Kac’s Program in Kinetic Theory

Kac’s Program in Kinetic Theory (Version of March 2, 2018)

Abstract.

This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean [42, 53] giving rise to the Boltzmann equation. We solve the conjecture raised by Kac [42], motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.

Motivated by the inspirative paper by Grünbaum [35], we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).

Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined in [10]. (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.

Keywords: Kac’s program; kinetic theory; master equation; mean-field limit; quantitative; uniform in time; jump process; collision process; Boltzmann equation; Maxwell molecules; non cutoff; hard spheres.

AMS Subject Classification: 82C40 Kinetic theory of gases, 76P05 Rarefied gas flows, Boltzmann equation, 54C70 Entropy, 60J75 Jump processes.

Contents:

Boltzmann is best known for the equation bearing his name in kinetic theory [7, 8]. Inspired by Maxwell’s discovery [51] of (what is now called) the Boltzmann equation and its “Maxwellian” (i.e. Gaussian) equilibrium, Boltzmann [7] discovered the “-theorem” (the entropy must increase under the time evolution of the equation), which explained why the solutions should be driven towards the equilibrium of Maxwell. In the same work he also proposed the deep idea of “stosszahlansatz” (molecular chaos) to explain how the irreversible Boltzmann equation can emerge from Newton’s laws of the dynamics of the particle system. Giving a precise mathematical meaning to this notion and proving this limit remains a tremendous open problem to this date; the best and astonishing result so far [45] is only valid for very short times.

In 1956, Kac [42] proposed the simpler, and seemingly more tractable, question of deriving the spatially homogeneous Boltzmann equation from a many-particle jump process. To do so, he introduced a rigorous notion of “molecular chaos”1 in this context. The “chaoticity” of the many-particle equilibrium with respect to the Maxwellian distribution, i.e. the fact that the first marginals of the uniform measure on the sphere converges to a Gaussian function as goes to infinity, has been known for a long time (at least since Maxwell).2 However in [42] Kac proposed the first proof of the propagation of chaos for a simplified collision evolution process for which series expansion of the solution exists, and he showed how the many-particle limit rigorously follows from this property of propagation of chaos. This proof was later extended to a more realistic collision model, the so-called cutoff Maxwell molecules, by McKean [53].

Since in this setting both the many-particle system and the limit equation are dissipative, Kac raised the natural question of relating their asymptotic behaviors. In his mind this program was to be achieved by understanding dissipativity at the level of the linear many-particle jump process and he insisted on the importance of estimating how its relaxation rate depends on the number of particles. This has motivated beautiful works on the “Kac spectral gap problem” [40, 50, 11, 13, 9], i.e. the study of this relaxation rate in a setting. However, so far this linear strategy has proved to be unsuccessful in relating the asymptotic behavior of the many-particle process and that of the limit equation (cf. the discussion in [10]).

In the time of Kac the study of nonlinear partial differential equations was rather young and it was plausible that the study of a linear many-dimension Markov process would be easier. However the mathematical developpement somehow followed the reverse direction and the theory of existence, uniqueness and relaxation to equilibrium for the spatially homogeneous Boltzmann is now well-developed (see the many references along this paper). This paper (together with its companion paper [58]) is thus an attempt to develop a quantitative theory of mean-field limit which strongly relies on detailed knowledge of the nonlinear limit equation, rather than on detailed properties of the many-particle Markov process.

The main outcome of our theory will be to find quantitative estimates to the propagation of chaos that are uniform in time, as well as propagation of entropic chaos. We also prove estimates on the relaxation rates, measured in the Wasserstein distance and relative entropy, that are independent of the number of particles. All this is done for the two important, realistic and achetypal models of collision with unbounded collision rates, namely hard spheres and the true (i.e. without cutoff) Maxwell molecules. This provides a first answer to the question raised by Kac. However, our answer is an “inverse” answer in the sense that our methodology is “top-down” from the limit equation to the many-particle system rather than “bottom-up” as was expected by Kac.


