what - university of massachusetts amhersthongkun/billiard.pdfa game of billia rd assume the billia...

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

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


Another haoti billiard - Sinai billiardQ

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

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

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

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?

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?

what - university of massachusetts amhersthongkun/billiard.pdfa game of billia rd assume the billia...

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