The Boltzmann-Grad limit of the periodic Lorentz gas. Andreas Strömbergsson Uppsala University astrombe

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1 The Boltzmann-Grad limit of the periodic Lorentz gas Andreas Strömbergsson Uppsala University astrombe joint work with Jens Marklof(Bristol) 4June2009 1

2 The Boltzmann-Grad limit of the periodic Lorentz gas Andreas Strömbergsson Uppsala University astrombe joint work with Jens Marklof(Bristol) 4June2009 2

3 Content 1. The periodic Lorentz gas 2. Main results 3.Thefreepathlength 4. Outline of proof of free path length distribution Key ingredient: Ratner s theorem 5. Outline of proof of the joint distribution of path segments 3

4 The periodic Lorentz gas b 2 b 1 FixalatticeLinR d,thatis L= { n 1 b 1 +n 2 b n d b d : n j Z }, where b 1, b 2,..., b d issomebasisofr d. 4

5 The periodic Lorentz gas 2ρ fixed Placeaballofradiusρ>0(small)ateachlatticepoint: { l+ x :l L, x <ρ } LetK ρ R d (our billiarddomain )bethecomplementofthisset. 5

6 The periodic Lorentz gas q 0 v 0 2ρ fixed The billiardflow :ϕ t :T 1 (K ρ ) T 1 (K ρ ), ( q 0, v 0 ) ( q(t), v(t)) K ρ =complementof { l+ x : l L, x <ρ }. T 1 (K ρ )=[unittangentbundleofk ρ ]=K ρ S d

7 Ergodic theory: Studies the dynamics in the limit of large times. Chaotic diffusion for large times BunimovichandSinai(CommMathPhys1980/81): Inthecaseofafinite horizonandindimensiond=2,thedynamicsisdiffusiveinthelimitoflarge times, and satisfies a central limit theorem. Here finite horizon means that the scatterers are sufficiently large so that the path length between consecutive collisions is bounded. Bleher(J Stat Phys 1992), Szasz-Varju(2007): Remove finite horizon hypothesis. Chernov(J Stat Phys 1994), Balint-Toth(2007): Extend the central limit theorem for finite horizon to higher dimensions.(some hypotheses still remain to be removed!) Kinetictheory:StudiesthedynamicsoftheLorentzgasinthelimitρ 0, appropriately rescaling the picture so that the average path length between two hits is finite(boltzmann-grad limit)... 7

8 The Boltzmann-Grad limit.(ρ 0) Q 0 V 0 Old picture ρ d 1! 2ρ d const ρ d 1 Macroscopic coordinates: Macroscopic billiard flow: ( Q(t), V(t))=(ρ d 1 q(ρ (d 1) t), v(ρ (d 1) t)). F t :T 1 (ρ d 1 K ρ ) T 1 (ρ d 1 K ρ ), ( Q 0, V 0 ) ( Q(t), V(t)). [ExtendtoallofT 1 (R d )bysettingf t =idonthecomplementoft 1 (ρ d 1 K ρ ).] 8

9 . MAIN RESULTS 9

10 Notation: Q 0 V 0 S 1 S 3 S 2 2ρ d const ρ d 1 10

11 Joint distribution of path segments. There exists a Markov process with memory two that describes the dynamics of the Lorentz gas in the Boltzmann-Grad limit: TheoremA.(Marklof,S 08) FixaBorelprobabilitymeasureΛonT 1 (R d ) whichisabsolutelycontinuouswithrespecttovol R d vol S d 1.Then,foreach 1 n Z + thereexistsaprobabilitydensityψ n,λ onr nd suchthat,foranyset A R nd withboundaryoflebesguemeasurezero, lim Λ({ ( Q 0, V 0 ) T 1 (ρ d 1 K ρ ):( S 1,..., S n ) A }) ρ 0 = Ψ n,λ ( S 1,..., S n)dvol R d( S 1) dvol R d( S n). Furthermore,forn 3, A Ψ n,λ ( S 1,..., S n )=Ψ 2,Λ ( S 1, S 2 ) n Ψ( S j 2, S j 1, S j ), j=3 whereψisafunctiononr 3d whichisindependentofλ. 11