Acknowledgments. We thank the mathematics departement of Chalmers University for the invitation in November 2008, where the abstract method was devised and the related joint work [58] with Bernt Wennberg was initiated. We thank Ismaël Bailleul, Thierry Bodineau, François Bolley, Anne Boutet de Monvel, José Alfredo Cañizo, Eric Carlen, Nicolas Fournier, François Golse, Arnaud Guillin, Maxime Hauray, Joel Lebowitz, Pierre-Louis Lions, Richard Nickl, James Norris, Mario Pulvirenti, Judith Rousseau, Laure Saint-Raymond and Cédric Villani for fruitful comments and discussions, and Amit Einav for his careful proofreading of parts of the manuscript. We would also like to mention the inspiring courses by Pierre-Louis Lions at Collège de France on “Mean-Field Games” in 2007-2008 and 2008-2009, which triggered our interest in the functional analysis aspects of this topic. Finally we thank the anonymous referees for helpful suggestions on the presentation.


1. Introduction and main results

1.1. The Boltzmann equation

The Boltzmann equation (Cf. [17] and [18]) describes the behavior of a dilute gas when the only interactions taken into account are binary collisions. It is given by

where is the bilinear collision operator acting only on the velocity variable, is the spatial domain and is the dimension. Some appropriate boundary conditions need to be imposed.

In the case where the distribution function is assumed to be independent of the position , we obtain the so-called spatially homogeneous Boltzmann equation:

(1.1)

which will be studied in this paper.

Let us now focus on the collision operator . It is defined by the bilinear symmetrized form

(1.2)

where we use the shorthands , , and . Moreover, and are parametrized by

(1.3)

Finally, is the deviation angle between and defined by

and is the collision kernel determined by the physical context of the problem.

The Boltzmann equation has the following fundamental formal properties: first, it conserves mass, momentum and energy, i.e.

Second, it satisfies Boltzmann’s celebrated -theorem:

We shall consider collision kernels of the form

where are nonnegative functions. In dimension , we give a short overview of the main collision kernels appearing in physics, highlighting the key models we consider in this paper.

  • Short (finite) range interaction are usually modeled by the hard spheres collision kernel

    (1.4)
  • Long-range interactions are usually modeled by collision kernels derived from interaction potentials

    They satisfy the formula

    and

    (b is in away from ). More informations about this type of interactions can be found in [17]. This general class of collision kernels includes the true Maxwell molecules collision kernel when and , i.e.

    (1.5)

    It also includes the so-called Grad’s cutoff Maxwell molecules when the singularity in the variable is removed. For simplicity we will consider the model where

    (1.6)

    which is an archetype of such collision kernels.

1.2. Deriving the Boltzmann equation from many-particle systems

The question of deriving the Boltzmann equation from particle systems (interacting via Newton’s laws) is a famous problem. It is related to the so-called -th Hilbert problem proposed by Hilbert at the International Congress of Mathematics at Paris in 1900: axiomatize mechanics by “developing mathematically the limiting processes […] which lead from the atomistic view to the laws of motion of continua”.

At least at the formal level, the correct limiting procedure has been identified by Grad [33] in the late fourties and a clear mathematical formulation of the open problem was proposed in  [16] in the early seventies. It is now called the Boltzmann-Grad or low density limit. However the original question of Hilbert remains largely open, in spite of a striking breakthrough due to Lanford [45], who proved the limit for short times (see also Illner and Pulvirenti [39] for a close-to-vacuum result). The tremendous difficulty underlying this limit is the irreversibility of the Boltzmann equation, whereas the particle system interacting via Newton’s laws is a reversible Hamiltonian system.

In 1954-1955, Kac [42] proposed a simpler and more tractable problem: start from the Markov process corresponding to collisions only, and try to prove the limit towards the spatially homogeneous Boltzmann equation. Kac’s jump process runs as follows: consider particles with velocities . Assign a random time for each pair of particles following an exponential law with parameter , and take the smallest. Next, perform a collision according to a random choice of direction parameter, whose law is related to , and start again with the post collision velocities. This process can be considered on ; however it has some invariant submanifolds of (depending on the number of quantities conserved under the collision), and can be restricted to them. For instance, in the original simplified model of Kac (scalar velocities, i.e. ) the process can be restricted to , the sphere with radius , where is the energy. In the more realistic models of the hard spheres or Maxwell molecules (when ) the process can be restricted to the following submanifold of associated to elastic collisions invariants for , :

(1.7)

Without loss of generality, we will consider the case , using Galilean invariance. We will denote by and refer to these submanifolds as Boltzmann spheres.

