the ehrenfest model and entropy zero deterministic random walksmb13434/pad14/... · 2014. 4....

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

the ehrenfest model and entropy zero deterministic random walksmb13434/pad14/... · 2014. 4....

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