Billiards with Time-Dependent Boundaries Alexander Loskutov, Alexey Ryabov and Leonid Akinshin Moscow State University Some publications
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Alexander Loskutov, Alexey Ryabov
and Leonid Akinshin
Moscow State University
Examples: Lorentz gas, Sinai billiard
Examples: stadium, ellipseBilliards
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.
• 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.
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.
Billiard map: (n, n, Vn, tn) (n+1, n +1, Vn +1, tn +1)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 ¶Qiand radius R embedded at sites of an infinite lattice with period a.
Billiard in Q=D\ri=1Biis called a regular 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.
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.
The boundary perturbation: focusing components are perturbed periodically in the normal direction, i.e. U(t)=U0 p(w(t+t0)), where wis a frequency oscillation and p( · )is a 2p/wperiod function.
Background color: the velocity change is transient
Vr corresponds to resonance between boundary perturbations and rotation near a fixed point in (x, y) coordinates
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.
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.)
n-th and (n+1)-th reflections of
the narrow beam of trajectories
from a boundary Q =const
n-th and (n+1)-th reflections of the narrow beam of trajectories from a moving boundary Q(t)
boundary with transversally intersect components dispersing billiard with
has the exponential divergence of trajectories.
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.
n-th and (n+1)-th reflections of the narrow beam of trajectories for the billiard on a sphere
Result 3. Dispersing billiard with transversally intersect components for which is chaotic.