The free path in a high velocity random flight process associated to a Lorentz gas in an external field
We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.
- 1 Introduction
- 2 The free path of the particle
- 3 Convergence of the free path process
- 4 The Process on its natural time scale
- 5 Classifying the boundaries of
- 6 Higher dimensions and general potentials
We study the behavior of a random flight process that arises as the Boltzmann-Grad limit of a random scatterer Lorentz gas in an external field. The Lorentz gas model was introduced in 1905 by H.A. Lorentz [Lor05] as a model for the motion of electrons in metallic bodies. Since its introduction, the model has been widely studied by both mathematicians and physicists (see [Det14] for a recent survey). In this model, a point particle travels in an array of fixed convex scatterers. When the particle comes in contact with a scatterer it reflects specularly. This variant is referred to as the “hard core” version. There is also a “soft core” version in which the scatterers deflect the moving particle via a potential instead of a hard boundary. Of the many variations on the hard core model, our study is motivated by the version in which an array of spherical scatterers is chosen randomly and the flight of the moving particle is determined by the action of an external field. The random scatterer Lorentz gas is difficult to study directly even in the absence of an external field. One approach that has proven tractable, however, is the Boltzmann-Grad limit. The Boltzmann-Grad limit is a low density limit in which the number of scatterers in a fixed box goes to infinity while, at the same time, the size of each scatterer goes to zero in such a way that the total volume of the scatterers in the box goes to zero. If the centers of scatterers are placed according to a Poisson process and the rates are chosen appropriately, the asymptotic behavior of the moving particle is described by a Markovian random flight process [Gal78, Spo78, Spo88]. The Markovian nature of the Boltzmann-Grad limit is due to the following two observations: re-collisions with scatterers become unlikely as the size of each scatterer goes to zero and the Poisson nature of the scatterer locations which means that knowing the location of one scatterer does not give information about the locations of the other scatterers. Since analyzing the random Lorentz gas directly is beyond the capability of current techniques, this random flight model is commonly studied in both the mathematics literature [RT99, Vys06, BFS00, BCLDM02, BR14] and the physics literature [ADBLP10, dWB04, vBM05, MvB04] to gain insight into the behavior of random Lorentz gas models.
We are interested in the regime in which the particle’s velocity is (typically) large. There are several natural examples of this setting. The first is an external field which accelerates the particle towards infinity. For example, the influence of a constant gravitational field has been studied for the random flight process in both a constant density of scatterers [PW79, RT99], a variable density of scatterers [BR14], and for the periodic Lorentz gas in two dimensions [CD09]. The random flight model in a gravitational field has also been used as a model for a particle percolating through a porous medium [WE82]. A second example of a high velocity limit is particle motion in a centered, mean zero, isotropic force field [DK09, KR06]. The soft core Poisson Lorentz gas is an example of this situation. A discussion of how much of the past work on particles moving in random force fields fits into the high velocity framework can be found in [DK09].
A unique contribution of this work, is that we study fields that are in general not mean zero, unlike prior work on random force fields. For example, the constant field directed towards the origin fits into our setting. In such fields, conservation of energy implies that particles have bounded trajectories. We are interested in the behavior of the system as the velocity of the particle, or equivalently the energy of the system, is sent to infinity. Our primary focus is to study the free path of the particle, which is the path of the particle between two reflection times. The study of the free path has played a key role in much of the work on the hard core Lorentz gas. To highlight a few examples, the free path played an important role in the recent derivation of the Boltzmann-Grad limit of the periodic Lorentz gas [MS11a, MS11b], which is substantially more complicated than the corresponding result for the random Lorentz gas. Furthermore, it has also played a central role in the derivation of a diffusive limit for the two dimensional periodic Lorentz gas in a gravitational field [CD09]. For these reasons, the free path has become an object of interest in its own right, see e.g [MS10, MS14a, MS14b, Wen12, BZ07, BGW98, GW00, CG03]. We determine precisely the asymptotic behavior of the free path of a particle whose domain of motion is predominantly in the high velocity regime. The primary complication that arises in our study is that the domain of the particle is determined by the external field and, as a result, the particle’s velocity is not bounded away from zero. If the particle enters a part of the domain where it is moving slowly, then there can be a long time before a reflection occurs, and we must control this in order to analyze the asymptotic behavior of the mean free path. As a consequence of our analysis of the free path, we obtain several diffusion approximations that illustrate different aspects of the particle’s behavior.