12 Consequence: A limiting random process. A cloud of particles with initial densityf( Q, V)evolvesintimetto TheintegralkernelofL t ρis [L t ρf]( Q, V)=f ( F 1 t ( Q, V) ). K t ρ( Q, V; Q 0, V 0 )=δ ( ( Q, V) F t ( Q 0, V 0 ) ). TheoremB.(Marklof,S 08) Foreveryt 0,thereexistsa(nice,explicit) linearoperatorl t suchthatforeverycont.comp.supp.fct.f andanyset A T 1 (R d )withboundaryoflebesguemeasurezero, lim ρ 0 A [L t ρf]( Q, V)dvol R d( Q)dvol S d 1( V) 1 = [L t f]( Q, V)dvol R d( Q)dvol S d 1( V). A 1 TheoperatorL t thusdescribesthemacroscopicevolutionofthelorentzgas intheboltzmann-gradlimitρ 0. 12

13 Analogous results are known for the Lorentz gas with a random(=poisson distributed) configuration of scatterers, see Galavotti(Phys Rev 1969) Spohn(CommMathPhys1978) Boldrighini, Bunimovich, Sinai(J Stat Phys 1983) In this case of a random configuration of scatterers, the function f(t, Q, V):=[L t f]( Q, V) is also seen to satisfy the linear Boltzmann equation: ( t + V Q ) f(t, Q, V)=C ( f(t, Q, ) ) ( V) withthe collisionintegral Cgivenby C(φ)( V)= ω S d 1 1 ω V >0 [ φ ( V 2( V ω) ω ) φ( V) ] ( ω V)d ω. 13

14 Back to periodic Lorentz gas: Hidden variables. Extension of phase space: X:= { } ( Q, V,T, V + ) T 1 (R d ) R 0 S d 1 1 : V V + R ρ :T 1 (R d ) X, ( Q, V) ( Q, V,T, V + ), where T =T( Q, V)= S 1 ( Q, V;ρ) =freepathlengthuntilthenextscatterer, V + = V + ( Q, V)=velocityafterthecollision. TheoremB.(Marklof,S 08) Foreveryt 0,thereexistalinearoperator L t :L 1 loc(x) L 1 loc(x)andanon-negativefunctionp( V,T, V + )suchthatfor everycont.comp.supp.fct.f ont 1 (R d )andanyseta Xwithboundary of Lebesgue measure zero, lim ρ 0 R 1 ρ (A) = A [L t ρf]( Q, V)dvol R d( Q)dvol S d 1( V) 1 [ L t [f p]]( Q, V,T, V + )dvol R d( Q)dvol S d 1 1 ( V)dTdvol S d 1( V + ). 1 14

15 The Fokker-Planck-Kolmogorov equation. Fact.(Marklof,S 08) Thefamily{ L t : t 0}formsasemigroup: L t+s = L t L s, t,s 0. ({ L t :t 0}iscontractingwhenactingonL 1 (X)). Theorem C.(Marklof, S 08) There is an explicit non-negative function p v0 ( V,ξ, V + )suchthat,forany nice f:x R,thefunctionf(t, ):= L t f is the unique solution of the differential equation [ t + V Q ] f(t, Q, V,ξ, V + ) ξ with = S d 1 1 f(t, Q, v 0,0, V)p v0 ( V,ξ, V + )dvol S d 1( v 0 ) 1 f(0, Q, V,ξ, V + ) f( Q, V,ξ, V + ). 15

