 and zero otherwise. For Φ ∈ MT , t ≥ 0 and ε > 0, define the gain operator, and define the loss operator, For someĈ(ε) > 0 depending on t and Φ of o(1) as ε tends to zero detailed later. Finally define the operatorQ ε t as follows,Q ε t =Q ε,+ t −Q ε,− t .
Theorem 4.10. For ε sufficiently small and for all Φ ∈ G(ε),P ε t solves the following The functions γ ε and ζ ε are given by We prove this theorem by breaking it into several lemmas proving the initial data, gain and loss term separately using the definition ofP t . Firstly, the initial condition requirement forP ε t . Definition 4.11. Let ω 0 ∈ U × R 3 be the random initial position and velocity of the tagged particle. For 1 ≤ j ≤ N let ω j be the random initial position and velocity of the jth background particle. By our assumptions ω 0 has distribution f 0 and each ω j has distribution g 0 . Finally let ω = (ω 1 , . . . , ω N ). Proof. If n(Φ) > 0,P 0 (Φ) = 0, because the tree involves collisions happening at some positive time and as such cannot have occurred at time 0. Else n(Φ) = 0, so Φ contains only the root particle. The probability of finding the root at the given initial data (x 0 , v 0 ) is f 0 (x 0 , v 0 ). But this must be multiplied by a factor less than one because we rule out situations that give initial overlap of the root particle with a background particle. Firstly we calculate, Proof. The proof is unchanged from the proof of [28, lemma 4.16].
Before we can prove the loss term we require a number of lemmas that are used to justify thatP ε t is differentiable for t > τ and has the required derivative.
Definition 4.14. Let ε > 0 and Φ ∈ G(ε). For h > 0 define That is W ε t,h (Φ) contains all possible initial points for a background particle to start such that, if it travels with constant velocity, it will collide the tagged particle at some time in (t, t + h). Further define, I ε t,h (Φ) :=
Indeed by (2.2) and (2.6) this only excludes a set of zero measure. As in the previous lemma we have, Combining (6.3) and (6.4) we have for C as in the previous lemma, Substituting R = ε −α/6 gives the required result.