We remark that our techniques may be of interest beyond the study of the Boltzmann-Grad limit of a Lorentz gas. The random flight process we study falls into a general class of models known as transport process. Models similar to the ones we study here are used in many areas of physics and chemistry as microscopic models for particle behavior. For example, they arise in neuron transport [AB99], the study of semiconductors [BAD96], electron transfer dynamics [HW13, WW05], and numerous other fields [BR00, Deg98, DGL81, DLLS13, HLB09].
1.1. The Model
We are primarily interested in the process in dimension three, which is the most interesting spatial dimension and also a dimension where certain technical simplifications occur. Let us start with a more detailed description of the Boltzmann-Grad limit. Let and be differentiable. The function will serve as the potential for a conservative force and will be the density of scatterers. Fix an energy level and consider a particle moving in the potential with total energy . By conservation of energy
where is the particle’s position and is its velocity. In this setting, the Boltzmann-Grad limit can be obtained as follows. Assume spherical scatterers with radius are placed so their centers are the points of a Poisson process with intensity . Further assume that for almost every initial condition the trajectory of a particle moving in the potential with total energy is open and has infinite length. The arguments of [Spo78, Spo88] can easily be adapted to include the external field and produce the following result: if the initial position and velocity of the particle is absolutely continuous with respect to Lebesgue measure on the constant energy surface then the distribution of the position and velocity process of the particle converges as , in the sense of convergence of finite dimensional distributions, to a Markovian random flight process with generator
where is the normalized surface measure on the unit sphere , see [Spo88]. A similar result holds in dimensions other than , but instead of the integral being against the normalized surface measure it is against a kernel that depends on .
The assumption that for almost every initial condition the trajectory of a particle with mass moving in the potential with total energy is open and has infinite length is essential to the derivation of the Boltzmann-Grad limit. The intuition for its necessity is that it is possible for a periodic orbit to avoid all of the scatterers in a random arrangement. If this happens, the effect on the limiting process is that one must keep track of whether or not an orbit has closed, which creates long range dependencies in the limiting process. The effect of this phenomenon on the Boltzmann-Grad limit has been the subject of study in both the mathematics and physics literature [BHHP96, BMHH97, BHPH01, DR04, KS98]. In these cases one may still obtain a limiting process with generator of the form (1.1), but additional randomness must be added to array of scatterers. For example, they can be allowed to move (but not transfer energy), or they can be arranged according to a space-time Poisson process where a scatterer comes into existence, exists for a certain amount of time, and then disappears.
The process can be constructed in the following way, which explains the name “random flight process”. Let with and denote the solution to the initial value problem
We construct our process recursively as follows. Set and let . For , assuming we have defined , we let be independent of this part of the path and uniformly distributed on and let satisfy
For we then define
We remark that under very mild assumptions a.s. and thus this defines the path of the particle for all times. Intuitively, defines the th reflection of our particle by a scatterer. By studying the free path of the particle, we mean to study, in particular, the conditional law of given under an appropriate scaling of the parameters in the model. Note that the path of the particle on the time interval is the path between two consecutive reflections. This is why we call it the free path.
At this point we make a simple but important observation. By conservation of energy,
so that, if we define
Since is independent of , this implies that if we define , then is a Markov chain. That the index in this chain starts at is an artifact of our deterministic choice of . If instead of choosing in the construction above we took , with uniformly distributed on , then is a Markov chain and its transition operator is
where is a random variable with distribution
and conditional on , is distributed like .