16 Explicitformulaforp v0 ( V,ξ, V + )indimensiond=2. p v0 ( V,ξ, V + )= 1 ( 2 V V + Φ 0 ξ, cos ( ϕ( V, V + )) (ϕ( V, v 0 )) ), cos 2 2 where ϕ( v 1, v 2 ) (0,2π)istheangle from v 1 to v 2 ; Φ 0 (ξ,w,z)= 6 π 2 1+ ξ 1 max( w, z ) 1 w+z if w+z 0 0 if w+z=0,ξ 1 <1+ w 1 if w+z=0,ξ 1 1+ w, with x = 0 ifx 0 x if0<x<1 1 if1 x. (Marklof, S 08; cf. also Golse, Caglioti 08) 16

17 The free path length. Q 0 V 0 S 1 S 2 2ρ d const ρ d 1 S 1 =thefreepathlength. Limitdistributionof S 1 asρ 0? 17

18 The free path length. q 0 v 0 s 1 s 2 2ρ const Inmicroscopiccoordinates:Freepathlength= S 1 =ρ d 1 s 1. Limitdistributionofρ d 1 s 1 asρ 0? 18

19 Previousstudiesonthelimitdistributionof S 1 (=freepathlength). d=2: BocaandZaharescu(CMP2007) exactlimitdistrfor q 0 random Boca,Zaharescu,Gologan(CMP2003) exactlimitdistrfor q 0 = 0 Earlier studies: Polya(Arch Math Phys 1918); Dahlquist(Nonlinearity 1997); Boca, Cobeli, Zaharescu(CMP 2000); Boca, Gologan, Zaharescu(CMP 2003); Caglioti and Golse(CMP 2003) d 2,meanfreepathlengths: Dumas, Dumas, Golse(J Stat Phys 1997) d 2,boundsonpossibleweaklimits: Bourgain, Golse, Wennberg(CMP 1998); Golse, Wennberg(Math. Model. Numer. Anal. 2000) See also Golse s ICM review(madrid 2006). 19

20 . Limit distribution of the free path length. (Assume covol(l)= 1.) TheoremD.(Marklof,S 07) Forevery q 0 R d,thereexistsacontinuous distributionfunctionf L, q0 :R 0 [0,1]suchthat,foreveryBorelprobability measureλons d 1 1 (abs. cont. wrt. Lebesgue measure), we have: lim λ({ v 0 S d 1 1 :ρ d 1 s 1 ξ }) =F L, q0 (ξ). ρ 0 Explicitformulawhen q 0 = 0(or q 0 L): [ ] F L, 0 (ξ)=prob L ((0,ξ) B1 d 1 ) }{{}, cylinder of height ξ wherelisarandomlatticeofcovolumeone. For irrational q 0 thelimitingdensityisuniversal: F(ξ):=F L, q0 (ξ) isindependentofland q 0,when q 0 / QL. [ ] F(ξ)=ProbL ((0,ξ) B1 d 1 ) }{{}, cylinder of height ξ wherelisarandomaffinelatticeofcovolumeone. 20

21 The space of lattices. Random lattices. Let b 1, b 2,..., b d beabasisofr d.alatticeinr d : L= { n 1 b 1 +n 2 b n d b d :n j Z } =Z d g, with g= covolume(l)=vol ( parallellogram[ b 1,..., b d ] ) = detg. b 1 b 2. M d,d(r). b d 0 b b 1 21

22 The space of lattices. Random lattices. Let b 1, b 2,..., b d beabasisofr d.alatticeinr d : L= { n 1 b 1 +n 2 b n d b d :n j Z } =Z d g, with g= covolume(l)=vol ( parallellogram[ b 1,..., b d ] ) = detg. b 1 b 2. M d,d(r). b d G=SL(d,R)= { g M d,d (R) : detg=1 } Γ=SL(d,Z)=SL(d,R) M d,d (Z) X=Γ\G=SL(d,Z)\SL(d,R)= { Γg :g G } thespeciallineargroup (adiscretesubgroup) (ahomogeneousspace) TheHaarmeasureonG: µ, normalizeby µ(x)=1. Identification:X [thespaceofalllatticesofcovolumeone]; Γg Z d g. ArandompointinXw.r.t.µ def Arandomlatticeofcovolumeone. 22

