The propagation of chaos for a rarefied gas of hard spheres

The propagation of chaos for a rarefied gas of hard spheres Ryan Denlinger 1 1 University of Texas at Austin 35th Annual Western States Mathematical Physics Meeting Caltech February 13, 2017 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 1 / 30

Outline 1 Introduction & Motivation 2 Lanford s theorem 3 Dispersive inequalities for hard spheres 4 Ideas behind the proof 5 References Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 2 / 30

Introduction & Motivation D. Hilbert, J.C. Maxwell and L. Boltzmann (Wikipedia) Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 3 / 30

Introduction & Motivation Hilbert s Sixth Problem To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. (Wikipedia) Certain areas of continuum mechanics require dissipation: compressible fluid dynamics (shock formation), incompressible avier-stokes (viscosity), etc. In any case kinetic energy is converted into thermal energy in an irreversible manner until some steady state is achieved (hence turning back the clock would violate the 2nd law of thermodynamics). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 4 / 30

Introduction & Motivation Hilbert s Sixth Problem To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. (Wikipedia) Certain areas of continuum mechanics require dissipation: compressible fluid dynamics (shock formation), incompressible avier-stokes (viscosity), etc. In any case kinetic energy is converted into thermal energy in an irreversible manner until some steady state is achieved (hence turning back the clock would violate the 2nd law of thermodynamics). The laws of physics are time-reversible! How should we resolve this apparent paradox? Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 4 / 30

Introduction & Motivation Physical systems contain many degrees of freedom, e.g. = 10 23. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 5 / 30

Introduction & Motivation Physical systems contain many degrees of freedom, e.g. = 10 23. We can observe O(1) degrees of freedom. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 5 / 30

Introduction & Motivation Physical systems contain many degrees of freedom, e.g. = 10 23. We can observe O(1) degrees of freedom. Boltzmann claims that - for a macroscopic system in equilibrium - all microscopic configurations that are consistent with observations are equally likely. Corollary: dynamical processes take us from less probable states to more probable states. This is quantified using entropy, i.e. the log of the number of microstates consistent with our observations. Heuristically, this idea explains irreversibility (toy example: free expansion of a perfect gas; the reverse process is never observed). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 5 / 30

Introduction & Motivation Physical systems contain many degrees of freedom, e.g. = 10 23. We can observe O(1) degrees of freedom. Boltzmann claims that - for a macroscopic system in equilibrium - all microscopic configurations that are consistent with observations are equally likely. Corollary: dynamical processes take us from less probable states to more probable states. This is quantified using entropy, i.e. the log of the number of microstates consistent with our observations. Heuristically, this idea explains irreversibility (toy example: free expansion of a perfect gas; the reverse process is never observed). But how do you justify this conclusion from ewton s laws? ot obvious! Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 5 / 30

Introduction & Motivation Physical systems contain many degrees of freedom, e.g. = 10 23. We can observe O(1) degrees of freedom. Boltzmann claims that - for a macroscopic system in equilibrium - all microscopic configurations that are consistent with observations are equally likely. Corollary: dynamical processes take us from less probable states to more probable states. This is quantified using entropy, i.e. the log of the number of microstates consistent with our observations. Heuristically, this idea explains irreversibility (toy example: free expansion of a perfect gas; the reverse process is never observed). But how do you justify this conclusion from ewton s laws? ot obvious! Ergodicity (if it holds) gives some insight but does not necessarily help us explain, e.g., compressible fluid flows. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 5 / 30

Lanford s theorem Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 6 / 30

Lanford s theorem Consider identical hard spheres in R d, of diameter ε > 0 (small), with the Boltzmann-Grad scaling ε d 1 = l 1 (l > 0 fixed). The spheres undergo free transport, interrupted by elastic collisions precisely when x i x j = ε for some 1 i < j. Elastic collision of hard spheres (ε = 1 in the figure) Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 7 / 30

Lanford s theorem Assume that the spheres are nearly independent at t = 0; that is, the joint probability f (0) factorizes to the maximum extent possible: f (0, x 1, v 1,..., x, v ) = Z 1 ( ) 1 xi x j >ε f 0 (x i, v i ) 1 i<j Here Z is a normalization factor ( partition function ), and f 0 (x, v) is nice (say, Schwartz class). i=1 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 8 / 30