Kac then formulated the following notions of chaos and propagation of chaos: Consider a sequence of probabilities on , where is some given Polish space (e.g. ): the sequence is said to be -chaotic if

for some given one-particle probability on . The convergence is to be understood as the convergence of the -th marginal of to , for any , in the weak measure topology. This is a low correlation assumption.

It is an elementary calculation to see that if the probability densities of the -particle system were perfectly tensorized during some time interval (i.e. had the form of an -fold tensor product of a one particle probability density ), then the latter would satisfy the nonlinear limit Boltzmann equation during that time interval. But generally interactions between particles instantaneously destroy this “tensorization” property and leave no hope to show its propagation in time. Nevertheless, following Boltzmann’s idea of molecular chaos, Kac suggested that the weaker property of chaoticity can be expected to propagate in time, in the correct scaling limit.

This framework is our starting point. Let us emphasize that the limit performed in this setting is different from the Boltzmann-Grad limit. It is in fact a mean-field limit. This limiting process is most well-known for deriving Vlasov-like equations. In a companion paper [58] we develop systematically our new functional approach to the study of mean-field limits for Vlasov equations, McKean-Vlasov equations, and granular gases equations.

1.3. The notion of chaos and how to measure it

Our goal in this paper is to set up a general and robust method for proving the propagation of chaos with quantitative rates in terms of the number of particles and of the final time of observation .

Let us briefly discuss the notion of chaoticity. The original formulation in [42] is: A sequence of symmetric3 probabilities on is -chaotic, for a given probability , if for any and any there holds

Together with additional assumptions on the moments, this weak convergence can be expressed for instance in Wasserstein distance as:

where denotes the marginal on the first variables. We call this notion finite-dimensional chaos.

In contrast with most previous works on this topic, we are interested here in quantitative chaos: Namely, we say that is -chaotic with rate , where as (typically , or , ), if for any there exists such that

(1.8)

Similar statements can be formulated for other metrics. For instance, a convenient way to measure chaoticity is through duality: for some normed space of smooth functions (to be specified) and for any there exists such that for any , , there holds

The Wasserstein distance is recovered when is the space of Lipschitz functions.

Observe that in the latter statements the number of variables considered in the marginal is kept fixed as goes to infinity. It is a natural question to know how the rate depends on . As we will see, the answer to this question is essential to the estimation of a relaxation time that will be uniform in the number of particles. We therefore introduce a stronger notion of infinite-dimensional chaos, based on extensive4 functionals. We consider the following definition:

(with corresponding quantitative formulations…). This amounts to say that one can prove a sublinear control on in terms of in (1.8). Variants for other extensive metrics could easily be considered as well.

Finally one can formulate an even stronger notion of (infinite-dimensional) entropic chaos (see [10] and definition (1.9) of the relative entropy below):

where is an invariant measure of the -particle system, which is -chaotic with an invariant measure of the limit equation. This notion of chaos obviously admits quantitative versions as well. Moreover, it is particularly interesting as it corresponds to the derivation of Boltzmann’s entropy and -theorem from the entropies of the many-particle system. We shall come back to this point.

Now, considering a sequence of symmetric -particle densities

and a -particle density of the expected mean field limit

we say that there is propagation of chaos on some time interval , , if the -chaoticity of the initial family implies the -chaoticity of the family for any time , according to one of the definitions of chaoticity above.

Note that the support of the -particle distributions matters. Indeed the energy conservation implies that the evolution is entirely decoupled on the different subspaces associated with this invariant, e.g. each for the different values of (we consider here centered distributions). On each such subspace the -particle process is ergodic and admits a unique invariant measure , which is constant. However when considered on the -particle process admits infinitely many invariant measures. Therefore, in the study of the long-time behavior we will often consider -particle distributions that are supported on for appropriate energy . We shall discuss the construction of such chaotic initial data in Section 4.

