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Lanford's theorem is the best known mathematical justification of Boltzmann's equation starting from deterministic classical mechanics. Unfortunately, Lanford's landmark result is only known to hold on a short time interval, whose size is comparable to the mean free time for a particle of gas. This limitation has only been overcome in restrictive perturbative regimes, most notably the case of an extremely rarefied gas of hard spheres in vacuum, which was studied by Illner and Pulvirenti in the 1980s. We give a complete proof of the convergence result due to Illner and Pulvirenti, building on the recent complete proof of Lanford's theorem by Gallagher, Saint-Raymond and Texier. Additionally, we introduce a notion that we call nonuniform chaoticity (classically known as strong one-sided chaos) which is propagated forwards in time under the microscopic dynamics, at least for the full time interval upon which uniform L∞ estimates are available for a specific ("tensorized'') solution of the BBGKY hierarchy.

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The propagation of chaos for a rarefied gas of hard spheres in vacuum by Ryan Denlinger A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University May, 2016 Nader Masmoudi Pro Q ue st Num b e r: 10139568 All rig hts re se rve d INFO RMATIO N TO ALL USERS The q ua lity o f this re p ro d uc tio n is d e p e nd e nt up o n the q ua lity o f the c o p y sub m itte d . In the unlike ly e ve nt tha t the a utho r d id no t se nd a c o m p le te m a nusc rip t a nd the re a re m issing p a g e s, the se will b e no te d . Also , if m a te ria l ha d to b e re m o ve d , a no te will ind ic a te the d e le tio n. Pro Q ue st 10139568 Pub lishe d b y Pro Q ue st LLC (2016). Co p yrig ht o f the Disse rta tio n is he ld b y the Autho r. All rig hts re se rve d . This wo rk is p ro te c te d a g a inst una utho rize d c o p ying und e r Title 17, Unite d Sta te s Co d e Mic ro fo rm Ed itio n © Pro Q ue st LLC. Pro Q ue st LLC. 789 Ea st Eise nho we r Pa rkwa y P.O . Bo x 1346 Ann Arb o r, MI 48106 - 1346 © Ryan Denlinger All Rights Reserved, 2016 Dedication I dedicate this work to my mother. iv Acknowledgements It would be futile to try to properly thank every person whose caring and dedication have brought me to this point. However, special thanks must go to Nader Masmoudi, my advisor, for giving me a problem to think about, and for his immeasurable patience and support in completing this thesis. Additionally, special thanks go to David Bories, my high school physics and mathematics teacher, without whose guidance I would never have found the career which has meant so much to me. Over the course of my time at NYU, I have been fortunate enough to learn from physicists and mathematicians who have shaped the contours of human knowledge. I would particularly like to thank Leslie Greengard, Pierre Germain, and S.R.S. Varadhan, who have taught me valuable and sometimes difficult lessons which will serve me throughout my life as an academic. I would also like to thank Pierre Germain, Fanghua Lin, Nader Masmoudi, Paul Bourgade, and S.R.S. Varadhan, for agreeing to serve on my thesis committee. I would like to express my gratitude to the outstanding kinetic and PDE theorists without whose advice I would never have been able to complete this work. Therefore, thanks go to Laure Saint-Raymond, Pierre Germain, Cl´ement Mouhot, Pierre-Emmanuel Jabin, and again to Nader Masmoudi, for many delightful discussions and insights which I will carry with me forever. Thanks go as well to C´edric Villani for having written his excellent textbooks on kinetic theory and optimal transport, which I have found very helpful throughout the writing of this thesis. v Many thanks go to my undergraduate research mentors at Caltech, Oscar Bruno and Sandra Troian, for all their support and advice over the years. The applied physics and applied mathemat- ics faculties at Caltech gave me a solid grounding for graduate-level research, and I am grateful for their efforts. Last but certainly not least, let me thank my mother and father, as well as my sister, Lindsay, for the constant love and encouragement they have given me; and, let me express my deep appreciation to my fianc´ee, Xinlin Yu, who has been incredibly understanding every step of the way. vi Abstract Lanford’s theorem is the best known mathematical justification of Boltzmann’s equation starting from deterministic classical mechanics. Unfortunately, Lanford’s landmark result is only known to hold on a short time interval, whose size is comparable to the mean free time for a particle of gas. This limitation has only been overcome in restrictive perturbative regimes, most notably the case of an extremely rarefied gas of hard spheres in vacuum, which was studied by Illner and Pulvirenti in the 1980s. We give a complete proof of the convergence result due to Illner and Pulvirenti, building on the recent complete proof of Lanford’s theorem by Gallagher, Saint-Raymond and Texier. Additionally, we introduce a notion that we call nonuniform chaoticity (classically known as strong one-sided chaos) which is propagated forwards in time under the microscopic dynamics, at least for the full time interval upon which uniform L∞ estimates are available for a specific (“tensorized”) solution of the BBGKY hierarchy. vii Table of Contents Dedication iv Acknowledgements v Abstract vii 1 The hard sphere gas 1 1.