Lanford s theorem Assume that the spheres are nearly independent at t = 0; that is, the joint probability f (0) factorizes to the maximum extent possible: f (0, x 1, v 1,..., x, v ) = Z 1 ( ) 1 xi x j >ε f 0 (x i, v i ) 1 i<j Here Z is a normalization factor ( partition function ), and f 0 (x, v) is nice (say, Schwartz class). Define f (t) as the pushforward of f (0) through the dynamics, and also introduce the marginals f (s) (t) for 1 s : f (s) (t, Z s) = f (t, Z )dz s+1... dz R d( s) R d( s) i=1 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 8 / 30

Lanford s theorem Recall f (0, x 1, v 1,..., x, v ) = Z 1 1 i<j 1 xi x j >ε Then we have (see (GSRT2014) for a quantitative version): ( ) f 0 (x i, v i ) Theorem (Lanford 1975; Gallagher, Saint-Raymond & Texier 2014) Consider the Boltzmann-Grad limit of hard spheres, ε d 1 = l 1, and let the particles be distributed initially like the function f (0) defined above, where f 0 is a function in the Schwartz class. Then for 0 t T (l, f 0 ), and any s, we have f (s) s (t) ft as pointwise a.e.; here f t (x, v) solves Boltzmann s kinetic equation, ( t + v x ) f t (x, v) = l 1 Q(f t, f t )(x, v) i=1 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 9 / 30

Lanford s theorem Further Remarks Lanford s proof only holds on a short time interval T (l, f 0 ) = O(l). o hydrodynamic limit! Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 10 / 30

Lanford s theorem Further Remarks Lanford s proof only holds on a short time interval T (l, f 0 ) = O(l). o hydrodynamic limit! The time has been improved for a tagged particle in an equilibrium background (Lebowitz & Spohn 1982). This result has been further improved by Bodineau, Gallagher & Saint-Raymond (2015), in order to derive the Brownian motion of the tagged particle. The total volume occupied by particles is negligible, so Lanford s theorem only applies to perfect gases (hence, no phase transitions). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 10 / 30

Lanford s theorem Further Remarks Lanford s proof only holds on a short time interval T (l, f 0 ) = O(l). o hydrodynamic limit! The time has been improved for a tagged particle in an equilibrium background (Lebowitz & Spohn 1982). This result has been further improved by Bodineau, Gallagher & Saint-Raymond (2015), in order to derive the Brownian motion of the tagged particle. The total volume occupied by particles is negligible, so Lanford s theorem only applies to perfect gases (hence, no phase transitions). Lanford s proof is extremely sensitive to the specific form of f (0), and that ansatz is not (!!) propagated forwards in time. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 10 / 30

Lanford s theorem Further Remarks Lanford s proof only holds on a short time interval T (l, f 0 ) = O(l). o hydrodynamic limit! The time has been improved for a tagged particle in an equilibrium background (Lebowitz & Spohn 1982). This result has been further improved by Bodineau, Gallagher & Saint-Raymond (2015), in order to derive the Brownian motion of the tagged particle. The total volume occupied by particles is negligible, so Lanford s theorem only applies to perfect gases (hence, no phase transitions). Lanford s proof is extremely sensitive to the specific form of f (0), and that ansatz is not (!!) propagated forwards in time. Improving the time = Difficult open problem! Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 10 / 30

Lanford s theorem Problem: The convergence at time t > 0 in Lanford s theorem is almost everywhere, whereas at t = 0 we have uniform convergence. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 11 / 30

Lanford s theorem Problem: The convergence at time t > 0 in Lanford s theorem is almost everywhere, whereas at t = 0 we have uniform convergence. In fact it is possible to prove that uniform convergence fails at positive times. Hence we cannot iterate the theorem taking the evolved state as initial data. The inability to propagate the initial strong factorization means we cannot use the full structure of Boltzmann s equation (e.g., H-theorem, etc.) This may present a significant obstacle in any attempt to improve Lanford s theorem. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 11 / 30

Lanford s theorem Problem: The convergence at time t > 0 in Lanford s theorem is almost everywhere, whereas at t = 0 we have uniform convergence. In fact it is possible to prove that uniform convergence fails at positive times. Hence we cannot iterate the theorem taking the evolved state as initial data. The inability to propagate the initial strong factorization means we cannot use the full structure of Boltzmann s equation (e.g., H-theorem, etc.) This may present a significant obstacle in any attempt to improve Lanford s theorem. (Then again there may exist more probabilistic approaches which avoid using Boltzmann s equation explicitly - we don t know!) Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 11 / 30

Lanford s theorem A partial remedy is the notion of strong chaos, which first appears in F. King s PhD thesis (1975), and also in Appendix A of (vblls1980). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 12 / 30

