Propagation of Chaos for a Thermostated Kinetic Model

Propagation of Chaos for a Thermostated Kinetic Model

Abstract

We consider a system of point particles moving on a -dimensional torus . Each particle is subject to a uniform field and random speed conserving collisions with . This model is a variant of the Drude-Lorentz model of electrical conduction [1]. In order to avoid heating by the external field, the particles also interact with a Gaussian thermostat which keeps the total kinetic energy of the system constant. The thermostat induces a mean-field type of interaction between the particles. Here we prove that, starting from a product measure, in the limit , the one particle velocity distribution satisfies a self consistent Vlasov-Boltzmann equation, for all finite time . This is a consequence of “propagation of chaos”, which we also prove for this model.

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Dedicated to Herbert Spohn in friendship and appreciation

1 Introduction

The derivation of autonomous kinetic equations describing the time evolution of the one particle phase-space distribution function in a macroscopic system of interacting particles, , is a central problem in non-equilibrium statistical mechanics. Examples of such equations are the Boltzmann (BE), Vlasov (VE) and Boltzmann-Vlasov (BVE) equations [15]. All of these equations have a quadratic non-linearity, and their derivation from the -particle microscopic dynamics requires proving (or assuming) that the two particle distribution function, factorizes (in an appropriate sense) for times , , as a product when given that the -particle distribution has a factorized form at . Such a property of the -particle dynamics, which goes under the name of “propagation of chaos” (POC), was first proved by Marc Kac for a spatially homogeneous stochastic model leading to a BE [11]. In his model, one picks a pair of particles at random and changes their velocities as if they had undergone an energy conserving collision. Kac’s proof makes essential use of a boundedness property of the generator of the master equation (Kolmogorov forward equation) corresponding to the Kac process. This boundedness property is valid for a Maxwellian collision model in which the rate of collision is independent of the speed of the particles. It is not valid for hard-sphere collisions, for which the rate depends on single-particle speeds, which can become arbitrarily large with large even when the energy per particle is of unit order.

Propagation of chaos for the Kac model with hard sphere collisions was studied by McKean and his student Grunbaum [10]. The first complete proof was given by Sznitman [18] in 1984. These authors used a more probabilistic methodology, and avoided direct analytic estimates on the master equation. This method has been extended to more general collisions and to higher dimensions by many authors: for the latest results, which are quantitative, see [14]. This method was also used by Lanford in deriving the BE, for short times , for a system model of hard spheres evolving under Newtonian dynamics, in the Boltzmann-Grad limit, see [12] and book and articles by H. Spohn [16, 17].

The derivation of the VE for a weakly interacting Hamiltonian system was first done by Braun and Hepp [7]. They considered a system of -particles in which there is a smooth force on each particle that is a sum of contributions from particles with each term being of , see also Lanford [12]. There is then, in the limit , an essentially deterministic smooth force on a given particle and the evolution of the one particle distribution is then described by the VE.

There are no rigorous derivations of the VBE where, in addition to the smooth collective force, there are also discontinuous collisions. Actually, even the appropriate scaling limit in which the VBE could be derived is not obvious: the collision part is obtained in the Grad-Boltzmann limit which means low density; the Vlasov force is obtained in the mean field limit. The two regimes could be incompatible. In [2], it is shown that this is not the case, by proposing a scaling limit in which the VBE is derived formally from the BBGKY hierarchy.

In this paper we go some way towards deriving such a VBE. The system we study is a variation of the Drude-Lorentz model of electrical conduction. In this model, the electrons are described as point particles moving in a constant external electric field and undergoing energy conserving collisions with much heavier ions. In order to avoid the gain of energy by the electrons from the field, we use a Gaussian thermostat that fixes the total kinetic energy of the -particle system.