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Notation and physical estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 The BBGKY hierarchy 21 2.1 Derivation of the BBGKY hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Derivation of the dual BBGKY hierarchy . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Local a priori bounds on observables . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Global a priori bounds on observables . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Pseudo-trajectories 44 3.1 Representation of marginals via pseudo-trajectories . . . . . . . . . . . . . . . . . 44 viii 3.2 Stability of pseudo-trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 The Boltzmann hierarchy 59 4.1 The Boltzmann hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Small solutions of the Boltzmann hierarchy . . . . . . . . . . . . . . . . . . . . . 63 5 The convergence proof 73 5.1 Construction of the initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Local-in-time convergence proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 Propagation of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6 Generalizations and open problems 90 Bibliography 92 ix Chapter 1 The hard sphere gas 1.1 Review We are interested in the system of N elastic hard spheres of diameter ε > 0, which move through d- dimensional Euclidean space according to the laws of Newtonian mechanics. This is an important model in mathematical physics because the rules are relatively simple and yet they capture in a realistic way the macroscopic behavior of many physical systems. Usually the number of particles is quite large, say N = 1023 , so it seems hopeless to follow the microscopic dynamics directly. The great insight, originally due to Boltzmann and Maxwell, is that we should expect “typical” microscopic states to evolve as if the particles are completely uncorrelated with one another. In modern terms, this means that the N-particle state factorizes as a product of N one-particle states. Boltzmann used this idea as the heuristic basis for the equation bearing his name, though his arguments were quite controversial in his day. Half a century after Boltzmann’s work, H. Grad used more precise reasoning in an attempt to give Boltzmann’s equation a firm physical footing. He devised a special scaling limit, known today 1 as the Boltzmann-Grad limit, in which the microscopic dynamics heuristically reduce to the Boltz- mann equation under an ill-defined “molecular chaos” assumption. [Grad, 1949] However, this did not resolve the question of deriving Boltzmann’s equation because there was no mathematical argument linking the microscopic Liouville equation to the Boltzmann equation. O.E. Lanford provided such a link in the 1970’s, by describing the reduced dynamics arising from low-order correlations, and showing that the high-order correlations have negligible influence on the behav- ior of the gas, at least for a short time. [Lanford, 1975] Unfortunately, Lanford’s proof was not completely rigorous because he omitted a careful analysis of the convergence of “typical” reduced trajectories. Many authors considered this “convergence proof” to be obvious, though in fact the issues surrounding the convergence are quite subtle and physically important. The first fully rig- orous derivation of Boltzmann’s equation for hard spheres on a short time was recently provided by I. Gallagher, L. Saint-Raymond and B. Texier [Gallagher et al., 2014], who followed Lanford’s strategy and provided careful estimates where they were needed. We remark on several related developments. The major limitation in Lanford’s theorem is the small time, which so far has not been lifted except in very restrictive perturbative regimes. R. Illner and M. Pulvirenti were able to overcome the time restriction and prove global conver- gence for a very rare gas in vacuum, using inequalities related to the dispersive nature of the system. [Illner, 1989, Illner and Pulvirenti, 1986, Illner and Pulvirenti, 1989] H. van Beijeren, O. E. Lanford, J. Lebowitz and H. Spohn studied the evolution of N particles at equilibrium in a box; they considered an initial perturbation which alters just one particle’s state, while leaving all the other particles unperturbed at the initial time (though all particles interact under the dynam- ics). They found that the “tagged” particle evolves according to the linear Boltzmann equation, while the remaining particles remain at equilibrium; this was eventually proven by J. Lebowitz and H. Spohn for arbitrary time intervals. [van Beijeren et al., 1980, Lebowitz and Spohn, 1982] 2 More recently, T. Bodineau, I. Gallagher and L. Saint-Raymond were able to prove quantitative estimates and thereby pass to a diffusive scaling regime, showing that the mutual interaction of a tagged particle with a gas at equilibrium would converge