Lanford s theorem A partial remedy is the notion of strong chaos, which first appears in F. King s PhD thesis (1975), and also in Appendix A of (vblls1980). Define the sets } B s = {(X s, V s ) R 2ds 1 i < j s : x i x j (1) } B s = {(X s, V s ) R 2ds t 0, (X s V s t, V s ) B s (2) Theorem (King (1975); van Beijeren et al (1980)) Consider the Boltzmann-Grad limit of hard spheres, ε d 1 = l 1, with initial distribution f (0). Assume that s the marginal f (s) (0) converges as uniformly on compact subsets of B s to f0 s, and f (s) (0) are nice enough. Then for 0 t T (l, f 0 ), the evolved marginals f (s) (t) also converge uniformly on compact subsets of B s, to a function f t, which solves Boltzmann s equation. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 12 / 30

Lanford s theorem The theorem on the previous slide lets us iterate as long as no singularities appear in the dynamics. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 13 / 30

Lanford s theorem The theorem on the previous slide lets us iterate as long as no singularities appear in the dynamics. It does not give us any information at points where x i x j ε, which is a significant limitation because the physical interaction is confined to the set x i x j = ε. In (BGSRS2016), quantitative estimates are employed to obtain a uniform error at distance scales x i x j ε log 1 ε, and their notion of chaos is propagated and the theorem may be iterated (absent singularities). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 13 / 30

Lanford s theorem The theorem on the previous slide lets us iterate as long as no singularities appear in the dynamics. It does not give us any information at points where x i x j ε, which is a significant limitation because the physical interaction is confined to the set x i x j = ε. In (BGSRS2016), quantitative estimates are employed to obtain a uniform error at distance scales x i x j ε log 1 ε, and their notion of chaos is propagated and the theorem may be iterated (absent singularities). Our contribution gives uniform estimates at all scales x i x j > ε, after deletion of a small set of velocities (absent singularities). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 13 / 30

Lanford s theorem Let ψs t be the flow of s hard spheres of diameter ε. Introduce the sets { } D s = Z s = (X s, V s ) R 2ds 1 i j < s, x i x j > ε (3) K s = { Z s = (X s, V s ) D s t > 0, ψ t s Z s = (X s V s t, V s ) } (4) { } Us η = Z s = (X s, V s ) D s inf v i v j > η (5) 1 i<j s Definition { Let F (0) = f (s) }1 s (0). The sequence {F (0)} is nonuniformly f 0 -chaotic if there exists κ (0, 1) such that, for all R > 0, lim were η(ε) = ε κ. ( f (s) ) s (0) f 0 1 Ks U s η(ε) 1 L s i=1 v i 2 2R 2 = 0 (6) Zs Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 14 / 30

Theorem (D. 2016) Consider the Boltzmann-Grad limit of hard spheres, ε d 1 = l 1, with initial distribution f (0). Assume that the initial marginals f (s) (0) are nice enough. Then if the sequence {F (0)} is nonuniformly f 0 -chaotic, then for 0 t T (l, f 0 ), the sequence {F (t)} is nonuniformly f t -chaotic, where f t solves Boltzmann s equation. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 15 / 30

Lanford s theorem Further Remarks Here nice enough always means bounded by Gaussian functions in v, uniformly in. This is an extremely strong bound; most experts expect these bounds to fail on large time intervals for large data. It is possible to propagate (weak) chaos with weaker bounds, e.g. a recent result of Bodineau, Gallagher & Saint-Raymond uses an L 2 framework to derive linear hydrodynamic equations. Can we prove strong chaos in L 2? (Open question!) Even better: strong chaos at the level of energy and entropy. This seems far out of reach, but if true, it would resolve Lanford s theorem once and for all. (Part of the problem is that uniqueness for Boltzmann s equation is unknown at this regularity!) Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 16 / 30

Dispersive inequalities for hard spheres Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 17 / 30

Dispersive inequalities for hard spheres Consider indentical hard spheres in R d, each with diameter ε > 0. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 18 / 30

Dispersive inequalities for hard spheres Consider indentical hard spheres in R d, each with diameter ε > 0. Suppose Z = (x 1, v 1,..., x, v ) is the initial configuration, and denote ψ t Z = Z (t) = (X (t), V (t)) the configuration at time t R. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 18 / 30

Dispersive inequalities for hard spheres Assume t > 0 and consider the following virial function: r Z (t) = ( xi (t) v i (t) v i (t) 2 t ) i=1 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 19 / 30