A model in which one particle moves on a two dimensional torus with fixed disc scatterers subject to a field and a Gaussian thermostat was originally introduced by Moran and Hoover [13]. Exact results for this system were derived in [8]. More information, both analytic and numerical, on this system for can be found in [4, 3]. There are no rigorous results for the deterministic case when , which was first studied in [6], and further studied in [5]

To obtain a many-particle model model that is amenable to analysis, it is natural to replace the deterministic collisions with the fixed disc scatterer by stochastic collisions. This too was done in [6] which introduced a variant of the model in which the deterministic collisions are replaced by “virtual collisions” in which each particle undergoes changes in the direction of its velocity, as in a collision with a disc scatterer, but now with the redirection of the velocity being random, and the “collision times” arriving in a Poisson stream, with a rate proportional to the particle speed. This variant is amenable to analysis for , as shown in [6]. In particular, [6] presented an heuristic derivation of a BE for this many-particle system. To make this derivation rigorous, one must prove POC for the model, which is the aim of the present paper.

In this class of models the collisions involve only one particle at a time, and are not the source of any sort of correlation. The particles interact only through the Gaussian thermostat. Dealing with this is naturally quite different from dealing with the effects of binary collisions. On a technical level, the novelty of the present paper is to provide means to deal with the correlating effects of the Gaussian thermostat. To keep the focus on this issue, we make one more simplifying assumption on the collisions: In the model we treat here, the virtual collision times arrive in a Poisson stream that is independent of particle speeds. In other words, our virtual collisions are “Maxwellian”. However, one still cannot use Kac’s strategy for proving POC in this case since the terms in the generator of the process representing the Gaussian thermostat involve derivatives, and are unbounded. Because the virtual collisions are not responsible for any correlation between the particles, the assumption of Maxwellian collisions is for technical reasons only. Later in the paper, when it is clear how we are using the Maxwellian property, we shall explain how we expect to be able to build on our present results and treat head-sphere virtual collisions as well. Interesting generalizations of the Gaussian thermostat problem that involve binary collisions have been studied by Wennberg and his students [19].

We now turn to a careful specification of the model we consider.

1.1 The Gaussian thermostatted particle system

The microscopic configuration of our system is given by , with , . The time evolution of the system between collisions is given by:

(1.1)

where

(1.2)

The term in (1.1) represents the Gaussian thermostat. It ensures that the kinetic energy per particle is constant in time. Each particle also undergoes collisions with “virtual” scatterers at random times according to a Poisson process with unit intensity which is independent of its velocity or position. A collision changes the direction but not the magnitude of the particle velocity. When a “virtual” collision takes place with incoming velocity , a unit vector is randomly selected according to a distribution , where , and the velocity of the particle is changed from to , where

For this means that .

Remark 1.1.

Observe that since and both yield the same outgoing velocity, we may freely suppose that is an even function on . Since , the two collision and are equally likely if

Moreover, for any two vectors and satisfying there is a collision that takes into : Simply take .

In the absence of an electric field, , each particle would move independently keeping its kinetic energy constant while the collisions would cause the velocity to become uniformly distributed on the dimensional sphere of radius . The force exerted by the Gaussian thermostat on the -th particle, induces, through , a mean-field type of interaction between the particles. When a particle speeds up (slows down) due to the interaction with the field some other particles have to slow down (speed up).

The probability distribution for the position and velocity of the particles will satisfy the master equation

(1.3)

where

(1.4)

and the integration is over the angles such that . Eq.(1.3) is to be solved subject to some initial condition . It is easy to see that the projection of on any energy surface will evolve autonomously. This evolution will satisfy the Döblin condition [9] for . Hence for any finite the effect of the stochastic collisions combined with the thermostated force is to make the approach a unique stationary state on each energy surface as . In fact this approach will be exponential for any finite . However in this paper we rather focus on the time evolution of the one particle marginal distribution for fixed as .

Now it seems reasonable to believe that in this limit one can use the law of large numbers to replace in (1.3) by its expectation value with respect to . When this is true, starting from a product measure

(1.5)

the system will stay in a product measure

and will satisfy an autonomous non-linear Vlasov-Boltzmann equation (VBE)

(1.6)

where and are given by

(1.7)

It is easy to check that and so independent of .

An important feature of the VBE (1.6) is that the current given by (1.7) satisfies an autonomous equation:

Lemma 1.2.

