1. Billiards with Time-Dependent Boundaries
Alexander Loskutov, Alexey Ryabov
and Leonid Akinshin
Moscow State University

2. 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 (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 (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
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
A.Loskutov and A.Ryabov. Chaotic time-dependent billiards.- Int. J. of Comp. Anticipatory Syst., 2001, v.8, p
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 (Russian).
A.Loskutov, A.Ryabov. Particle dynamics in time-dependent stadium-like billiards.- J. Stat. Phys., 2002, v.108, No5-6, p

3. Billiards Dispersing billiards Focusing billiards
Billiards are systems of statistical mechanics corresponding to the free motion of a mass point inside of a region QM with a piecewise-smooth boundary ¶ Q with the elastic reflection from it.
Dispersing billiards
Examples: Lorentz gas, Sinai billiard
Focusing billiards
Examples: stadium, ellipse

4. 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.

5. 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.

6. Lorentz Gas
Lorentz gas is a real physical application of billiard problems.
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.
Two cases of the boundary perturbation:
stochastic oscillation
periodic (and phase-synchronized) oscillations
Billiard map: (n, n, Vn, tn)  (n+1, n +1, Vn +1, tn +1)

7. Fermi acceleration for time-dependent Lorentz gas
These are the average velocity of the ensemble of 5000 trajectories with different initial velocity directions. These directions have been chosen as random ones.

8. 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.
The boundary perturbation: focusing components are perturbed periodically in the normal direction, i.e. U(t)=U0 p(w(t+t0)), where w is a frequency oscillation and p( · ) is a 2p/w period function.

9. The Billiard Map Focusing Components in the form of Circle Arcs

10. Phase Diagrams of the Velocity Change
V<Vr
V>Vr
Velocity increase
Velocity decrease
V=Vr
Inaccessible areas
Background color: the velocity change is transient
Vr corresponds to resonance between boundary perturbations and rotation near a fixed point in (x, y) coordinates

11. The Particle Velocity
Maximal velocity value reached by particle ensemble to the n-th iteration
Minimal velocity value reached by particle ensemble to the n-th iteration
Average velocity of the particle ensemble
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.

12. Particle Deceleration
Increase
Decrease
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 y 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 y 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.

13. 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.)

14. Dynamics of Time-Dependent Billiards
n-th and (n+1)-th reflections of
the narrow beam of trajectories
from a boundary Q =const

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

16. Result 1. For any sufficiently small oscillations of the
boundary with transversally intersect components dispersing billiard with
has the exponential divergence of trajectories.

17. 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.

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

19. Thus,
where
Result 3. Dispersing billiard with transversally intersect components for which is chaotic.