to a Brownian motion for the tagged par- ticle. [Bodineau et al., 2015b] The same authors have also analyzed a more symmetric N-particle distribution in order to derive the linearized Boltzmann equation. [Bodineau et al., 2015a] There are several other important results which are not directly related to Lanford’s theorem but are nevertheless foundational in kinetic theory. • Stochastic models. All models we have mentioned so far have been fully deterministic; this means that randomness is allowed in the choice of initial data, but the evolution for each initial state is fully determined. However, there is an important class of models in kinetic theory where the dynamics itself introduces randomness. We specifically mention the Kac model; in this model, the position coordinates are treated as “hidden variables,” and in particular the impact parameter for each collision is a random variable with some specified law. When the number of particles tends to infinity, the evolution is seen to converge to the (nonlinear) space-homogeneous Boltzmann equation with the appropriate collision kernel. These models were first analyzed in a couple of influential papers by M. Kac and H. McKean. [Kac, 1956, McKean, 1967a] There have been many papers dealing with similar models in the intervening years, and a very complete treatment has been given by S. Mischler and C. Mouhot. [Mischler and Mouhot, 2013] • Lorentz gases. We refer to a class of models first studied by G. Gallavotti. [Gallavotti, 1969] In these Lorentz gas-type models, the dynamics is indeed deterministic, but they differ from the case of Lanford in that all the particles but one are considered stationary obstacles, distributed like Poisson scatterers. Hence randomness appears both in the initial distribution of the tagged particle and in the positions of the background particles. The dynamics is 3 much simpler in this case because the background particles never move out of place; in the Boltzmann-Grad limit one recovers the linear Boltzmann equation for the evolution of the tagged particle. Note that it is not possible to enforce momentum conservation in a Lorentz gas, so these models are only physically realistic if the tagged particle is much lighter than the background particles. • Mean field limit. Physical limits in which each particle feels the influence of the entire gas are generally called mean-field limits; these models can be fully deterministic, or they can possess some stochasticity. The Boltzmann-Grad limit is, in some sense, the “opposite” of the mean field limit, since in the Boltzmann-Grad limit, a typical particle only interacts with a very small fraction of the surrounding particles. The mean field limit has a more pleasant mathematical structure because a typical particle’s trajectory is governed by the average of the other particles’ trajectories; this property is very helpful in controlling the correlations generated by the dynamics. Whereas the Boltzmann-Grad scaling leads to Boltzmann’s ki- netic equation, pure mean-field models lead to Vlasov-type equations in the limit N → ∞. The study of mean field limits is a vast field in its own right and we provide only a small sam- pling of the relevant literature. [Dobrushin, 1979, McKean, 1967b, Jabin and Hauray, 2011] There have been a few major results for space-inhomogeneous stochastic models in kinetic or hydrodynamic scalings. [Olla et al., 1993, Rezakhanlou, 2004] These so-called entropy methods have not been extended to the fully determinstic setting, mainly due to the absence of a microscopic entropy dissipation inequality for the Liouville equation. Henceforth in this work we will not be concerned with stochastic dynamics, Lorentz gases, or mean field models. The primary goal of this work is to review several major developments related to Lanford’s theorem, while filling in a few of the technical gaps apparent in the small-data result of Illner & Pulvirenti. [Illner and Pulvirenti, 1986, Illner and Pulvirenti, 1989] However, there is one notable 4 point of departure from the prior literature, and this is the fact that our main convergence result, Theorem 5.3.1, may be iterated forwards in time (subject to the smallness constraint which guar- antees global a priori bounds). It is well-known that any uniform convergence estimate across the entire phase space Ds ⊂ R2ds at positive times must be incorrect, because if such an estimate were true then we could use the reversibility of the microscopic dynamics to deduce a contradic- tion. [Cercignani, 1972] Though this problem is widely acknowledged, the absence of any rigorous convergence proof prevented progress towards a resolution. The situation was partially clarified by Gallagher, Saint-Raymond and Texier [Gallagher et al., 2014], who gave the first rigorous proof of Lanford’s theorem. However, in their result, convergence was only obtained after an average over velocities, so the theorem could not be iterated even if a priori bounds were known. A complicated scheme for avoiding iteration has been given [Bodineau et al., 2015b], though this “pruning” pro- cedure still relies on having perfect information at the initial time. Further progress