Kinetic transport in quasicrystals

Transcription

1 Kinetic transport in quasicrystals September 203 Jens Marklof* University of Bristol joint work with Andreas Strömbergsson (Uppsala) *supported by Royal Society and ERC

2 The Lorentz gas Arch. Neerl. (905) Hendrik Lorentz ( ) 2

3 The Boltzmann-Grad limit Consider the dynamics in the limit of small scatterer radius ρ ( q(t), v(t) ) = microscopic phase space coordinate at time t A dimensional argument shows that, in the limit ρ 0, the mean free path length (i.e., the average time between consecutive collisions) scales like ρ (d ) (= /total scattering cross section) We thus re-define position and time and use the macroscopic coordinates ( Q(t), V (t) ) = ( ρ d q(ρ (d ) t), v(ρ (d ) t) ) 3

4 The linear Boltzmann equation Time evolution of initial data (Q, V ): (Q(t), V (t)) = Φ t ρ(q, V ) Time evolution of a particle cloud with initial density f L : f t (ρ) (Q, V ) := f ( Φ t ρ (Q, V ) ) In his 905 paper Lorentz suggested that f t (ρ) linear Boltzmann equation: is governed, as ρ 0, by the [ ] t + V Q f t (Q, V ) = S d [ ft (Q, V 0 ) f t (Q, V ) ] σ(v 0, V )dv 0 where the collision kernel σ(v 0, V ) is the cross section of the individual scatterer. E.g.: σ(v 0, V ) = 4 V 0 V 3 d for specular reflection at a hard sphere 4

5 The linear Boltzmann equation rigorous proofs Galavotti (Phys Rev 969 & report 972): Poisson distributed hard-sphere scatterers Spohn (Comm Math Phys 978): extension to more general random scatterer configurations and potentials Boldrighini, Bunimovich and Sinai (J Stat Phys 983): prove convergence for almost every scatterer configuration Quantum: Spohn (J Stat Phys 977): Gaussian random potentials, weak coupling limit & small times; Erdös and Yau (Contemp Math 998, Comm Pure Appl Math 2000): General random potentials, weak coupling limit; Eng and Erdös (Rev Math Phys 2005): Low density limit 5

6 The periodic Lorentz gas 6

7 7

8 The Boltzmann-Grad limit Recall: We are interested in the dynamics in the limit of small scatterer radius (q(t), v(t)) = microscopic phase space coordinate at time t Re-define position and time and use the macroscopic coordinates (Q(t), V (t)) = (ρ d q(ρ (d ) t), v(ρ (d ) t)) 8

9 A limiting random process A cloud of particles with initial density f(q, V ) evolves in time t to f t (ρ) (Q, V ) = [L t ρf](q, V ) = f ( Φ t ρ (Q, V ) ). Theorem A [JM & Strömbergsson, Annals of Math 20]. For every t > 0 there exists a linear operator L t : L (T (R d )) L (T (R d )), such that for every f L (T (R d )) and any set A T (R d ) with boundary of Liouville measure zero, lim ρ 0 A [Lt ρf](q, V ) dq dv = A [Lt f](q, V ) dq dv. The operator L t thus describes the macroscopic diffusion of the Lorentz gas in the Boltzmann-Grad limit ρ 0. Note: The family {L t } t 0 does not form a semigroup. 9

10 A generalization of the linear Boltzmann equation Consider extended phase space coordinates (Q, V, ξ, V + ): (Q, V ) T (R d ) usual position and momentum ξ R + flight time until the next scatterer V + S d velocity after the next hit [ t + V Q ξ ] f t (Q, V, ξ, V + ) = S d f t (Q, V 0, 0, V )p 0 (V 0, V, ξ, V + )dv 0 with a new collision kernel p 0 (V 0, V, ξ, V + ), which can be expressed as a product of the scattering cross section of an individual scatterer and a ceratin transition probability for hitting a given point the next scatterer after time ξ. We obtain the original particle density via f t (Q, V ) = 0 S d f t (Q, V, ξ, V + ) dv + dξ. 0

11 Why a generalization of the linear Boltzmann equation? [ ] t + V Q ξ f t (Q, V, ξ, V + ) = S d f t (Q, V 0, 0, V )p 0 (V 0, V, ξ, V + )dv 0 Substituting in the above the transition density for the random (rather than periodic) scatterer configuration f t (Q, V, ξ, V + ) = g t (Q, V )σ(v, V + )e σξ, σ = vol(b d ), p 0 (V 0, V, ξ, V + ) = σ(v, V + )e σξ yields the classical linear Boltzmann equation for g t (Q, V ).