In order to avoid the notational difficulties and technical assumptions on the geometry of the set that come from working with a general potential , we restrict ourselves to the spherically symmetric case where and , where . In this case angular momentum is conserved, which implies that the motion of the particle between collisions with scatterers takes place in a plane. Consequently, we may restrict our attention to for some of the analysis of the free path. This lets us work in polar coordinates for these aspects of the analysis. Throughout let , , and be the set of real valued functions on having continuous derivatives. We let be the potential and we assume our particle has mass and total energy . Let denote the trajectory in polar coordinates of a particle with mass , total energy , initial position and whose velocity vector at time makes angle with the radial vector. Consequently is the solution to the equations of motion in polar coordinates for a particle of mass in the potential :
with initial conditions
where is the radial unit vector, is the angular unit vector and is the angle the initial velocity makes with the line segment from the origin to the point . We remark that in this setup it is generally possible for the trajectory to hit the origin in finite time, at which point the equations of motion are no longer well defined. In this case we continue the path in such a way that momentum is continuous, if possible and such that angular momentum is conserved. The extension is most easily described in Euclidean coordinates, since it may involve the angular component reentering from in polar coordinates. Suppose that is a trajectory that hits the origin for the first time at time . We extend the path to the interval by defining for . Since , we can continue the path by the equations of motion until it hits the origin again, at which point we repeat the extension procedure. A straightforward calculation using the radial symmetry of the potential shows that
so our extension solves (1.2) away from the origin and it is easy to see that is continuous at if exists. In this way, we guarantee that all initial conditions lead to trajectories that exist for all time.
Let denote the speed of the particle as a function of time. By conservation of energy, we have
and, as a result,
We can also write as a function of angular and radial velocity:
In this section, we give heuristic arguments to motivate our scaling. Consider the following simple situation that encapsulates the phenomena in which we are interested. In particular, suppose that , , so that produces a uniform acceleration towards the origin, and that . Suppose we start the particle at the origin with speed moving in direction . By conservation of energy the total energy of the particle will be
Since produces a uniform acceleration towards the origin, the maximum distance the particle can move away from the origin is . Consequently, there is no way to rescale the trajectory of the particle to obtain a diffusive limit: Any scaling of space will cause the trajectory to degenerate, and scaling time alone will produce jumps. However, we can obtain a diffusive limit if we let (and, consequently, ) go to infinity. This is the type of high velocity limit considered in [DK09, KR06]. In those papers, however, the high velocity was used to prevent the particle from being trapped in a bounded domain, while in our setting the particle is still trapped in a bounded domain, but the size of the domain goes to infinity as does.
Treating as a parameter, the energy becomes a function of , specifically is the maximum distance the particle can travel from the origin. Let be the trajectory of the random flight process with initial conditions and initial velocity , where is uniformly distributed on . In order to obtain a diffusive limit for , we rescale space so that its maximum distance from the origin remains constant. In particular, we look at , which travels at most distance from the origin. This is the process for which we analyze the behavior of the free paths.
Since the velocity of the particle is going to infinity, the time between reflections is going to , so we can expect a diffusive limit if we scale time appropriately. Since , the distance travels between reflections has an exponential distribution with rate 1, so the distance travels between reflections is of order . Writing down the equations of motion for between reflections, we find that its trajectories are the same as those governed by the potential moving with total energy , initial velocity , and the scattering rate as measured along these paths is .
By Brownian scaling, in order to obtain a diffusive limit, we expect that we should scale time so that there are approximately reflections per unit time. Typically, we expect the particle to be distance of order from the origin, and if and is a reflection time, the time until the next reflection is approximately
Thus the amount of time for reflections to occur is of the order . This suggests looking for a diffusive limit of as tends to infinity, which is what we undertake.
1.3. General scaling and assumptions
In general, we allow to be a small perturbation of the constant density and to produce a small perturbation of the constant field, small being relative to the speed of the particle. In particular, we will assume the following conditions.
(A1) The path of the particle evolves as the position of a random flight process where the density of scatters is
the potential energy of the field
and the total energy of the particle
where , , and are fixed and independent of and and . That is, the density of scatterers and potential are spherically symmetric. By (1.12) we can write the speed as a function of and ; consequently the speed is also rescaled as
The trajectory of the particle with these rescaled parameters we denote by
when we are working in with the path in 3 three dimensions and by
when we are considering the path in polar coordinates with respect to its plane of motion. When no confusion will arise, we will leave the dependence on initial conditions implicit.