Dispersive inequalities for hard spheres Assume t > 0 and consider the following virial function: r Z (t) = ( xi (t) v i (t) v i (t) 2 t ) i=1 Illner (1989) observed that this function is monotonic in time for almost every fixed data Z = (X, V ). One can prove the following identity: r Z (T ) r Z (0) = ε k ω k (v ik (t k ) v j k (t k )) 0 where the sum indexes collisions along {Z (t)} 0 t T. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 19 / 30

Dispersive inequalities for hard spheres Illner s identity immediately implies that r Z (t) r Z (0) for a.e. Z and all t 0. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 20 / 30

Dispersive inequalities for hard spheres Illner s identity immediately implies that r Z (t) r Z (0) for a.e. Z and all t 0. This inequality is the only information we need to prove Lanford s theorem globally in time for small data near vacuum. (IP1989) Indeed the inequality r Z (t) r Z (0) (t 0) allows us to compare the particle flow to free transport; this essentially forces the particles to undergo only finitely many collisions on average. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 20 / 30

Dispersive inequalities for hard spheres Recall f (0, x 1, v 1,..., x, v ) = Z 1 Then we have: 1 i<j 1 xi x j >ε ( ) f 0 (x i, v i ) Theorem (Illner & Pulvirenti 1986; Illner & Pulvirenti 1989) Consider the Boltzmann-Grad limit of hard spheres, ε d 1 = l 1, and let the particles be distributed initially like the function f (0) defined above, where f 0 is a function in the Schwartz class. Then if l is sufficiently large (depending on f 0 ) then for t 0, and any s, we have f (s) s (t) ft as pointwise a.e.; here f t (x, v) solves Boltzmann s kinetic equation, ( t + v x ) f t (x, v) = l 1 Q(f t, f t )(x, v) i=1 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 21 / 30

Ideas behind the proof Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 22 / 30

Ideas behind the proof The proof follows the strategy of Lanford; in particular, it relies on the BBGKY hierarchy. The letters stand for Bogoliubov-Born-Green-Kirkwood-Yvon. The BBGKY hierarchy is the evolution equation satisfied by the marginals f (s) (t) of f (t). Lanford solved this equation on a small time interval to prove his theorem. We prove uniform bounds on the marginals f (s) (t) in L ; these bounds are global in time if the mean free path is large compared to f 0. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 23 / 30

Ideas behind the proof The proof follows the strategy of Lanford; in particular, it relies on the BBGKY hierarchy. The letters stand for Bogoliubov-Born-Green-Kirkwood-Yvon. The BBGKY hierarchy is the evolution equation satisfied by the marginals f (s) (t) of f (t). Lanford solved this equation on a small time interval to prove his theorem. We prove uniform bounds on the marginals f (s) (t) in L ; these bounds are global in time if the mean free path is large compared to f 0. Finally, following the classical strategy, we use the uniform bounds to control the probability of pathological collision trees. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 23 / 30

Ideas behind the proof { Let F (t) = f (s) }1 s (t) sphere BBGKY hierarchy: ( t + V s Xs ) f (s) C s+1 f (s+1) (t) = ( s)ε d 1 be a sequence of densities solving the hard (t) = l 1 C s+1 f (s+1) (t) s i=1 R d S d 1 ω (v s+1 v i ) f (s+1) (t,..., x i, v i,..., x i + εω, v s+1 )dωdv s+1 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 24 / 30

Ideas behind the proof Lanford has shown that if the following bounds hold at t = 0, for some β > 0 and µ R: sup sup sup e µs e 1 2 β s i=1 v i 2 (s) f (0, Z s) < (7) 1 s Z s D s then for some small T we have the following bounds for some β 1 < β and µ 1 < µ: sup sup sup sup e µ 1s e 1 2 β s 1 i=1 v i 2 (s) f (t, Z s) < (8) 0 t T 1 s Z s D s More generally we can assume that these bounds hold up to some T > 0 which may not be small. Such bounds have been justified under restrictive assumptions (e.g. large mean free path or tagged particle). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 25 / 30

Ideas behind the proof Lanford s convergence proof is based on a series solution for the BBGKY hierarchy; this series counts all possible interactions, without any consideration for symmetries or cancellations. ( t + V s Xs ) f (s) (t) = l 1 C s+1 f (s+1) (t) Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 26 / 30