1.4. Kac’s program

As was mentioned before, Kac proposed to derive the spatially homogeneous Boltzmann equation from a many-particle Markov jump process with binary collisions, via its master equation (the equation on the law of the process). Intuitively, this amounts to considering the spatial variable as a hidden variable inducing ergodicity and markovian properties on the velocity variable. Although the latter point has not been proved so far to our knowledge, it is worth noting that it is a very natural guess and an extremely interesting open problem (and possibly a very difficult one). Hence Kac introduced artificial stochasticity as compared to the initial Hamiltonian evolution, and raised a fascinating question: if we have to introduce stochasticity, at least can we keep it under control through the process of deriving the Boltzmann equation and relate it to the dissipativity of the limit equation?

Let us briefly summarize the main questions raised in or motivated by [42]:

  1. Kac’s combinatorical proof (later to be extended to collision processes that preserve momentum and energy [53]) had the unfortunate non-physical restriction that the collision kernel is bounded. These proofs were based on an infinite series “tree” representation of the solution according to the collision history of the particles, as well as some sort of Leibniz formula for the -particle operator acting on tensor product. The first open problem raised was: can one prove propagation of chaos for the hard spheres collision process?

  2. Following closely the spirit of the previous question, it is natural to ask whether one can prove propagation of chaos for the true Maxwell molecules collision process? (this is the other main physical model showing an unbounded collision kernel). The difficulty here lies in the fact that the particle system can undergo infinite number of collisions in a finite time interval, and no “tree” representation of solutions is available. This is related to the interesting physical situation of long-range interaction, as well as the interesting mathematical framework of fractional derivative operators and Lévy walk.

  3. Kac conjectures [42, Eq. (6.39)] that (in our notations) the convergence

    is propagated in time, which would imply Boltzmann’s -theorem from the monotonicity of for the Markov process. He concludes with: “If the above steps could be made rigorous we would have a thoroughly satisfactory justification of Boltzmann’s -theorem.” In our notation the question is: can one prove propagation of entropic chaos in time?

  4. Eventually, Kac discusses the relaxation times, with the goal of deriving the relaxation times of the limit equation from those of the many-particle system. This requires the estimations to be independent of the number of particles on this relaxation times. As a first natural step he raises the question of obtaining such uniform estimates on the spectral gap of the Markov process. This question has triggered many beautiful works (see the next subsection), however it is easy to convince oneself (see the discussion in [10] for instance) that there is no hope of passing to the limit in this spectral gap estimate, even if the spectral gap is independent of . The norm is catastrophic in infinite dimension. Therefore we reframe this question in a setting which “tensorizes correctly in the limit ”, that is in our notation: can one prove relaxation times independent of the number of particles on the normalized Wasserstein distance or on the normalized relative entropy ?

This paper is concerned with solving these four questions.

1.5. Review of known results

Kac [42]-[43] has proved point (1) in the case of his one-dimensional toy model. The key point in his analysis is a clever combinatorial use of an exponential formula for the solution, expressing it in terms of an abstract derivation operator (reminiscent of Wild sums [78]). It was generalized by McKean [53] to the Boltzmann collision operator but only for “Maxwell molecules with cutoff”, i.e. roughly when the collision kernel is constant. In this case the convergence of the exponential formula is easily proved and this combinatorial argument can be extended. Kac raised in [42] the question of proving propagation of chaos in the case of hard spheres and more generally unbounded collision kernels, although his method seemed impossible to extend (the problem is the convergence of this exponential formula, as discussed in [53] for instance).

During the seventies, in a very abstract and compact paper [35], Grünbaum proposed another method for dealing with the hard spheres model, based on the Trotter-Kato formula for semigroups and a clever functional framework (partially reminiscent of the tools used for mean-field limits for McKean-Vlasov equations). Unfortunately this paper was incomplete for two reasons: (1) It was based on two “unproved assumptions on the Boltzmann flow” (page 328): (a) existence and uniqueness for measure solutions and (b) a smoothness assumption. Assumption (a) was recently proved in [31] using Wasserstein metrics techniques and in [28], adapting the classical DiBlasio trick [19]. Assumption (b), while inspired by the cutoff Maxwell molecules (for which it is true), fails for the hard spheres model (cf. the counterexample constructed by Lu and Wennberg [49]) and is somehow too ”rough” in this case. (2) A key part in the proof in this paper is the expansion of the “” function, which is a clever idea by Grünbaum — however, it is again too coarse for the hard spheres case (though adaptable to the cutoff Maxwell molecules). Nevertheless it is the starting point for our idea of developing a differential calculus in spaces of probability measures in order to control fluctuations.