12 The key theorem: 2

13 Joint distribution of path segments S 5 S 3 S 2 S 4 S 3

14 Joint distribution for path segments The following theorem proves the existence of a Markov process that describes the dynamics of the Lorentz gas in the Boltzmann-Grad limit. Theorem B [JM & Strömbergsson, Annals of Math 20]. Fix an a.c. Borel probability measure Λ on T (R d ). Then, for each n N there exists a probability density Ψ n,λ on R nd such that, for any set A R nd with boundary of Lebesgue measure zero, lim ρ 0 Λ({ (Q 0, V 0 ) T (R d ) : (S,..., S n ) A }) and, for n 3, = Ψ n,λ (S,..., S n ) = Ψ 2,Λ (S, S 2 ) A Ψ n,λ(s,..., S n) ds ds n, n j=3 Ψ(S j 2, S j, S j ), where Ψ is a continuous probability density independent of Λ (and the lattice). Theorem A follows from Theorem B by standard probabilistic arguments. 4

15 First step: The distribution of free path lengths 5

16 References Polya (Arch Math Phys 98): Visibility in a forest (d = 2) Dahlquist (Nonlinearity 997); Boca, Cobeli, Zaharescu (CMP 2000); Caglioti, Golse (CMP 2003); Boca, Gologan, Zaharescu (CMP 2003); Boca, Zaharescu (CMP 2007): Limit distributions for the free path lengths for various sets of initial data (d = 2) Dumas, Dumas, Golse (J Stat Phys 997): Asymptotics of mean free path lengths (d 2) Bourgain, Golse, Wennberg (CMP 998); Golse, Wennberg (CMP 2000): bounds on possible weak limits (d 2) Boca & Gologan (Annales I Fourier 2009), Boca (NY J Math 200): honeycomb lattice JM & Strömbergsson (Annals of Math 200, 20, GAFA 20): proof of limit distribution and tail estimates in arbitrary dimension 6

17 Lattices L R d euclidean lattice of covolume one recall L = Z d M for some M SL(d, R), therefore the homogeneous space X = SL(d, Z)\ SL(d, R) parametrizes the space of lattices of covolume one µ right-sl(d, R) invariant prob measure on X (Haar) 7

18 Affine lattices ASL(d, R) = SL(n, R) R d the semidirect product group with multiplication law (M, x)(m, x ) = (MM, xm + x ). An action of ASL(d, R) on R d can be defined as y y(m, x) := ym + x. the space of affine lattices is then represented by X = ASL(d, Z)\ ASL(d, R) where ASL(d, Z) = SL(d, Z) Z d, i.e., L = (Z d + α)m = Z d (, α)(m, 0) µ right-asl(d, R) invariant prob measure on X 8

19 Let us denote by τ = τ(q, v) the free path length corresponding to the initial condition (q, v). Theorem C [JM & Strömbergsson, Annals of Math 200]. Fix a lattice L 0 and the initial position q. Let λ be any a.c. Borel probability measure on S d. Then, for every ξ > 0, the limit F L0,q(ξ) := lim ρ 0 λ({v S d : ρ d τ ξ}) exists, is continuous in ξ and independent of λ. Furthermore F L0,q(ξ) = with the cylinder F 0 (ξ) := µ ({L X : L Z(ξ) = }) if q L 0 F (ξ) := µ({(l X : L Z(ξ) = }) if q / QL 0. Z(ξ) = { (x,..., x d ) R d : 0 < x < ξ, x x2 d < }. 9