23 The space of affine lattices. Arandomaffinelatticeofcovolumeone:L+ v, wherelisarandomlatticeofcovolumeone,i.e.l=z d gwithγg X pickedatrandomw.r.t.µ,and, v= xg,with xpickedatrandomin[0,1) d (independently of g). Identification with a homogeneous space: G=ASL(d,R)=SL(d,R) R d ; grouplaw: (g 1, x 1 ) (g 2, x 2 )=(g 1 g 2, x 1 g 2 + x 2 ). GactsonR d fromtheright: v(g, x):= vg+ x. Γ=ASL(d,Z) X=Γ\G=ASL(d,Z)\ASL(d,R) TheHaarmeasureonG: µ, normalizeby µ(x)=1. X [thespaceofallaffinelatticesofcovolumeone]; Γ(g, x) Z d (g, x). ArandompointinXw.r.t.µ def Arandomaffinelatticeofcovolumeone. 23

24 Consequencesoftheexplicitformula for q 0 / QL(or q 0 random) Recall:For q 0 / QLwehave F(ξ)=limλ ({ v 0 S d 1 1 :ρ d 1 s 1 ξ }) [ ] =ProbL ((0,ξ) B1 d 1 ) ρ 0 }{{}, wherelisarandomaffinelatticeofcovolumeone. cylinder of height ξ Using this one proves: Proposition. F(ξ)=1 π d d dγ( d+3 2 )ζ(d)ξ 1 +O ( ξ 1 2) d as ξ ; F(ξ)= π(d 1)/2 Γ((d+1)/2) ξ+o( ξ 2) as ξ 0. Compare with Bourgain, Golse, Wennberg(CMP 1998) and Golse, Wennberg (Math.Model.Numer.Anal.2000):TheyprovedC 1 ξ 1 <1 F(ξ)<C 2 ξ 1 (for large ξ) if the limit distribution exists. 24

25 Explicitformulaford=2 q 0 / QL(or q 0 random) F (x)= { 2 24 π 2 x if0 x 1/2 12 π 2 (Ψ(x) 2Ψ(2x) logx 2 2log2)+4 if1/2 x, where Ψ(x)= x 1 (1 t 1 ) 2 log 1 t 1 dt (thusψ C 2 (R + )\C 3 (R + )). 2,0 1,5 1,0 y=f (x) 0, x for q 0 randomthislimitwasprovedbybocaandzaharescu(cmp2007). 25

26 Consequencesoftheexplicitformula for q 0 = 0 Recall:For q 0 = 0wehave F L, 0 (ξ)=limλ({ v 0 S d 1 1 : ρ d 1 s 1 ξ }) [ ] =ProbL ((0,ξ) B1 d 1 ) ρ 0 }{{}, wherelisarandomlatticeofcovolumeone. cylinder of height ξ Using this one proves: Proposition. C d >0: ξ C d : F L, 0 (ξ)=1 F L, 0 (ξ)= π(d 1)/2 Γ( d+1 2 )ζ(d)ξ+o(ξ2 ) as ξ 0. 26

27 Explicitformulaford=2 q 0 = 0 F L, 0 (x)= 12 if0 x 1/2 π 2 12(1 x) log ex π 2 x 1 x if1/2 x 1 0 if1 x, 1,5 1,0 0,5 y=f L, 0 (x) 0 0 0,5 1,0 1,5 x This limit was proved by Boca, Zaharescu, Gologan(CMP 2003). 27