In general, subscripts or superscripts of refer to distributions relative to these scaled parameters. For example, corresponding to (1.8) is the random variable with distribution
and corresponding to (1.7) is the Markov chain with transition operator
where is uniformly distributed on and conditionally given , is distributed like . Similarly, will be constructed as in (1.6) we the rescaled parameters.
To make the connection with are heuristic arguments in Section 1.2, if we take , and then . Thus the scaling we do here accounts for increasing the initial speed and scaling space to keep the particle’s trajectory contained. This is the scaling under which we analyze the free path of the particle. We will scale time separately when we consider diffusive limits.
(A2) We assume that for all the process evolves in a domain where or , where . The domain is chosen so that for all . This is equivalent to for all .
We require the particle to have positive speed in the interior so the time between reflections approaches as goes to infinity. This way we obtain a nontrivial diffusion limit.
(A3) On the boundary we have the following assumptions.
The conditions on force the speed of the particle to be zero at the respective endpoints and , while the conditions on ensure the force field at the endpoints is pointing towards the interior . Having zero speed at the boundaries and having the force field point ‘inwards’ prevents the particle from leaving the domain .
This condition ensures the particle is not trapped at the origin.
If we require that for any ,
This condition implies that for any , which shows that the reflection rate does not go to as the process goes to infinity.
This smoothness assumption ensures that the velocity and the acceleration of the particle depend continuously on the position . If we can relax this condition to so that we can allow potentials of the form which are not defined at .
(A5) The density is spherically symmetric and satisfies
We require some smoothness from because the diffusion limit we get depends on the derivative . The second assumption is needed because we do not want to have regions where the the reflection rate goes to . If one allows to approach , then the free path may scale differently when the process is started in these regions. Examples of such situations are dealt with in Section 8 of [BR14].
Our first result considers the asymptotic behavior of the steps of the free path Markov chain , where is the location of the particle at the time of the ’th reflection and, anticipating our diffusion approximations, is the time of the ’th collision. This Markov chain has transition operator
The following result characterizes the asymptotic mean and covariance structure of the free path chain.
be the scaled drift of , and let be a compact subset of . Then
uniformly on . Furthermore, let
This theorem is a combination of Lemmas 2.10 and 2.11 below. Heuristically, this says that both and impart a drift towards areas where the corresponding function has a smaller value. That is, the particle prefers to move towards areas where it travels quickly and where there are few scatterers, which may be competing influences. It is interesting to note that the potential does not appear in the covariance terms. It can, and will, effect the diffusion coefficient of the limiting diffusion only through an overall time change.
With Theorem 6.1 in hand, it is straightforward to obtain the following diffusion approximation, which separates out the effects of the particle’s position at times of reflection from the effects of speed at which the particle is moving. Let .
Let be a diffusion on whose generator acts on functions with compact support in by
and killed if/when hits the boundary of . Consider any with and start the process at , where . Define the stopping times
Then, as , the family of continuous time processes converges in distribution on the Skorokhod space to the diffusion
with initial position . This convergence happens jointly with the convergence of the hitting times, .
This theorem is proved in Section 3, where it is restated for convenience.
The convergence above also holds without stopping near a boundary point of that is inaccessible for the diffusion . See Section 5 for how one can determine when a point is inaccessible. In particular, the boundaries are inaccessible for the constant acceleration towards the origin and for Newtonian gravity centered at the origin.
This type of limit theorem, that gives joint convergence of a process observed when it changes direction and the time spent between such changes up to that point, is typical of the continuous time random walk literature, see e.g. [MS14c]. Although our models can naturally be viewed as transport processes, our results on the free path of the particle suggest an approach to diffusive limits that has more in common with that used to study Continuous Time Random Walks (CTRW) [MS04, MS14c] than that typically used to study transport processes. CTRW models also arise in many physical applications (see e.g. [WZL14, MK00, BCDS06, SBMB03]) and have been particularly useful in studying anomalous diffusions. The advantage of this approach is that, on the diffusive scale, we may easily distinguish between the effects of the particle’s displacement between collisions and the effects of the speed at which the particle is traveling. In this way, our approach is similar to the one taken in [BR14], and our work can be seen as an extension of the methods used there. This approach is also similar to the approach in [CD09] to studying the two dimensional periodic Lorentz gas in a gravitational field, though, of course, our setting is far simpler.
