The Ehrenfest model and entropy zero deterministic random walksmb13434/pad14/... · 2014. 4. 26. · The Ehrenfest model TheEhrenfest windtree model: Z2-periodic array of rectangular

The Ehrenfest modeland entropy zero

deterministic random walks

Corinna Ulcigrai

University of Bristol

Probability, Analysis and Dynamics

Bristol, 23 April 2014

The Ehrenfest modelThe Ehrenfest windtree model:Z2-periodic array of rectangular scatterersBilliard trajectory: straight line + elastic reflections(angle of incidence = angle of reflection)

Paul and Tatjana Ehrenfest, 1912 (periodic version by Hardy-Weber)

I Is a typical trajectory recurrent?(i.e. do points come back?)

I Are there dense trajectories?

I Is the billiard ergodic?(i.e. no invariant sets)

I What is the diffusion speed?

Lorentz gas versus Ehrenfest model

I the periodic Lorentz Gas

H. A. Lorentz, 1905

Very well studied, many results fromthe 80s onwards(Buminovich, Sinai, Bleher, Szasz,Varju, Chernov, Dolgopyat, Melbourne,

Nicol, Dettmann, Marklof,

Strombergson, Toth...)

Hyperbolic: positive entropy

I the Ehrenfest windtree model

Paul and Tatjana Ehrenfest, 1912

(periodic version by Hardy-Weber)numerical simulations, almost no

rigourous results to now...

several recent breakthroughs (last 2-3

years) via Teichmueller dynamics

Flat: entropy zero!

H. A. Lorentz, 1905

(periodic version by Hardy-Weber)

numerical simulations, almost no

Flat: entropy zero!

H. A. Lorentz, 1905

Flat: entropy zero!

H. A. Lorentz, 1905

Flat: entropy zero!

H. A. Lorentz, 1905

Flat: entropy zero!

Some recent results on the Ehrenfest model

Notation: Let 0 0) divergent trajectories; (Delecroix)

Theorem (Delecroix-Hubert-Lelievre)For all a, b the billiard is superdiffusive: the distance d(bθt (p), p) from theinitial point p after time t satisfies

lim sup d(bθt (p), p) ∼ t2/3.

More precisely:lim sup log d(bθt (p),p)

log t = 2/3.

Ergodicity for bounded polygonal billiards

I Consider a bounded polygonal table, (e.g. cell of Ehrenfest)

I Remark: in a rational polygon (angles of the form pqπ) ⇒trajectories of bθt take finitely many directions.

E.g. angles π2 ,3π2 ⇒ direction in {±θ, π ± θ}.

I Fact: the flow bθt preserves µ = Leb ×∑k

i=1 δθi .

I Recall: The billiard flow bθt is ergodic if for any set A which isinvariant, i.e. bθt (A) = A, either µ(A) = 0 or µ(A

c) = 0.

Theorem (Kerkhoff-Masur-Smillie)Given any rational compact polygonal billiard, foralmost every direction θ the billiard flow bθt isergodic.

CorollaryBilliard trajectories in a random direction are dense.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

i=1 δθi .

c) = 0.

Non-ergodicity for the Ehrenfest model

Theorem (Fraczek-Ulcigrai)For any lengths 0 < a, b < 1 of the sides,for a.e. direction θ, bθt in the Ehrenfest model is:

I not transitive (i.e. no trajectory is dense)

I NOT ergodic (uncountably many erg. comp.).

[Ref: Fraczek-U, Inventiones for a.e. (a,b) + Eskin-Chaika for all (a,b)]

More in general: criterium for non ergodicitywhich holds for a class of periodic billiards

E.g.: billiard in a tube with periodic barriers

I Tools: Non-ergodicity and superdiffusion both exploit:

I deterministic random walks(driven by interval exchange transformations);

I Lyapunov exponents of product of matrices(given by the Kontsevich-Zorich cocycle).

Reduction to a straight-line flow

Fix θ. Consider 4 copies of the Ehrenfest table, one per direction.

Billiard trajectories become straight-line trajectories on 4 tables withpairs of sides glued together (=infinite flat surface).

A deterministic walk driven by a rotationConsider a simpler Z-periodic example: the staircaseE.g: a straight-line trajectory on the staircase (opposite sides are glued).

Consider the red section ∼= [0, 1]×Z.

Given x ∈ [0, 1], the next hittingT (x) is given by a rotation

T (x) = x − α mod 1,

α = cot θ

Associated deterministic walk on Z:

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

Snf (x) is the Z-displacement of x after n hittings; r.v. on ([0, 1], Leb)Rk: Snf are highly correlated r.v. (T is deterministic with zero entropy)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

T (x) = x − α mod 1,

α = cot θ

f (x) =

{+1 if x ∈ [0, 1/2)−1 if x ∈ [1/2, 1) Snf (x) =

n−1∑k=0

f (T ix)

Walks driven by interval exchange transformations

Similarly: the vertical (or horizontal) Z-motionin the Ehrenfest billiard is also given bya deterministic random walk, with:

Snf (x) =n−1∑k=0

f (T ix)

I T is an interval exchange transformation (orIET), i.e a piecewise isometry of the interval:

I f : [0, 1]→ Z is a piecewise constant functionon each exchanged subinterval with E(f ) = 0

Say: T random IET = any irreducible permuation, a.e. choice of lenghts

Snf (x) =n−1∑k=0

f (T ix)

Snf (x) =n−1∑k=0

f (T ix)

A B C D

ABD C

T

Snf (x) =n−1∑k=0

f (T ix)

A B C D

ABD C

T

Snf (x) =n−1∑k=0

f (T ix)

A B C D

ABD C

T

Limsup behaviour of walks driven by IETs

Theorem (Polynomial deviations, Zorich)For a random IET T and f piecewise constant with E(f ) = 0,there exists 0

Renormalization and Lyapunov exponents: a sketchMain tool to prove polynomial deviations: renormalization.