A completely different approach was undertaken by Sznitman in the eighties [68] (see also Tanaka [71] for partial results concerning non-cutoff Maxwell molecules). Starting from the observation that Grünbaum’s proof was incomplete, he took on to give a full proof of propagation of chaos for hard spheres by another approach. His work was based on: (1) a new uniqueness result for measures for the hard spheres Boltzmann equation (based on a probabilistic reasoning on an enlarged space of “trajectories”); (2) an idea already present in Grünbaum’s approach: reduce the question of the propagation of chaos to a law of large numbers on measures by using a combinatorical argument on symmetric probabilities; (3) a new compactness result at the level of the empirical measures; (4) the identification of the limit by an “abstract test function” construction showing that the (infinite particle) system has trajectories included in the chaotic ones. While Sznitman’s method proves propagation of chaos for the hard spheres, it doesn’t provide any rate for chaoticity as defined previously.

Let us also emphasize that in [52] McKean studied fluctuations around deterministic limit for 2-speed Maxwellian gas and for the usual hard spheres gas. In [34] Graham and Méléard obtained a rate of convergence (of order for the -th marginal) on any bounded finite interval of the -particle system to the deterministic Boltzmann dynamics in the case of Maxwell molecules under Grad’s cut-off hypothesis. Lastly, in [29, 30] Fournier and Méléard obtained the convergence of the Monte-Carlo approximation (with numerical cutoff) of the Boltzmann equation for true Maxwell molecules with a rate of convergence (depending on the numerical cutoff and on the number of particles).

Kac was raising the question of how to control the asymptotic behavior of the particle system in the many-particle limit. As a first step, he suggested to study the behavior of the spectral gap in of the Markov process as goes to infinity and conjectured it to be bounded away from zero uniformly in terms of . This question has been answered only recently in [40, 11] (see also [50, 13]). However the norm behaves geometrically in terms of for tensorized data; this leave no hope to use it for estimating the long-time behavior as goes to infinity, as the time-decay estimates degenerate beyond times of order . In the paper [10], the authors suggested to make use of the relative entropy for estimating the relaxation to equilibrium, and replace the spectral gap by a linear inequality between the entropy and the entropy production. They constructed entropically chaotic initial data, following the definition mentioned above, but did not succeed in proving the propagation in time of this chaoticity property. Moreover the linear inequality between the entropy and the entropy production is conjectured to be wrong for any physical collision kernels in [75], see for instance [23, 24] for partial confirmations to this conjecture.

After we had finished writing our paper, we were told about the recent book [44] by Kolokoltsov. This book focuses on fluctuation estimates of central limit theorem type. It does not prove quantitative propagation of chaos but weaker estimates (and on finite time intervals), and moreover we were not able to extract from it a full proof for the cases with unbounded collision kernels (e.g. hard spheres). However the comparison of generators for the many-particle and the limit semigroup is reminiscent of our work.

1.6. The abstract method

Our initial inspiration was Grünbaum’s paper [35]. Our original goal was to construct a general and robust method able to deal with mixture of jump and diffusion processes, as it occurs in granular gases (see our companion paper [58] for results in this direction). This lead us to develop a new theory, inspired from more recent tools such as the course of Lions on “Mean-field games” at Collège de France, and the master courses of Méléard [55] and Villani [73] on mean-field limits. One of the byproduct of our paper is that we make Grünbaum’s original intuition fully rigorous in order to prove propagation of chaos for the Boltzmann velocities jump process for hard spheres.

Like Grünbaum [35], we use a duality argument. We introduce the semigroup associated to the flow of the -particle system in , and the semigroup in in duality with it. We also introduce the (nonlinear) semigroup in associated to the mean-field dynamics (the exponent “NL” signifies the nonlinearity of the limit semigroup, due to the interaction) as well as the associated (linear) “pullback” semigroup in (see Subsection 2.3 for the definition). Then we prove stability and convergence estimates between the linear semigroups and as goes to infinity.

The preliminary step consists of defining a common functional framework in which the -particle dynamics and the limit dynamics make sense so that we can compare them. Hence we work at the level of the “full” limit space and, at the dual level, . Then we identify the regularity required in order to prove the “consistency estimate” between the generators and of the dual semigroups and , and finally prove a corresponding “stability estimate” on , based on stability estimates at the level of the limit semigroup .