By inverting the time process and showing that the free path is typically not far from the straight line between its endpoints, we can arrive at a diffusion approximation for , the trajectory of the particle. In particular, we obtain the following result.
Let denote the trajectory of the particle and fix with . Suppose that and define
We have the following convergence in distribution on the Skorokhod space :
where is as in Theorem 3.1 and is the time change given by
and is the inverse operator defined by . Furthermore, the time changed process is a diffusion process whose generator acts on functions with compact support in by
This theorem is proved in Section 4, where it is restated for convenience.
The cutoffs are necessary because of the generality the potentials and scattering densities we allow permit very different behaviors at the boundary. In some cases we the boundaries to be inaccessible (see Section 5), while in other cases we expect the boundaries to be reflecting as in [RT99] and in yet others, like , the origin should trap the particle. We leave a detailed investigation of the boundary as an open problem.
Our results have much in common with [Cos91], but there are key differences at both heuristic and technical levels and, as a result, there is no overlap between the precise results. For instance, our models do not satisfy the underlying technical assumptions of [Cos91]. The function from [Cos91], which controls the reflection rate, has to be differentiable up to the boundary of the region where the motion takes place (see assumption (H2) from [Cos91]). In our case we have singularities at the origin and when we also have singularities at the boundary of the domain. Furthermore, the speed of the particle must be bounded away from and in [Cos91], while there is no such restriction in our model. This is related to the fact that [Cos91] restricts the particle’s motion by imposing a reflecting boundary while in our case the domain of the particle’s motion is completely determined by and .
The form of the generator (1.1) puts the random flight process in the class of processes known as transport processes. There is a substantial literature on diffusion approximations of transport processes and, more generally random evolutions. Very general approaches and results can be found, for example in [Kur73, Pap75, BLP79, Cos91]. In order to apply these general techniques the generator needs to be smoothed to control for regions where the particle’s velocity is low. Nonetheless, it is possible to adapt these approaches to obtain Theorem 1.7. However, our results on the free path are stronger than the diffusion approximation results and cannot be recovered by these general methods. Moreover, these methods do not clarify the connection with the free path chain.
2. The free path of the particle
In this section we analyze the jumps of the free flight chain whose transition operator is given by (1.16). Our proofs are complicated by the fact that for a general potential it is not possible to explicitly find the trajectory of the particle in the absence of collisions and this makes it difficult to determine how long this path spends in domains where it is traveling slowly. Our first lemma will give us local control over the how the radial part of the path behaves.
For any and
for some depending on , and .
Throughout the remainder of this section we let be closed intervals satisfying
Also, for any , we define
to be the shortest time it takes for the particle’s displacement from its initial position to be at least when started inside .
then for all one has
Since , we have
where since is bounded away from .
The next lemma shows that the scaled trajectories converge to the starting point, uniformly over a fixed time interval.
Fix . Then
The claims about are slightly more complicated, so we prove those and leave the more obvious claims about to the reader. Let be such that . By Remark 2.2 there exists large enough such that whenever . Equivalently, for we have
for all . This proves the uniform convergence of to . Define
and note by (2.4) that
is bounded away from and imply that is uniformly continuous on . By construction for all and . Because converges uniformly to on compact sets, we have
by (2.8). This completes the proof. ∎
Since the free flight chain tracks the process at reflection times, we want to apply the estimates for (2.5) between reflections. By (1.3) and the rescaling, we know that for every , if at the particle is at position and reflects in the direction given by the angle , then is distributed like the random variable which we define by
Throughout, we will often suppress the and dependencies of and write or when no confusion will arise. To go back and forth between Euclidean and polar coordinates, note that we have the identity in distribution
where is defined as in (1.8) with the appropriately rescaled parameters. Consequently, for studying the asymptotic duration of the free path, there is no loss in studying in a uniform way over and .
For many of our proofs we require estimates that show the time between reflections approaches with high probability as the scaling factor goes to infinity. This will allow us to apply the local estimates from the expansion of in Lemma 2.1 and to show the free flight process does not undergo large jumps. We first prove some bounds on the moments of .
is bounded in for .
We will assume that or . The cases and can be treated similarly.