I I (n+1) ⊂ I (n) nested intervals;I T (n) induced IETs on I (n)

(same # exchanged intervals)

I I(n)1 , . . . , I

(n)d exchanged intervals;

I r(n)1 , . . . , r

(n)d return times;

I f = (f1, . . . , fd), fi value on I(0)i ;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

f (T k(xi )), where xi ∈ I (n)i .

I Growth of f (n): f (n+1) = An f(n), where An = A(T

(n)) ∈ SL(d ,Z)

⇒ f (n+1) = An An−1 . . .A1 f (Kontsevich-Zorich cocycle)

⇒ use Oseledetes Thm/Lyapunov exponents.

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

I I(n)1 , . . . , I

I r(n)1 , . . . , r

(n)d return times;

I set f (n) = (f(n)1 , . . . , f

(n)d ) where

f(n)j =

r(n)i∑

k=0

(n)) ∈ SL(d ,Z)

Deterministic walks on R with singularities of type 1x

Consider Xn = Snf , where

I Snf (x) =∑n−1

k=0 f (Tn(x)),

I T is an interval exchangetransformation;

I f is R-valued with 1x -type singularities,[i.e. f (x − xi ) ∼ cix−xinear (a subset of) discontinuities xi of T ]

A B C D

ABD C

T

Motivation:

ergodic theory of smooth area-preservingflows on surfaces:

limiting behaviour of Snf determineschaotic properties (in particular mixing)of locally Hamiltonian flows

I Snf (x) =∑n−1

k=0 f (Tn(x)),

A B C D

ABD C

T

Motivation:

I Snf (x) =∑n−1

k=0 f (Tn(x)),

Motivation:

I Snf (x) =∑n−1

k=0 f (Tn(x)),

Motivation:

I Snf (x) =∑n−1

k=0 f (Tn(x)),

Motivation:

I Snf (x) =∑n−1

k=0 f (Tn(x)),

Motivation:

Limit theorems for 1x -type of singularitiesBehaviour of Xn = Snf is different if

1x singuarities of f are:

asymmetrice.g. f = 1x

E(Xn) = ∞

Theorem (U’, EDTS ’07)T random IET,f with symmetric 1x -sing,

Xnn log n

P→ constant.

Key to prove: Mixing components inloc. Hamiltonian flows with traps

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

Limit theorems for 1x -type of singularities

Behaviour of Xn = Snf is different if1x singuarities of f are:

E(Xn) = ∞

Xnn log n

P→ constant.

symmetric, e.g.

f = 1x −1

1−x

E(Xn) =∞,but

limδ→0

∫ 1−δδ

f

Limiting behaviour for random walks driven by IETsI Back to: Snf (x) =

∑n−1k=0 f (T

n(x)), with:

I T interval exchange transformation,

I f piecewise constant, E(f ) = 0.

I Consider the rescaled r.v.

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

Bufetov (Annals of Math., ’13) has recently shown that:

I Xn do NOT converge in distribution;

I Consider exponential scales (given 0 < s < 1, consider Yn = X[sn]):

I Accumulation points of Yn contain both Dirac deltas (for dense setof s) and non-degenerate distributions.

I Convergence to a moving distribution (driven by the Teichmueller

geodesic flow, e.g. Ynd→ X (gln sS), S associated surface).

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

∑n−1k=0 f (T

n(x)), with:

Xn =Snf√

Var(Snf ).

A B C D

ABD C

T

Some results on existence of limiting distributions

I Tα(x) = x + α mod 1 rotation;

I f : I → R real valued r.v.;I Xn(x , α) = Snf (x ,Tα) =

∑n−1k=0 f (T

kαx)

as r.v. jointly in x and α.

Theorem (Kesten)f piecewise constant with 2 values, Ef = 0Xn(x ,α)

log nd→ X , with X Cauchy r.v.

E.g. f (x) = χI − |I |

E.g. f (x) = 1x =1

1−x

Theorem (Sinai-U’)If f with symmetric 1x singularities

Xn(x ,α)n

d→ X (limiting distribution) .

Open: Similar limit theorems for T IET? for random θ in Ehrenfest?

∑n−1k=0 f (T

kαx)

log nd→ X , with X Cauchy r.v.

E.g. f (x) = χI − |I |

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

∑n−1k=0 f (T

kαx)

log nd→ X , with X Cauchy r.v. E.g. f (x) = χI − |I |

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

∑n−1k=0 f (T

kαx)

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

∑n−1k=0 f (T

kαx)

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

∑n−1k=0 f (T

kαx)

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

∑n−1k=0 f (T

kαx)

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

∑n−1k=0 f (T

kαx)

E.g. f (x) = 1x =1

1−x

Xn(x ,α)n

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The Ehrenfest model and entropy zero deterministic random walksmb13434/pad14/... · 2014. 4. 26. · The Ehrenfest model TheEhrenfest windtree model: Z2-periodic array of rectangular