The latter crucial step leads us to introduce an abstract differential calculus for functions acting on measures endowed with various metrics associated with the weak or strong topologies. More precisely, we shall define functions of class on the space of probability measures by working on subspaces of the space of probability measures, for which the tangent space has a Banach space structure. This “stratification” of subspaces is related to the conservation properties of the flows and . This notion is related but different from the notions of differentiability developed in the theory of gradient flow by Ambrosio, Otto, Villani and co-authors in [2, 41, 62], and from the one introduced by Lions in [46].

Another viewpoint on our method is to consider it as some kind of accurate version (in the sense that it establishes a rate of convergence) of the BBGKY hierarchy method for proving propagation of chaos and mean-field limit on statistical solutions. This viewpoint is extensively explored and made rigorous in Section 8 where we revisit the BBGKY hierarchy method in the case of a collisional many-particle system, as was considered for instance in [3]. The proof of uniqueness for statistical solutions to the hierarchy becomes straightforward within our framework by using differentiability of the limit semigroup as a function acting on probabilities.

This general method is first exposed at an abstract level in Section 3 (see in particular Theorem 3.1). This method is, we hope, interesting per se for several reasons: (1) it is fully quantitative, (2) it is highly flexible in terms of the functional spaces used in the proof, (3) it requires a minimal amount of informations on the -particle systems but more stability informations on the limit PDE (we intentionally presented the assumptions in a way resembling the proof of the convergence of a numerical scheme, which was our “methodological model”), (4) the “differential stability” conditions that are required on the limit PDE seem (to our knowledge) new, at least for the Boltzmann equation.

1.7. Main results

Let us give some simplified versions of the main results in this paper. All the abstract objects will be fully introduced in the next sections with more details.

Theorem 1.1 (Summary of the main results).

Consider some centered initial distribution with finite energy , and with compact support or moment bounds. Consider the corresponding solution to the spatially homogeneous Boltzmann equation for hard spheres or Maxwell molecules, and the solution of the corresponding Kac’s jump process starting either from the -fold tensorization of or the latter conditioned to .

One can classify the results established into three main statements:

  1. Quantitative uniform in time propagation of chaos, finite or infinite dimensional, in weak measure distance (cf. Theorems 5.1-5.2-6.1-6.2):

    for some as . In the hard spheres case, the uniformity in time of this estimate is only proved when the distribution is conditioned to . Moreover, the proof provides explicit estimates on the rate , which are controlled by a power law for Maxwell molecules, and by a power of a logarithm for hard spheres.

  2. Propagation of entropic chaos (cf. Theorem 7.1-(i)): Consider the case where the initial datum of the many-particle system has support included in . If the initial datum is entropically chaotic in the sense

    with

    (1.9)

    where is the Gaussian equilibrium with energy and is the uniform probability measure on , then the solution is also entropically chaotic for any later time:

    This proves the derivation of the -theorem this context, i.e. the monotonic decay in time of , since for any , the functional is monotone decreasing in time for the Markov process.

  3. Quantitative estimates on relaxation times, independent of the number of particles (cf. Theorems 5.2-6.2 and Theorem 7.1-(ii)): Consider the case where the initial datum of the many-particle system has support included in . Then we have

    for some as . Moreover, the proof provides explicit estimates on the rate , which are controlled by a power law for Maxwell molecules, and by a power of a logarithm for hard spheres.

    Finally in the case of Maxwell molecules, if we assume furthermore that the Fisher information of the initial datum is finite:

    (1.10)

    then the following estimate also holds:

    for some function as , with the same kind of estimates.

1.8. Some open questions

Here are a few questions among those raised by this work:

  1. What about the optimal rate in the chaoticity estimates along time? Our method reduces this question to the chaoticity estimates at initial time, and therefore to the optimal rate in the quantitative law of large numbers for measures according to various weak measure distances.

  2. What about the optimal rate in the relaxation times (uniformly in the number of particles)? Spectral gap studies predict exponential rates, both for the many-particle system and for the limit system, however our rates are far from it!

