Lecture II: The Lorentz gas

Lecture II: The Lorentz gas Applications of measure rigidity: from number theory to statistical mechanics Simons Lectures, Stony Brook October 203 Jens Marklof University of Bristol http://www.maths.bristol.ac.uk supported by Royal Society and ERC

Boltzmann s statistical mechanics Boltzmann proposed to explain the motion of a gas cloud by using the dynamics of microscopic particles atoms and molecules, whose existence was highly disputed during Boltzmann s lifetime. In his 872 paper, Boltzmann derived the famous nonlinear Boltzmann equation in the limit of low particle densities, assuming that the dynamics of the colliding gas molecules is chaotic. Ludwig Boltzmann (844-906) 2

The Boltzmann gas: Sensitive dependence in two-molecule collision. The first rigorous justification of the Boltzmann equation was given by Oscar Lanford in 975 for the dynamics over very short time intervals (a fraction of the mean collision time). The problem for longer time scales is still wide open. 3

The Lorentz gas In an attempt to describe the evolution of a dilute electron gas in a metal, Lorentz proposed in 905 a model, where the heavier atoms are assumed to be fixed, whereas the electrons are interacting with the atoms but not with each other. For simplicity, Lorentz assumed like Boltzmann that the atoms can be modeled by elastic spheres. The Lorentz gas is still one of the iconic models for chaotic diffusion, both in a random and periodic configuration of scatterers. Hendrik Lorentz (853-928) 4

The Lorentz gas with randomly positioned scatterers. 5

The periodic Lorentz gas 6

The periodic Lorentz gas 7

The Boltzmann-Grad limit consider dynamics of macroscopic particle cloud in the limit of small scatterer radius ρ (q(t), v(t)) = microscopic phase space coordinate at time t the mean free path length and the time between collisions scales like ρ (d ) rescale position and time to the macroscopic coordinates (Q(t), V (t)) = (ρ d q(ρ (d ) t), v(ρ (d ) t)) define macroscopic Billiard flow Φ t ρ by (Q(t), V (t)) = Φ t ρ(q 0, V 0 ) 8

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 = L t ρ f, [L t ρf](q, V ) := f ( Φ t ρ (Q, V ) ) In his 905 paper Lorentz suggested that f t is governed, as ρ 0, by the linear Boltzmann equation: [ ] 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 9

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 0

Failure of the Boltzmann equation: The periodic Lorentz gas A cloud of particles with initial density f(q, V ) evolves in time t to f t (Q, V ) = [L t ρf](q, V ) = f ( Φ t ρ (Q, V ) ). Theorem A (JM & Strömbergsson, Annals 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 Lebesgue 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.

A generalization of the linear Boltzmann equation In the case of the periodic Lorentz gas L t does not form a semigroup, and hence in particular the linear Boltzmann equation does not hold. This problem is resolved by considering extended phase space coordinates (Q, V, ξ, V + ) where (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 We prove the following generalization of the linear Boltzmann equation in the extended phase space: [ ] 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 certain transition probability for hitting a given point the next scatterer after time ξ. 2

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 ). 3

The key theorem: 4

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

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 technical probabilistic arguments. 6

First step: The distribution of free path lengths 7

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 8

Polya s forest 9

The distribution of free path lengths in any dimension τ = τ (q, v) free path length for initial condition (q, v) Theorem C (JM & Strömbergsson, Annals Math 200) Fix a lattice L and the initial position q. Let λ be any a.c. Borel probability measure on S d. Then, for every ξ > 0, the limit lim λ({v Sd ρ 0 : ρ d τ ξ}) exists, is continuous in ξ, independent of λ. If q L, it is given by the probability F 0 (ξ) that a random lattice L avoids the cylinder Z(ξ) = { (x,..., x d ) R d : 0 < x < ξ, (x 2,..., x d ) < }. If q / QL, it is given by the probability F (ξ) that a random affine lattice L α avoids the cylinder Z(ξ). 20

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 ) 2

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). /ζ(d) is the relative density of primitive lattice points (i.e., the lattice points visible from the origin). 22

The distribution of free path lengths Theorem C (JM & Strömbergsson, Annals Math 200) Fix a lattice L and the initial position q. Let λ be any a.c. Borel probability measure on S d. Then, for every ξ > 0, the limit lim λ({v Sd ρ 0 : ρ d τ ξ}) exists, is continuous in ξ, independent of λ. If q L, it is given by the probability F 0 (ξ) that a random lattice L avoids the cylinder Z(ξ) = { (x,..., x d ) R d : 0 < x < ξ, (x 2,..., x d ) < }. If q / QL, it is given by the probability F (ξ) that a random affine lattice L α avoids the cylinder Z(ξ). 23

The distribution of free path lengths Theorem C (JM & Strömbergsson, Annals Math 200) Fix a lattice L and the initial position q. Let λ be any a.c. Borel probability measure on S d. Then, for every ξ > 0, the limit lim λ({v Sd ρ 0 : ρ d τ ξ}) exists, is continuous in ξ, independent of λ. If q L, it is given by the probability F 0 (ξ) that a random lattice L avoids the cylinder Z(ξ) = { (x,..., x d ) R d : 0 < x < ξ, (x 2,..., x d ) < }. If q / QL, it is given by the probability F (ξ) that a random affine lattice L α avoids the cylinder Z(ξ). 24

What is a random lattice? 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) F 0 (ξ) := µ ({M X : Z d M Z(ξ) = }) 25

What is a random affine lattice? ASL(d, R) = SL(d, 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 α := (Z d + α)m µ right-asl(d, R) invariant prob measure on X F (ξ) := µ({(m, x) X : (Z d M + x) Z(ξ) = }) 26

Outline of proof of Theorem C (in the case q L and L = Z d ) 27

ρ (d ) ξ λ ({ v S d : ρ d τ ξ }) =... 28

ρ (d ) ξ = λ ({ v S d : no scatterer intersects ray(v, ρ (d ) ξ) }) 29

ρ (d ) ξ 2ρ λ ({ v S d : Z d Z(v, ρ (d ) ξ, ρ) = }) 30

ρ (d ) ξ 2ρ ( Rotate by K(v) SO(d) such that v e ) 3

2ρ ρ (d ) ξ λ ({ v S d : Z d K(v) Z(e, ρ (d ) ξ, ρ) = }) 32

( Apply Dρ = diag(ρ d, ρ,..., ρ ) SL(d, R) ) 33

ξ 2 λ ({ v S d : Z d K(v)D ρ Z(e, ξ, ) = }) 34

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(d, Z)\ SL(d, R) via right translation by ( ) Φ t e (d )t 0 = t 0 e t. Theorem D. Fix any M 0 SL(d, 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). 35

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 (measure rigidity). 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. 36

Other scatterer 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(d, 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. 37

Free path lengths for unions of lattices Theorem E (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. 38

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 F (JM & Strömbergsson, 203). Assume that M,..., M N SL(d, 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 ). 39