20 Idea of proof (q = 0) ρ (d ) ξ = λ ({ v S d : no scatterer intersects ray(v, ρ (d ) ξ) }) 20

21 Idea of proof (q = 0) ρ (d ) ξ 2ρ λ ({ v S d : Z d Z(v, ρ (d ) ξ, ρ) = }) 2

22 Idea of proof (q = 0) ρ (d ) ξ 2ρ ( Rotate by K(v) SO(d) such that v e ) 22

23 Idea of proof (q = 0) 2ρ ρ (d ) ξ λ ({ v S d : Z d K(v) Z(e, ρ (d ) ξ, ρ) = }) 23

24 Idea of proof (q = 0) ( Apply Dρ = diag(ρ d, ρ,..., ρ ) SL(n, R) ) 24

25 Idea of proof (q = 0) ξ 2 λ ({ v S d : Z d K(v)D ρ Z(e, ξ, ) = }) 25

26 The following Theorem shows that in the limit ρ 0 the lattice ( ) Z d ρ d 0 K(v) t 0 ρ behaves like a random lattice with respect to Haar measure µ. Define a flow on X = SL(n, Z)\ SL(n, R) via right translation by ( ) Φ t e (d )t 0 = t 0 e t. Theorem D. Fix any M 0 SL(n, R). Let λ be an a.c. Borel probability measure on S d. Then, for every bounded continuous function f : X R, lim t S d f(m 0 K(v)Φ t )dλ(v) = X f(m)dµ (M). 26

27 Theorem D is a direct consequence of the mixing property for the flow Φ t. This concludes the proof of Theorem C when q L = Z d M 0. The generalization of Theorem D required for the full proof of Theorem C uses Ratner s classification of ergodic measures invariant under a unipotent flow. We exploit a close variant of a theorem by N. Shah (Proc. Ind. Acad. Sci. 996) on the uniform distribution of translates of unipotent orbits. 27

28 Limiting densities for d = 3 Φ(ξ) = d dξ F (ξ) Φ 0(ξ) = d dξ F 0(ξ) For random scatterer configuarions: Φ(ξ) = σe σξ, Φ 0 (ξ) = σe σξ with σ = vol(b d ) 28

29 Tail asymptotics [JM & Strömbergsson, GAFA 20]. Φ(ξ) = π d 2 2 d d Γ( d+3 2 ) ζ(d) ξ 2 + O ( ξ 2 2 d ) as ξ Φ 0 (ξ) = 0 for ξ sufficiently large Φ(ξ) = σ σ2 ζ(d) ξ + O( ξ 2 ) Φ 0 (ξ) = σ ζ(d) as ξ 0 + O(ξ) as ξ 0. with σ = vol(b d ) = π(d )/2 Γ((d+)/2). 29

30 Quasicrystals 30

31 Cut and project R n = R d R m, π and π int orthogonal projections onto R d, R m L R n an (in general) affine lattice of full rank A := π int (L) is an abelian subgroup of R m, with Haar measure µ A W A a regular window set (i.e. bounded with non-empty interior, µ A ( W) = 0) P(W, L) = {π(y) : y L, π int (y) W} R d is called a regular cut-and-project set P(W, L) defines the locations of scatterers in our quasicrystal 3

32 Example: The Penrose tiling (from: de Bruijn, Kon Nederl Akad Wetensch Proc Ser A, 98) 32

33 Density We have the following well known facts: P(W, L) is a Delone set, i.e., uniformly discrete and relatively dense in R d For any bounded D R d with boundary of Lebesgue measure zero, lim T (the constant c L is explicit) #(P T D) T d = c L vol(d)µ A (W) 33

34 Recall τ = τ(q, v) denotes the free path length corresponding to the initial condition (q, v). Theorem E [JM & Strömbergsson, 203]. Fix a regular cut-and-project set P 0 and the initial position q. Let λ be any a.c. Borel probability measure on S d. Then, for every ξ > 0, the limit F P0,q(ξ) := lim ρ 0 λ({v S d : ρ d τ ξ}) exists, is continuous in ξ and independent of λ. In analogy with Theorem C and the space of lattices, we will express F P0,q(ξ) in terms of a random variable in the space of quasicrystals. 34

35 Spaces of quasicrystals (I) Set G = ASL(n, R), Γ = ASL(n, Z). For g G, define an embedding of SL(d, R) in G by the map (( ) ) A 0 ϕ g : SL(d, R) G, A g, 0 g. 0 m It follows from Ratner s theorems that there exists a closed connected subgroup H g ofc G such that Γ H g is a lattice in H g ϕ g (SL(d, R)) H g the closure of Γ\Γϕ g (SL(d, R)) in Γ\G is given by Γ\ΓH g. Denote the unique right-h g invariant probability measure on Γ\ΓH g by µ g. 35