was made by Pulvirenti, Saffirio & Simonella [Pulvirenti et al., 2014], who proved uniform convergence across certain “non-interacting” points in the phase space; however, even this result was too weak to iter- ate. Building on previous work [Gallagher et al., 2014, Pulvirenti et al., 2014], we have identified a set of “good” configurations for which uniform convergence can be proven and which contains enough information to iterate the theorem, conditional on the existence of as-yet-unproved a priori estimates. Due to the structure of the “good” set, the result can be iterated forwards in time but not backwards, so our result does not contradict the reversibility of the microscopic dynamics. In Section 1.2, we describe the ideas behind our proof, and we present a simplified version of our main convergence result. Section 1.3 gives the precise physical setting for our problem, along with crucial conservation and monotonicity properties that hold without any conditions on the initial data. One of these results, namely Proposition 1.3.5, is a dispersive interaction estimate which appears to be new. Sections 2.1 & 2.2 are devoted to careful formal derivations of the 5 BBGKY and dual BBGKY hierarchies. Sections 2.3 & 2.4 give proofs of a priori bounds on the BBGKY hierarchy by a duality argument; bounds are proven both locally in time for large data, and globally in time for data sufficiently close to vacuum. (These a priori estimates are not new, but we use a different approach for the proofs.) Chapters 3 and 4 present a number of important technical tools and results; our main technical contribution is the stability result in Section 3.2. The detailed convergence analysis is given in Chapter 5. We conclude with generalizations and open problems in Chapter 6. 1.2 Statement of main results The main focus of this paper is a complete formal treatment of the Boltzmann-Grad limit globally in time under a “small data” assumption. [Illner and Pulvirenti, 1986, Illner and Pulvirenti, 1989] We will use the BBGKY hierarchy (Bogoliubov-Born-Green-Kirkwood-Yvon), which is a set of (s) equations describing the evolution of marginals fN (t) under the hard sphere flow. The proof re- lies on a dispersive decay estimate, and therefore only holds in the whole space Rd . Our approach differs from the prior literature in two ways: first, we have chosen to use the dual BBGKY hier- archy in proving a priori estimates; and second, in our convergence proof, we employ a notion of chaoticity which is roughly analogous to the classical idea of “strong one-sided chaos.” Let us first discuss the role of the dual BBGKY hierarchy in our proof. One of the classical problems in the Boltzmann-Grad limit is the lack of a well-suited functional space for a priori estimates on the BBGKY hierarchy for hard spheres. Indeed, the only physical bounds available without restriction on the time or initial data are of type L1 ∩ L log L, whereas the interaction term (collision operator) only makes sense given higher regularity, e.g. continuity of the marginals. On a short time, or for small data, one can prove L∞ bounds on the marginals if such bounds exist at 6 the initial time, but this bound does not follow from a fixed point iteration in L∞ because the colli- sion operator is not well-defined on L∞ . The regularity issue has classically been avoided by using density arguments or series expansions. (See [Gallagher et al., 2014, Cercignani et al., 1997] and references therein; also, see [Pulvirenti and Simonella, 2015, Simonella, 2014] for a more direct approach towards the series expansion.) We have instead chosen to prove a priori estimates on the dual BBGKY hierarchy; note that the dual BBGKY hierarchy has so far only received limited attention in the literature. [Gerasimenko, 2013, Cercignani et al., 1997] As it turns out, the dual BBGKY hierarchy generator is meaningful even without strong regularity conditions, and we are able to prove bounds in a weighted L 1 space; then, the classical L∞ bounds on marginals fol- low by duality. This approach also allows us to easily employ the dispersive inequalities which characterize decay in the whole space. We now turn to the content of Theorem 5.3.1, which is the main result we will prove. Theorem 5.3.1 uses dispersive decay estimates [Illner and Pulvirenti, 1989, Bardos and Degond, 1985] to achieve convergence globally in time, but here we will discuss a non-quantitative though slightly more general result which follows from the same techniques. Essentially the result states that if a priori bounds are known then chaoticity is propagated forwards in time; the novel aspect of this result is that the strength of convergence we require at time t = 0 is, in some sense, equivalent to the strength of convergence we prove at time t > 0. The direction of time is built into our notion of chaoticity, so the theorem cannot be applied to prove propagation of chaos backwards in time. We now state a simplified version of our main result. To this end, let us introduce two sets which together determine “good” configurations of particles. We will regard η > 0 as a “small”

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