  3. Can uniform in time propagation of chaos be proved for non-reversible jump processes (such as inelastic collision processes) for which the invariants measures and are not explicitely known (e.g. granular gases)?

1.9. Plan of the paper

In Section 2 we set the abstract functional framework together with the general assumption and in Section 3 we state and prove our main abstract theorem (Theorem 3.1). In Section 4 we present some tools and results on weak measure distances, on the construction of initial data with support on the Boltzmann sphere for the -particle system, and on the sampling process of the limit distribution by empirical measures. In Section 5 we apply the method to (true) Maxwell molecules (Theorems 5.1 and 5.2). In Section 6 we apply the method to hard spheres (Theorems 6.1 and 6.2). Section 7 is devoted to the study of entropic chaos. Lastly, in Section 8 we revisit the BBGKY hierarchy method for the spatially homogeneous Boltzmann equation in the light of our framework.

2. The abstract setting

In this section we shall state and prove the key abstract result. This will motivate the introduction of a general functional framework.

2.1. The general functional framework of the duality approach

Let us set up the framework. Here is a diagram which sums up the duality approach (norms and duality brackets shall be specified in Subsections 2.4):

In this diagram:

- denotes a Polish space:

This is a separable completely metrizable topological space. We shall denote by the distance on this space in the sequel.

- denotes the -permutation group.

- denotes the set of symmetric probabilities on :

Given a permutation , a vector

a function and a probability , we successively define

and

and finally

We then say that a probability on is symmetric if it is invariant under permutations:

- The probability measure denotes the empirical measure:

where denotes the Dirac mass on at point .

- denotes the subset of empirical measures of .

- denotes the space of probabilities on the Polish space (endowed for instance with the Prokhorov distance), and this is again a Polish space.

- denotes the space of continuous and bounded functions on :

This space shall be endowed with either the weak or strong topologies (see Subsection 2.4), and later with some metric differential structure.

- The map from to is defined by

- The map from to is defined by

- The map from to is defined by:

- The map from to is defined by:

for any and any , where the first bracket means and the second bracket means .

Let us now discuss the “horizontal” arrows:

- The arrows pointing from the first column to the second one consists in writing the Kolmogorov equation associated with the many-particle stochastic Markov process.

- The arrows pointing from the second column to the third column consists in writing the dual evolution semigroup (note that the -particle dynamics is linear). As we shall discuss later the dual spaces of the spaces of probabilities on the phase space can be interpreted as the spaces of observables on the original systems.

Our functional framework shall be applied to weighted spaces of probability measures rather than directly in . More precisely, for a given weight function we shall use affine subsets of the weighted space of probability measures

as our basis functional spaces. Typical examples are for some fixed with or , . More specifically when , we shall use or , .

We shall sometimes abuse notation by writing for when or in the examples above. We shall denote by the space of finite signed measures endowed with the total variation norm, and the space of finite signed measures whose variation satisfies , and endowed with the total variation norm. Again we contract the notation as when or .

2.2. The -particle semigroups

Let us introduce the mathematical semigroups describing the evolution of objects living in these spaces, for any .

Step 1. Consider the trajectories , , of the particles (Markov process viewpoint). We make the further assumption that this flow commutes with permutations:

For any , the solution at time starting from is .

This mathematically reflects the fact that particles are indistinguishable.

Step 2. This flow on yields a corresponding semigroup acting on for the probability density of particles in the phase space (statistical viewpoint), defined through the formula

where the bracket obviously denotes the duality bracket between and and denotes the expectation associated to the space of probability measures in which the process is built. In other words, is nothing but the law of . Since the flow commutes with permutation, the semigroup acts on : if the law of belongs to , then for later times the law of also belongs to . To the -semigroup on one can associate a linear evolution equation with a generator denoted by :

which is the forward Kolmogorov (or Master) equation on the law of .

Step 3. We also consider the Markov semigroup acting on the functions space of observables on the evolution system on (see the discussion in the next remark), which is in duality with the semigroup , in the sense that:

To the -semigroup on we can associate the following linear evolution equation with a generator denoted by :

which is the backward Kolmogorov equation.


2.3. The mean-field limit semigroup

We now define the evolution of the limit mean-field equation.