36 Spaces of quasicrystals (II) Given an affine lattice L R n as above, choose δ > 0 and g G so that L = δ /n Z n g. Then A = π int (Z n g). Since Γ\ΓH g is the closure of Γ\Γϕ g (SL(d, R)), it follows that π int (Z n hg) A for all h H g, and π int (Z n hg) = A for almost all h H g. 36

37 Spaces of quasicrystals (III) For each fixed δ > 0 and window W A, the map from Γ\ΓH g to the set of point sets in R d, Γ\Γh P(δ /n Z n hg, W) gives a natural parametrisation of a space of quasicrystals by Γ\ΓH g. Denote the image of this map by Q g = Q g (W, δ), and define a probability measure on Q g as the push-forward of µ g (for which we will use the same symbol). This defines a random point process in R d. In the topology inherited from Γ\ΓH g, the space Q g is the closure of the set {P(Lh, W) : h SL(d, R)}, with the (in general) affine lattice L = δ /n Z n g. 37

38 Theorem F [JM & Strömbergsson, 203]. For P 0 = P(L, W) and initial position q, pick g G and δ > 0 such that L (q, 0) = δ /n Z n g. Then F P0,q(ξ) = µ g ({P Q g : Z ξ P = }). 38

39 Examples If P 0 = P(L, W), then for almost every L in the space of lattices and q P 0, we have H g = SL(n, R), Γ H g = SL(n, Z). F P0,q(ξ) is independent of P 0 and q and has compact support. If P 0 = P(L, W), then for almost every L in the space of lattices and almost every q, we have H g = ASL(n, R), Γ H g = ASL(n, Z). F P0,q(ξ) ξ (ξ ) where the implied constants depend on n, W, δ. Again F P0,q(ξ) is independent of P 0 and q. If P 0 is the Penrose quasicrystal and q P 0, we have H g = SL(2, R) 2, Γ H g = a congruence subgroup of the Hilbert modular group SL(n, O K ), with O K the ring of integers of K = Q( 5). F P0,q(ξ) work in progress... 39

40 Other aperiodic scatterer configurations: unions of lattices Consider now scatterer locations at the point set P 0 = N i= L j, L i = n /d i (Z d + ω i )M i with ω i R d, M i SL(d, R) and n i > 0 such that n n N = Let S be the commensurator of SL(d, Z) in SL(d, R): S = {(det T ) /d T : T GL(d, Q), det T > 0}. We say that the matrices M,..., M N SL(n, R) are pairwise incommensurable if M i Mj / S for all i j. A simple example is M i = ζ i/d ( ζ i 0 0 d ), i =,..., N, where ζ is any positive number such that ζ, ζ 2,..., ζ N / Q. 40

41 Free path lengths for unions of lattices Theorem G [JM & Strömbergsson, 203]. Let P 0 as on the previous slide with M i SL(d, R) pairwise incommensurable. Let λ be any a.c. Borel probability measure on S d. Then, for every ξ > 0, the limit F P0,q(ξ) := lim ρ 0 λ({v S d : ρ d τ ξ}) exists. If for instance ω i n /d i qmi / Q d for all i, then F P0,q(ξ) = N i= F (n i ξ) where F (ξ) is the distribution of free path length corresponding to a single lattice and generic initial point (as in Theorem C). Recall F (ξ) Cξ for ξ. Thus F P0,q(ξ) C N ξ N. 4

42 Key step in the proof: equidistribution in products... is as before an equidistribution theorem that follows from Ratner s measure classification (again via Shah s theorem): Theorem H [JM & Strömbergsson, 203]. Assume that M,..., M N SL(n, R) are pairwise incommensurable, and α,..., α N / Q d (for simplicity). Let λ be an a.c. Borel probability measure on S d. Then lim f ( ( t S d d, α )(M K(v)Φ t, 0),...,..., ( d, α N )(M N K(v)Φ t, 0) ) dλ(v) = X N f(g,..., g N ) dµ(g ) dµ(g N ). 42

43 Future work 43