28 OutlineofproofofTheoremD λ ({ v S d 1 1 :ρ d 1 s 1 ξ }) 2ρ fixed ξρ 1 d q 0 28

29 OutlineofproofofTheoremD =λ ({ v S d 1 1 : C v L }) fixed 2ρ C v ξρ 1 d v q 0 29

30 OutlineofproofofTheoremD =λ ({ v S d 1 1 : C v L }) fixed 2ρ Forgiven v: C v TRANSLATE q 0 ξρ 1 d q 0 v ROTATE BY K v SO(d) vk v =e 1. RESCALE: x 1 : ρ d 1, x 2,...,x d : ρ 1. 30

31 OutlineofproofofTheoremD =λ ({ v S d 1 1 : C 0 ( ) }) L(0, q 0 )K v D }{{} ρ ρ d D ρ = 0 ρ ρ C 0 (0,ξ) B d 1 1 ξ 31

32 OutlineofproofofTheoremD =λ ({ v S d 1 1 : C 0 ( L(0, q 0 )K v D ρ ) }{{} }) RATNER S THEOREM! = a random affine lattice of covolume one (when q 0 / QLandρis small) 1 1 C 0 (0,ξ) B d 1 1 ξ 32

33 OutlineofproofofTheoremD =λ ({ v S d 1 1 : C 0 ( L(0, q 0 )K v D ρ ) }{{} }) RATNER S THEOREM! = a random affine lattice of covolume one (when q 0 / QLandρis small) [ ] ρ 0 Prob((0,ξ) B1 d 1 ) L, wherelisarandomaffinelatticeofcovolumeone. Q.E.D. 33

34 Ratner s theorem on unipotent flows G aconnectedliegroupwithhaarmeasureν. Γ adiscretesubgroupofg. { } SPACE:X=Γ\G= Γg :g G. Assumeν(X)< (normalizesothatν(x)=1). U= { u(t) : t R } anad-unipotentsubgroupofg. FLOW:F t :X Γg Γg u(t) X. RATNER STHEOREM(1990): Foreachp=Γg Xthereexistsaconnected closedliesubgrouph Gsuchthat U H, {F t (p)}=γ\γgh and ν H ( (H g 1 Γg)\H ) < (normalizeto=1!). Theorbit{F t (p)}isasymptoticallyequidistributedinγ\γghw.r.t.ν H,i.e. 1 T f C c (X): f(f t (p))dt fdν H as T. T 0 Γ\ΓgH Ratner deduces this from an even more fundamental theorem characterizing 34 the ergodic invariant measures under U.

35 Ratner s theorem example( easy ) Linearflowonatorus: SPACE:X=Z 2 \R 2 (thusg=r 2,Γ=Z 2 ). FLOW:F t ((x 1,x 2 ))=(x 1,x 2 )+t(1,α) (thusu={u(t)=t(1,α) :t R}) p 35

36 Ratner s theorem example ( fairly easy ) Horocycle flow on a hyperbolic surface (G = SL(2, R), Γ = SL(2, Z), u(t) = ( 10 1t )) 36

37 Other applications of Ratner s theorem: OPPENHEIM S CONJECTURE (First proved by Margulis ) LetF(x 1,...,x n )beanon-degenerate,indefinitequadraticforminn 3variables. Assume that F is not proportional to a quadratic form with rational coefficients. ThenF(Z n )isdenseinr. Proof:Assumen=3.TakeG=SL(3,R),Γ=SL(3,Z),X=Γ\G. { ( 1 ) ( 1 ) }. LetU=SO(2,1) = u SL(3,R) :u 1 u t = 1 WriteF=g ( ) g t (g SL(3,R)). Ratner= Theorbit(Γg)UisasymptoticallyequidistributedinΓ\ΓgH. CASE1:H=G:ThenΓgU=G,impliesF(Z 3 )=R! CASE2:H=U:ThenFhastobeproportionaltoarationalform