Step 1. Consider a semigroup acting on associated with an evolution equation and some operator :

For any (assuming possibly some additional moment bounds), then where is the solution to

(2.1)

This semigroup and the operator are typically nonlinear for mean-field models, namely bilinear in case of Boltzmann’s collisions interactions.

Step 2. Consider then the associated pullback semigroup acting on :

(Again additional moment bounds can be required on in order to make this definition rigorous.) Note carefully that is always linear as a function of , although of course is not linear in general as a function of . We shall associate (when possible) the following linear evolution equation on with some generator denoted by :

Remark 2.1.

The semigroup can be interpreted physically as the semigroup of the evolution of observables of the nonlinear equation (2.1). Let us give a short heuristic explanation. Consider a nonlinear ordinary differential equation

with divergence-free vector field for simplicity. One can then define formally the linear Liouville transport partial differential equation

where is a time-dependent probability density over the phase space , whose solution is given (at least formally) by following the characteristics backward . Now, instead of the Liouville viewpoint, one can adopt the viewpoint of observables, that is functions depending on the position of the system in the phase space (e.g. energy, momentum, etc …). For some observable function defined on , the evolution of the value of this observable along the trajectory is given by and is solution to the following dual linear PDE

Now let us consider a nonlinear evolution system

By analogy we define two linear evolution systems on the larger functional spaces and : first the abstract Liouville equation

and second the abstract equation for the evolution of observables

However in order to make sense of this heuristic, the scalar product have to be defined correctly as duality brackets, and, most importantly, a differential calculus on has to be defined rigorously. Taking , this provides an intuition for our functional construction, as well as for the formula of the generator below (compare the previous equation with formula (2.9)). Be careful that when , the abstract Liouville and observable equations refers to trajectories in the space of probabilities (i.e. solutions to the nonlinear equation (2.1)), and not trajectories of a particle in . Note also that for a dissipative equation at the level of (such as the Boltzmann equation), it seems more convenient to use the observable equation rather than the Liouville equation since “forward characteristics” can be readily used in order to construct the solutions to this observable equation.

Summing up we obtain the following picture for the semigroups:

Hence a key point of our construction is that, through the evolution of observables, we shall “interface” the two evolution systems (the nonlinear limit equation and the -particle system) via the applications and . From now on we shall write .

2.4. The metric issue

is our fundamental space in which we shall compare (through their observables) the semigroups of the -particle system and the limit mean-field equation. Let us make the topological and metric structures used on more precise. At the topological level there are two canonical choices (which determine two different sets ):

  1. The strong topology which is associated to the total variation norm, denoted by ; the corresponding set shall be denoted by .

  2. The weak topology, i.e. the trace on of the weak topology on (the space of Radon measures on with finite mass) induced by ; the corresponding set shall be denoted by .

It is clear that

The supremum norm does not depend on the choice of topology on , and induces a Banach topology on the space . The transformations and satisfy:

(2.2)

The transformation is well defined from to , but in general, it does not map into since the map

is not continuous.

In the other way round, the transformation is well defined from to , and therefore also from to : for any and for any sequence weakly, we have weakly, and then .

There are many different possible metric structures inducing the weak topology on . The mere notion of continuity does not require discussing these metrics, but any subspace of with differential regularity shall strongly depend on this choice, which motivates the following definitions.

Definition 2.2.

For a given weight function , we define the subspaces of probabilities:

As usual we contract the notation as when and , , .

We also define the corresponding bounded subsets for

For a given constraint function such that is well defined for any and a given space of constraints , for any , we define the corresponding (possibly empty) constrained subsets

and the corresponding (possibly empty) bounded constrained subsets

We also define the corresponding space of increments

Be careful that the space of increments is not a vector space in general. Let us now define the notion of distances over probabilities that we shall consider.

Definition 2.3.

Consider a weight function , a constraint function and a set of constraints . We shall use for the associated spaces of the previous definition the following simplified contracted notation: for , for , for , for , for and for .

We shall consider a distance which

(1) either is defined on the whole space (i.e. whatever the values of the constraints),
(2) or such that there is a Banach space endowed with a norm such that is defined for any on , by setting

Let us finally define a quantitative Hölder notion of equivalence for the distances over probabilities.

Definition 2.4.

Consider some weight and constraint functions , . We say that two metrics and defined on are Hölder equivalent on bounded sets if there exists and, for any