Ideas behind the proof Lanford s convergence proof is based on a series solution for the BBGKY hierarchy; this series counts all possible interactions, without any consideration for symmetries or cancellations. ( t + V s Xs ) f (s) (t) = l 1 C s+1 f (s+1) (t) f (s) f (s) t (t) = T s(t)f (s) (0) + l 1 T s (t t 1 )C s+1 f (s+1) (t 1 )dt 1 0 s (t) = t l j j=0 0 tj 1 T s (t t 1 )C s+1... T s+j (t j )f (s+j) (0)dt j... dt 1 0 Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 26 / 30

Ideas behind the proof Lanford s convergence proof is based on a series solution for the BBGKY hierarchy; this series counts all possible interactions, without any consideration for symmetries or cancellations. ( t + V s Xs ) f (s) (t) = l 1 C s+1 f (s+1) (t) f (s) f (s) t (t) = T s(t)f (s) (0) + l 1 T s (t t 1 )C s+1 f (s+1) (t 1 )dt 1 0 s (t) = t l j j=0 0 tj 1 T s (t t 1 )C s+1... T s+j (t j )f (s+j) (0)dt j... dt 1 0 Problem: The series is only known to converge on a short time interval; this is why Lanford s proof breaks down after a short time. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 26 / 30

Ideas behind the proof Similar to (BGSR2015), we use the uniform estimate on the marginals, f (s) (t), to break the interval [0, T ] into manageable pieces. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 27 / 30

Ideas behind the proof Similar to (BGSR2015), we use the uniform estimate on the marginals, f (s) (t), to break the interval [0, T ] into manageable pieces. Let h > 0 be a small time interval. The BBGKY series expansion defines (informally) a branching process running backwards in time. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 27 / 30

Ideas behind the proof Similar to (BGSR2015), we use the uniform estimate on the marginals, f (s) (t), to break the interval [0, T ] into manageable pieces. Let h > 0 be a small time interval. The BBGKY series expansion defines (informally) a branching process running backwards in time. Good branching histories are those growing sub-exponentially at scale h: The branch points refer to fictitious collisions with fictitious particles! Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 27 / 30

Ideas behind the proof Following the scheme set forth by Lanford, we aim to show that typical branching histories converge (as ) to some limiting branching process governed by Boltzmann s equation. The missing link is the estimation of pathological collision trees. These are the trees which exhibit super-exponential growth of collisions, or contain recollisions (i.e. collisions which are not associated with a collision operator C s+k ). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 28 / 30

Ideas behind the proof Following the scheme set forth by Lanford, we aim to show that typical branching histories converge (as ) to some limiting branching process governed by Boltzmann s equation. The missing link is the estimation of pathological collision trees. These are the trees which exhibit super-exponential growth of collisions, or contain recollisions (i.e. collisions which are not associated with a collision operator C s+k ). We estimate all the pathological trees using the uniform bounds, and show that the remaining trees are close to those generated by the Boltzmann process. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 28 / 30

Ideas behind the proof Following the scheme set forth by Lanford, we aim to show that typical branching histories converge (as ) to some limiting branching process governed by Boltzmann s equation. The missing link is the estimation of pathological collision trees. These are the trees which exhibit super-exponential growth of collisions, or contain recollisions (i.e. collisions which are not associated with a collision operator C s+k ). We estimate all the pathological trees using the uniform bounds, and show that the remaining trees are close to those generated by the Boltzmann process. This concludes the proof. Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 28 / 30

References T. Bodineau, I. Gallagher & L. Saint-Raymond. arxiv:1511.03057 (2015). (BGSR2015) T. Bodineau, I. Gallagher & L. Saint-Raymond. Invent. math. (2015). (BGSRS2016) T. Bodineau, I. Gallagher, L. Saint-Raymond, & S. Simonella. arxiv:1612.03722 (2016) R. Denlinger. arxiv:1605.00589 (2016). (GSRT2014) I. Gallagher, L. Saint-Raymond & B. Texier. Zurich Lec. Adv. Math (2014). R. Illner. Transport Theory and Stat. Phys. 18(1), 71-86 (1989). (IP1989) R. Illner & M. Pulvirenti. Comm. Math. Phys. 121(1), 143-146 (1989). F. King, Ph.D. thesis, 1975. O.E. Lanford. In: J. Moser (ed.), Dynamical Systems, Theory and Applications, Lec. otes in Physics, vol. 38, pp. 1-111. (1975) (vblls1980) H. van Beijeren, O. E. Lanford, J. L. Lebowitz, and H. Spohn. J. Stat. Phys. 22 no. 2, 237-257 (1980). Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 29 / 30

Thank you! Ryan Denlinger (UT Austin) Propagation of chaos for hard spheres Caltech 30 / 30