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Billiards with Time-Dependent Boundaries

Billiards with Time-Dependent Boundaries

Alexander Loskutov, Alexey Ryabov

and Leonid Akinshin

Moscow State University

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Some publications• L.G.Akinshin and A.Loskutov. Dynamical properties of some two-dimensional

billiards with perturbed boundaries.- Physical Ideas of Russia, 1997, v.2-3, p.67-86 (Russian).

• L.G.Akinshin, K.A.Vasiliev, A.Loskutov and A.B.Ryabov. Dynamics of billiards with perturbed boundaries and the problem of Fermi acceleration.- Physical Ideas of Russia, 1997, v.2-3, p.87-103 (Russian).

• A.Loskutov, A.B.Ryabov and L.G.Akinshin. Mechanism of Fermi acceleration in dispersing billiards with perturbed boundaries.- J. Exp. and Theor. Physics, 1999, v.89, No5, p.966-974.

• A.Loskutov, A.B.Ryabov and L.G.Akinshin. Properties of some chaotic billiards with time-dependent boundaries.- J. Phys. A, 2000, v.33, No44, p.7973-7986.

• A.Loskutov and A.Ryabov. Chaotic time-dependent billiards.- Int. J. of Comp. Anticipatory Syst., 2001, v.8, p.336-354.

• A.Loskutov, L.G.Akinshi and A.N.Sobolevsky. Dynamics of billiards with periodically time-dependent boundaries.- Applied Nonlin. Dynamics, 2001, v.9, No4-5, p.50-63 (Russian).

• A.Loskutov, A.Ryabov. Particle dynamics in time-dependent stadium-like billiards.- J. Stat. Phys., 2002, v.108, No5-6, p.995-1014.

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Billiards Billiards are systems of statistical mechanics corresponding to the free motion of a mass point inside of a region QM with a piecewise-smooth boundary Qwith the elastic reflection from it.

V

QQ

+Q

0

Q -

Dispersing billiards

Q +

Examples: Lorentz gas, Sinai billiard

Focusing billiards

Q -

Examples: stadium, ellipse

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Importance of the billiard problem:

• very useful model of non-equilibrium statistical mechanics;

• the problem of mixing in many-particle systems the basis of the L.Boltzmann ergodic conjecture;

• ergodic properties of some billiard problems are often important for the theory of differential equations.

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Billiards with Time-Dependent Boundaries

If Q is not perturbed with time billiards with fixed (constant) boundary. In the case of Q= Q(t) we have billiard with time-dependent boundaries.

Two main questions:• description of statistical properties of billiards with Q= Q(t)

• study of trajectories for which the particle velocity can grow infinitely

The last problem goes back to the question concerning the origin of high energy cosmic particles and known as Fermi acceleration.

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Lorentz Gas

a

aV n

U ( t)

A system consisting of dispersing Qi+

components of the boundary Q is said to be a dispersing billiard. A system defined in an unbounded domain D containing a set of heavy discs Bi (scatterers) with boundaries Qi and radius R embedded at sites of an infinite lattice with period a.

Billiard in Q=D\ri=1Bi is called a regular Lorentz gas.

Lorentz gas is a real physical application of billiard problems.

Two cases of the boundary perturbation:•stochastic oscillation•periodic (and phase-synchronized) oscillations

)()( )()( trrUtrRtR

Billiard map: (n, n, Vn, tn) (n+1, n +1, Vn +1, tn +1)

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0,0 2,0x105 4,0x105 6,0x105 8,0x105 1,0x1060

2

4

6

8

10

12

14

16 regular oscillations

analytic

stochastic oscillations

V

n, number of collisions0,0 5,0x104 1,0x105 1,5x105

0

2

4

6

8

10

12

V

t, time

These are the average velocity of the ensemble of 5000 trajectories with different initial velocity directions. These directions have been chosen as random ones.

Fermi acceleration for time-dependent Lorentz gas

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Stadium-like Billiards

Stadium-like billiard a closed domain Q with the boundary Q consisting of two focusing curves.Mechanism of chaos: after reflection the narrow beam of trajectories is defocused before the next reflection.

Billiard dynamics determined by the parameter b:• b << l, a. The billiard is a near integrable system.• b =a/2. The billiard is a K-system.

n + 1Q

The boundary perturbation: focusing components are perturbed periodically in the normal direction, i.e. U(t)=U0 p((t+t0)), where is a frequency oscillation and p( · ) is a 2/ period function.

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The Billiard Map Focusing Components in the form of Circle Arcs

|2| if

cos

cos2coscos,

,cossinarcsin

,2modtan

,sinsincos

,

|| if

,cos2

,2mod)2(

,

,sinarcsin

,4cos4

*

11

11

1

11

11

1*

11

1*

1

*1

*1

1

1

*

1

*1

*1

1

*1

221

nn

nn

nnnn

nnn

nn

nn

nnn

nnn

n

nnn

n

n

nnn

nnn

nn

nn

nn

nnnnnn

v

lRtt

R

xlxx

Rx

v

Rtt

v

v

uuvvv

n + 1

R

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Phase Diagrams of the Velocity Change

V<Vr

V>Vr

V=Vr

Velocity decreaseVelocity increase

Inaccessible areas

Vr corresponds to resonance between boundary perturbations and rotation near a fixed point in coordinates

Background color: the velocity change is transient

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0,0 2,0x105 4,0x105 6,0x105 8,0x105 1,0x106

0

1

2

3

4

20

40

60

80

<V>

VmaxV

n

Particle velocity for different initial values V01=1 and V02=2 .