Let be a solution of the VBE (1.6), and let the corresponding current be defined by (1.7). Then satisfies the equation

(1.8)

where, for ,

(1.9)

in which denotes the area of in . In particular, for , , and . For , we have .

Proof.

We treat the case . By the definition of it suffices to show that

(1.10)

Since , the left hand side of (1.10) equals

Note that

To evaluate this last integral, introduce coordinates in which , and parameterize by where and . Then , and since ,

Combining calculations yields (1.10), and completes the proof. ∎

The initial value problem for the current equation (1.8) is readily solved: Write where is the component of parallel to . Then defining ,

(1.11)

The equation for is easily solved once the equation for is solved. Rescaling , and writing the equation for in the form , so that , we see that at

and hence

Since , , and by the definitions of and , . Hence the equation for has a unique solution with . The equation is solvable by separating variables, but we shall not need the explicit form here, though we use the uniqueness below.

The fact that solves an autonomous equation that has a unique solution allows us to proves existence and uniqueness for the spatially homogeneous for of the VBE (1.6); see the appendix.

In the large limit, the mean current of the particle model will satisfy this same equation as a consequence of POC: It follows from (1.3) that

where

Propagation of chaos then shows that, as expected, in the limit ,

Our main result is contained in the following theorem. For its proof we will need to control the fourth moment of . We thus set

We also remind the reader that a function on is -Lipschitz if for any we have .


Theorem 1.3.

Let be a probability density on with

(1.12)

Let be the solution of (1.3) with initial data given by (1.5). Then for all -Lipschitz function on , and all ,

(1.13)

where is the solution of (1.6) with initial data .

Because the interaction between the particles does not depend at all on their locations, the main work in proving this theorem goes into estimates for treating the spatially homogeneous case. Once the spatially homogeneous version is proved, it will be easy to treat the general case.

We now begin the proof, starting with the introduction of an auxiliary process that propagates independence.

2 The B process.

By integrating (1.3) on the position variables we see that the marginal of on the velocities

satisfies the autonomous equation

(2.1)

This implies that, starting from an initial data in (1.5) independent from the solution of (1.3) will remain independent of and will satisfy (2.1). We will thus ignore the position and study propagation of chaos for (2.1). Theorem 1.3 will be an easy corollary of the analysis in this section, as we show at the end of Section 4.

We start the system with an initial state

(2.2)

Given we can solve the BE equation (1.6) and find , see Appendix A.

To compare the solution of the Boltzmann equation to the solution of the master equation (2.1) we introduce a new stochastic process. Consider a system of particles with the following dynamics between collisions

(2.3)

where is obtained from the solution of (1.8). The particles also undergo “virtual” collision with the same distribution described after (1.1). It is easy to see that the master equation associated to this process is

(2.4)

where we have assumed that is independent of . Clearly, under this dynamics each particle evolves independently so that if the initial condition for this system is of the form of (2.2) then will also have that form,

(2.5)

where solves the spatially homogeneous BE (1.6).

We shall call the system described by the dynamics (2.3) the “Boltzmann” or B-system and the original system described by the dynamics (1.1) the A-system. We will prove that, for fixed and every trajectory of the A-system with the dynamics (1.1) can be made close (in an appropriate sense) to that of the B-system with dynamics (2.3). It will then follow that, starting the B-system and A-system with the same product measure, the A-system pair correlation function will be close to that of the B-system, and since the B-dynamics propagates independence, the latter will be a product. In this way we shall see that (1.6) is satisfied.

2.1 Comparison of the A and B processes.

To compare the path of the two processes, we first observe that a collision is specified by its time , the index of the particle that collides, and the unit vector that specifies the post-collisional velocity. We will call , with , a collision history in with collisions.

Let be the flow generated by the autonomous dynamics (1.1) and the flow generated by the non autonomous dynamics (2.3) both starting at at time . Given a collision history in with collisions the A-process starting from at time , has the path

(2.6)

while the B-process has the path

(2.7)

where , respectively , is the updating of the velocity of the -th particle generated by the -th collision in the A (resp. B) -dynamics. To simplify notation, here and in what follows, we will use and for the A and B processes.

