What - University of Massachusetts Amherst

What is a billiard system?Hong-Kun ZhangDepartment of Mathemati s and Statisti sUniversity of Massa husettsAmherst, MA 01003, USAMay, 2011Hong-Kun Zhang What is a billiard system?

Overview of the talkWhat are billiards?Let's play a game!Why billiards are important?Appli ations

Hong-Kun Zhang What is a billiard system?

What are billiards?Billiards appear as natural models in many problems ofmathemati s and physi s. The me hani of billiard system is verysimilar to that of the ue sport of billiards.The billiard parti le moves alongstraight lines inside the table;When it rea hes the wall, theangle of in iden e is equal to theangle of re�e tion (elasti ollision).Hong-Kun Zhang What is a billiard system?

What are billiards?Billiards appear as natural models in many problems ofmathemati s and physi s. The me hani of billiard system is verysimilar to that of the ue sport of billiards.

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PSfrag repla ements α α

The billiard parti le moves alongstraight lines inside the table;When it rea hes the wall, theangle of in iden e is equal to theangle of re�e tion (elasti ollision).Hong-Kun Zhang What is a billiard system?

A game of billiardAssume the billiard table is smooth � no fri tion, but does nothave to be re tangular;Assume the parti le is tinyThere is a hole somewhere on the tableGoal: Hit the parti le and make it ollide with the wall at leaston e before running into the hole. Can we win the game?xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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Re tangular billiards

If the initial error is small, we an win the game easily.Hong-Kun Zhang What is a billiard system?





Dispersing billiardx


Dispersing billiard


Dispersing billiard


Dispersing billiard

The error grows very fast, it is hopeless to win mostly even if theinitial error is small. We all this billiard haoti , sin e we an notpredi t a traje tory by knowing its adja ent ones.Hong-Kun Zhang What is a billiard system?

Another haoti billiard - Sinai billiardQ

b

a

c



b

a

c



b

a

c

It seems there is NO way that we an win the game sin e it is a haoti billiard!Hong-Kun Zhang What is a billiard system?

The game of billiards

Wait: did we assume that the table is smooth? .....Can we see any hope in the long run?Yes, we an win the game ertainly even for the haoti Sinaibilliard! Hong-Kun Zhang What is a billiard system?

The game of billiardst=4 Wait: did we assume that the table is smooth? .....Can we see any hope in the long run?Yes, we an win the game ertainly even for the haoti Sinaibilliard! Hong-Kun Zhang What is a billiard system?

The game of billiardst=4

t=19.1416

Wait: did we assume that the table is smooth? .....Can we see any hope in the long run?Yes, we an win the game ertainly even for the haoti Sinaibilliard! Hong-Kun Zhang What is a billiard system?

regular billiardsA regular billiard has properties "opposite" to haoti one, i.e. we an predi t a path very well a ording to its neighbors. Forexample, billiards on a ir le or ellipse are regular.


Cir le, ellipse, hyperbola and parabolaCir le, ellipse, hyperbola and parabola an be obtained by utting a one using a plane. So these urves are also alled oni se tions.


ir ular billiard


Regular system � ir ular billiards

Any billiard path has a ir ular austi ; reversely, any on entri ir le inside the table is the austi of a path to abilliard path. We all a system regular if it has the aboveproperty.If two initial onditions are very lose, then the two pathremain lose and have similar austi s.Hong-Kun Zhang What is a billiard system?

Another regular billiard � ellipti billiards

Billiards in an ellipse has two families of austi s ( onfo al ellipsesand onfo al hyperbolas).Hong-Kun Zhang What is a billiard system?









Another regular billiard � ellipti billiards3


Birkho� onje tureIt was onje tured by Birkho� in 1927 that the only regular billiardsare ellipti (in luding ir ular) billiards.This onje ture is still unsolved � so anyone who is interested inmath is en ouraged to think about it.


Another questionPi k a region A in the Sinai billiard table, what is the frequen ythat a typi al billiard traje tory visits A in the long run?xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

A

Sin e eventually theimages of x will be almostuniformly distributed inthe phase spa e,Our onje ture:frequen y of visiting A ≈ area AAny system has thisproperty is alled anergodi system.Hong-Kun Zhang What is a billiard system?

Another questionPi k a region A in the Sinai billiard table, what is the frequen ythat a typi al billiard traje tory visits A in the long run?

t=4

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A



t=4


A



t=4


A


Our onje ture:

t=4

A

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The Sinai billiard is ergodi Hong-Kun Zhang What is a billiard system?

Boltzmann's Hypothesis - a Conje ture for enturies?Boltzmann's Ergodi Hypothesis, 1870'sFor any losed system of large number of intera ting parti les inequilibrium, time averages are lose to the spa e average. (Physi almodel)Sinai Ergodi Hypothesis, 1961The n− dimensional Sinai billiard is ergodi for any n ≥ 2.(Mathemati al model)


Ya. G. Sinai

Professor in Prin eton.Proved the ergodi ity of 2-d Sinai billiard in 1970's.Hong-Kun Zhang What is a billiard system?

Dis rete billiard models and appli ationsBilliards appear as natural models for many problems in physi s, hemi al engineering, and other �eldsFermi a elerationBrownian motionUsed to prove the Ohm's lawPSfrag repla ementsxFx Hong-Kun Zhang What is a billiard system?

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What - University of Massachusetts Amherst