In the first case the particle velocity in ensemble is bounded.

In the second one there are particles with high velocities.

Maximal velocity value reached by particle ensemble to the n-th iteration

Average velocity of the particle ensemble

Minimal velocity value reached by particle ensemble to the n-th iteration

The Particle Velocity

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Particle Deceleration

Decrease

Increase

U n

V n

n V n+ 1

•The probability of the collision with the right part of the component is more than with the left its side. For the fixed component we have the dotted line.• If at the moment of the collision the focusing component moves outside the billiard table then in some cases after the collision the angle will be the same.• When the time of free path is multiple to the period of the boundary oscillation then the billiard particle should undergoes only decelerated collisions. In the Fig.b: for a large one can see areas with the decreasing velocity corresponding the angle of the particle motion for which the time of free path is multiple to the oscillation period of the focusing component.

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Concluding remarks

For billiards with the developed chaos (the Lorentz gas and the stadium with the focusing components in the form of semicircles), the dependence of the particle velocity on the number of collisions has the root character. At the same time, for a near-rectangle stadium an interesting phenomena is observed. Depending on the initial values, the particle ensemble can be accelerated, or its velocity can decrease up to quite a low magnitude. However, if the initial values do not belong to a chaotic layer then for quite high velocities the particle acceleration is not observed. Analytical description of the considered phenomena requires more detailed analysis and will be published soon (A. Loskutov and A. B. Ryabov, To be published.)

( )

( ) ( )

1 1 1

1 1 1

1 2sin

2 sign 1 2sin sin

nn nn n n

n

n nn n n nn n n

n n

h a h a

h h a h a

kd d dg

g

k kdg d d d dg

g g

+ + +

+ + +

æ ö÷ç ÷ç= + +÷ç ÷÷ççè ø

æ öæ ö æ ö ÷÷ ç ÷ç ç ÷÷ ÷çç ç= + × + + ÷÷ ÷çç ç ÷÷ ÷ç÷ ÷ç çç ç ÷ç ÷è ø è øè ø

nhd

1nhd

+

Dynamics of Time-Dependent Billiards

n-th and (n+1)-th reflections ofthe narrow beam of trajectoriesfrom a boundary Q =const

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n-th and (n+1)-th reflections of the narrow beam of trajectories from a moving boundary Q(t)

V

U

*21 1

2

32 2

2*

1 2 sin

4 41 sin

1 2 sinsin

1 2 sin sin

nn

nn n

n nn

n n

n nn

n n nn n

n nn n

n n

UV

a aU UV V

U UV V

U UV V

g

g

gg

k kg g

+ +

+=

+ +

æ ö÷ç ÷+ +ç ÷ç ÷ç ÷è ø=

+ +

Denotations:

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Result 1. For any sufficiently small oscillations of the

boundary with transversally intersect components dispersing billiard with

has the exponential divergence of trajectories.

min 0Q

k k+¶

$ = >

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Result 2. Consider a time-dependent billiard consisting of focusing (with constant curvature) and neutral components (for example, stadium). Suppose that in this billiard

Then for small enough boundary perturbations this

billiard is chaotic.

2.

ak- >

n-th and (n+1)-th reflections of the narrow beam of trajectories for the billiard on a sphere

( )

( )

1 1 11

1 1 11

2cos sin sin

sin

2 1cos sin cos

sin

n n nnn nn

n

n n nnn nn

n

a a ah R h R

R R R

a a ah

R R R R

kd d dg

g

kdg d dg

g

+ + ++

+ + ++

æ öæ ö æ ö æ ö÷ç ÷ ÷ ÷÷ç ç çç= + +÷ ÷ ÷÷ç ç çç ÷ ÷ ÷ç ç ç÷è ø è ø è ø÷ççè ø

æ öæ öæ ö æ ö æ ö ÷ç ÷ç ÷÷ ÷ ÷÷ç ç ççç= - + ×÷÷ ÷ ÷÷ç ç ççç ÷ ÷ ÷ç ç ç ÷÷è ø è ø è øç ÷çç ÷ç ÷è øè ø

( )1 1 12

cos sin sinsin

sn

s n n nnn n n

n

S

a a aS sign R h R

R R Rk

d dgg

+ + +æ öæ öæ ö æ ö æ ö ÷ç ÷ç ÷÷ ÷ ÷÷ç ç çççº + + ÷÷ ÷ ÷÷ç ç ççç ÷ ÷ ÷ç ç ç ÷÷è ø è ø è øç ÷çç ÷ç ÷è øè øwhere

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( )

( )

1 1 1 1

2 1

1

1 1 11

2cos sin sin

sin

1 tan1

1tan 1 2 tan tan

sin

n n n nn n

n n n

n

n

n n nn nn

n n

h a a aR R

h R R R h

a

Ra a ah R R RR R R h

d k dgd g d

dgk dgd

g d

+ + + +

+

+

+ + ++

æ ö æ ö æ ö÷ ÷ ÷ç ç ç= + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø

æ æ öç ÷çç + ÷ç ÷ç çè øçç= -çæ ö æ ö æ öç÷ ÷ ÷çç ç ç+ +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è øè

ö÷÷÷÷÷÷÷÷÷÷ç ÷÷çç ÷øç

where

Thus,

Result 3. Dispersing billiard with transversally intersect components for which is chaotic.min 0

Qk k

+¶$ = >