38 Other application of Ratner s theorem: The number of solutions to certain diophantine systems of equations E.g.: Eskin, Mozes, Shah, Example: Givenp(λ)=λ n +a n 1 λ n 1 + +a 0 (a j Z,pirreducibleoverQ),write { } V p (Z)= A M n (Z):det(λI A)=p(λ), and let { } N(T,V p )= A V p (Z) : A T. Then N(T,V p ) c p T n(n 1) as T. (c p >0) 38

39 Other applications of Ratner s theorem: Fine scale statistics for certain explicit sequences of numbers Forfixedα,β Rwith4β α 2 >0: m 2 +αmn+βn 2 (m,n Z): 0=λ 0 <λ 1 λ 2 λ 3... Eskin, Margulis, Mozes 01: Pair correlation= Poisson for explicit α, β! (Sarnak 97:foralmostallα,β.) Theanalogfor(m γ) 2 +(n δ) 2 :Marklof 00. nmod1,n=1,2,3,...:elkies,mcmullen 04. αnmod1,n=1,2,3,...,(α=randomnumber):marklof 00, S&Venkatesh 05. OPENPROBLEM:αn 2 mod1;paircorrelation= Poisson forexplicitα??? (Rudnick&Sarnak 98:Trueforalmostallα!) 39

40 PROOF OF THE LIMITING JOINT DISTRIBUTION OF PATH SEGMENTS... 40

41 Recall: TheoremA.(Marklof,S 08) FixaBorelprobabilitymeasureΛonT 1 (R d ) whichisabsolutelycontinuouswithrespecttovol R d vol S d 1.Then,foreach 1 n Z + thereexistsaprobabilitydensityψ n,λ onr nd suchthat,foranyset A R nd withboundaryoflebesguemeasurezero, lim Λ({ ( Q 0, V 0 ) T 1 (ρ d 1 K ρ ):( S 1,..., S n ) A }) ρ 0 = Ψ n,λ ( S 1,..., S n)dvol R d( S 1) dvol R d( S n). Furthermore,forn 3, A Ψ n,λ ( S 1,..., S n )=Ψ 2,Λ ( S 1, S 2 ) n Ψ( S j 2, S j 1, S j ), j=3 whereψisafunctiononr 3d whichisindependentofλ. 41

42 Theorem A ingredients of proof. h TheoremD STRONG Take q 0 L.Fixapoint honthehalfsphere: Fixξ>0.Then: lim λ ({ v 0 S d 1 1 : ρ d 1 s 1 =ξand ρ 0 h {}}{ hitpoint = h }) = Prob [ L C ξ = and h L ], wherelisarandomlatticeofcovolumeone,and: h proj h C ξ ξ

43 Theorem A ingredients of proof. TheoremD STRONG Alsoallowtoreplaceinitialcondition( 0, v 0 ): withmoregeneral(ρ β( v 0 ), v 0 ): ρ ( β:s d 1 1 S d 1 1 anyfixed, nice function). The limit lim λ ({ v 0 S d 1 1 :ρ d 1 s 1 =ξand hitpoint = h }) ρ 0 still exists, but more complicated formula(depends on β and on λ). 43

44 Theorem A ingredients of proof. TheoremD STRONG makeuniform overλ(inan equismoothfamily ) over β:s d 1 1 S d 1 1 (in an equicontinuous family) over the test function(in a uniformly bounded and equicontinuous family) 44

45 OutlineofproofofTheoremA(hereforn=3) Addoverallpairsofobstacles,likeO 1,O 2 : O 2 O 1 q 0 45

46 OutlineofproofofTheoremA(hereforn=3) O 1,O 2 -contribution: O 2 O 1 q 0 CanapplyTheoremD STRONG tounderstandtheo 1,O 2 -contributionwhenρis small. 46

47 OutlineofproofofTheoremA(hereforn=3) FixedO 1,addingoverO 2 : O 2 O 1 q 0 ApplyTheoremD STRONG again, oneleveldown,tocapturethissum! 47