We want to compare the trajectories of the two processes. We first observe that, given two incoming velocities and and a collision vector for we can always find a corresponding collision vector for such that:

  • The probability of selecting for an incoming velocity is equal to the probability of selecting for an incoming velocity . By Remark 1.1, this is equivalent to having

    (2.8)
  • The distance between the two outgoing velocities is equal to the distance between the incoming velocities, i.e. . Since and , this is equivalent to

    (2.9)

Our claim amounts to:

Proposition 2.1.

Given any three unit vectors , and , we can find a fourth unit vector such that both (2.8) and (2.9) are satisfied.

Proof.

For , a solution is given by , i.e. . For , let be any planar rotation such that , and define . Then (2.8) and (2.9) reduce to and , so that is a solution.

For , we reduce to the planar case as follows: Let be the orthogonal projection onto the plane spanned by and . Then is a solution of (2.8) and (2.9) if and only if

(2.10)

Let us seek a solution with . If , we simply choose so that . Otherwise, divide both (2.10) by , and we have reduced to the planar case: If is a rotation in the plane spanned by and such that , then we define , and finally

where is any unit vector orthogonal to and . ∎

From the proposition it follows that to every collision history for the A process we can associate a collision history for the B process such that for every collision time we have that

(2.11)

where , resp. , is the path of the A-process just before, resp. just after, the -th collision, and similarly for the B-process, and given a vector we set

This means that we can see the A and B processes as taking place on the same probability sample space defined by the initial distribution and the collision histories . We will call the probability measure on this space.

The result of this comparison is summarized in the following:

Theorem 2.2 (Pathwise Comparison of the Two Processes).

Let satisfy (2.2) with satisfying (1.12). Then for all ,

where and is defined in (3.5) below.

3 Proof of Theorem 2.2.

To prove this theorem we need to find an expression for . First observe that between collisions we have

Differentiating the above expression with respect to and calling the differential of the flow , i.e.

we get

(3.1)

If the two flows start from different initial condition we get

(3.2)

where is the segment from to .

The main idea of the proof is to iterate (3.2) collision after collision, using the fact that the distance between the A and B trajectory is preserved by the collisions.

To control (3.2) we need first to show that

(3.3)

can be bounded uniformly in . We will obtain such an estimate in Section 3.3 but our estimate will depend on . Since the B -process does not preserve the total energy of the particles we will need to show that is, with a large probability uniformly in , bounded away from 0. This is the content of the technical Lemma in Section 3.1.

Having bounded the differential of we will need to show that

(3.4)

is small uniformly in . This is the content of Section 3.4 and it is based on the Law of Large Numbers.

Thus we will have that the distance between the corresponding paths of the two processes at time can be estimated in term of the norm (3.3) of the differential of , the integral (3.4) and the distance just after the last collision at . Using (2.11) we can reduce the problem to an estimate of the distance before the last collision. Finally we can get a full estimate iterating the above procedure over all the collisions.

3.1 A Technical Lemma

While the propagation of independence is an important advantage of the master equation (2.4), one disadvantage is that, while the A-evolution (1.1) conserves the energy , the B evolution does not: it only does so in the average.

In what follows we will need a bound on showing that the event that this quantity is large has a small probability.

Lemma 3.1.

Let be the B process with initial data given by (2.2) with satisfying (1.12). Define

(3.5)

with given by (1.12) and for any given let denote the least integer such that . Then

(3.6)
Proof.

Observe that the collisions do not affect . Thus is path-wise differentiable, and

By the Schwarz inequality, and , Therefore,

(3.7)

The differential equation

(3.8)

is solved by

Let denote the time at which :

Let us choose . Then

(3.9)

Note that the length of the interval is independent of .

It now follows from (3.7) and what we have proven about the differential equation (3.8) that if , then for all on the interval , . Then, provided that

we have

Moreover, for any given , there is only a small (for large ) probability that . Indeed since , we have

(3.10)

It follows that

Combining this estimate with (3.10) and (3.11) from the following Lemma leads to (3.6).

Lemma 3.2.

Let be a probability density on satisfying (1.12) then for all ,

(3.11)
Proof.

Using (1.6), we compute