Billiards with Time-Subordinate Limits.


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Dynamical properties of some two-dimensional billiards with irritated limits. ... Progress of billiards with irritated limits and the issue of Fermi speeding up. ...
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Billiards with Time-Dependent Boundaries Alexander Loskutov, Alexey Ryabov and Leonid Akinshin Moscow State University

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Some productions L.G.Akinshin and A.Loskutov. Dynamical properties of some two-dimensional billiards with irritated limits.- Physical Ideas of Russia , 1997, v.2-3, p.67-86 (Russian). L.G.Akinshin, K.A.Vasiliev, A.Loskutov and A.B.Ryabov. Progression of billiards with annoyed limits and the issue of Fermi quickening.- Physical Ideas of Russia , 1997, v.2-3, p.87-103 (Russian). A.Loskutov, A.B.Ryabov and L.G.Akinshin. Instrument of Fermi increasing speed in scattering billiards with irritated limits.- J. Exp. also, Theor. Material science , 1999, v.89, No5, p.966-974. A.Loskutov, A.B.Ryabov and L.G.Akinshin. Properties of some disorganized billiards with time-subordinate limits.- J. Phys. A , 2000, v.33, No44, p.7973-7986. A.Loskutov and A.Ryabov. Disorderly time-subordinate billiards.- Int. J. of Comp. Expectant Syst. , 2001, v.8, p.336-354. A.Loskutov, L.G.Akinshi and A.N.Sobolevsky. Progression of billiards with occasionally time-subordinate limits.- Applied Nonlin. Elements , 2001, v.9, No4-5, p.50-63 (Russian). A.Loskutov, A.Ryabov. Molecule flow in time-subordinate stadium-like billiards.- J. Detail. Phys. , 2002, v.108, No5-6, p.995-1014.

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Dispersing billiards Focusing billiards Examples: Lorentz gas, Sinai billiard Examples: stadium, circle Billiards are frameworks of factual mechanics comparing to the free movement of a mass point within an area Q M with a piecewise-smooth limit ¶ Q with the versatile reflection from it.

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Importance of the billiard issue: • exceptionally helpful model of non-harmony measurable mechanics; • the issue of blending in numerous molecule frameworks  the premise of the L.Boltzmann ergodic guess; • ergodic properties of some billiard issues are regularly essential for the hypothesis of differential conditions.

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Billiards with Time-Dependent Boundaries If ¶ Q is not irritated with time  billiards with altered (steady) limit. On account of ¶ Q= ¶ Q(t) we have billiard with time-subordinate limits . Two principle questions: portrayal of measurable properties of billiards with ¶ Q= ¶ Q(t) investigation of directions for which the molecule speed can become interminably The last issue backpedals to the inquiry concerning the root of high vitality inestimable particles and known as Fermi quickening .

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Two instances of the limit bother: stochastic swaying occasional (and stage synchronized) motions Billiard guide: (  n ,  n , V n , t n )  (  n+1 ,  n +1 , V n +1 , t n +1 ) Lorentz Gas Lorentz gas is a genuine physical utilization of billiard issues. A framework comprising of scattering ¶ Q i + segments of the limit ¶ Q is said to be a scattering billiard.  A framework characterized in an unbounded area D containing an arrangement of substantial plates B i (scatterers) with limits ¶ Q i and span R inserted at destinations of a boundless cross section with period a . Billiard in Q=D\  r i=1 B i is known as a normal Lorentz gas .

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Fermi speeding up for time-subordinate Lorentz gas These are the normal speed of the troupe of 5000 directions with various beginning speed headings. These bearings have been picked as arbitrary ones.

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Stadium-like Billiards Stadium-like billiard  a shut space Q with the limit ¶ Q comprising of two centering bends. System of confusion: after reflection the restricted light emission is defocused before the following reflection. Billiard flow dictated by the parameter b : b << l, a. The billiard is a close integrable framework. b =a/2. The billiard is a K-framework. The limit irritation: centering parts are bothered intermittently in the typical heading, i.e. U ( t ) =U 0 p ( w ( t+t 0 )), where w is a recurrence wavering and p ( · ) is a 2 p/w period capacity.

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The Billiard Map Focusing Components as Circle Arcs

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Phase Diagrams of the Velocity Change V<V r V>V r Velocity expand Velocity diminish V=V r Inaccessible zones Background shading: the speed change is transient V r relates to reverberation between limit irritations and turn almost an altered point in ( x, y ) facilitates

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The Particle Velocity Maximal speed esteem came to by molecule outfit to the n-th emphasis Minimal speed esteem came to by molecule gathering to the n-th cycle Average speed of the molecule troupe Particle speed for various introductory qualities V 01 =1 and V 02 =2 . In the principal case the molecule speed in troupe is limited. In the second one there are particles with high speeds.

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Particle Deceleration Increase Decrease The likelihood of the crash with the right part of the segment is more than with the left its side. For the altered part we have the dabbed line . In the event that right now of the impact the centering part moves outside the billiard table then now and again after the crash the edge y will be the same. At the point when the season of free way is numerous to the time of the limit swaying then the billiard molecule ought to experiences just decelerated impacts. In the Fig.b : for a vast y one can see regions with the diminishing speed comparing the point of the molecule movement for which the season of free way is different to the wavering time of the centering segment.

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Concluding comments For billiards with the created mayhem (the Lorentz gas and the stadium with the centering segments as half circles), the reliance of the molecule speed on the quantity of crashes has the root character. In the meantime, for a close rectangle stadium an intriguing wonders is watched. Contingent upon the underlying qualities, the molecule gathering can be quickened, or its speed can diminish up to a significant low greatness. Be that as it may, if the underlying qualities don\'t have a place with a tumultuous layer then for entirely high speeds the molecule increasing speed is not watched. Scientific portrayal of the considered marvels requires more point by point examination and will be distributed soon ( A. Loskutov and A. B. Ryabov, To be distributed . )

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Dynamics of Time-Dependent Billiards n - th and ( n +1)- th impressions of the thin light emission from a limit  Q =const

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Denotations: n - th and ( n +1)- th impressions of the slender light emission from a moving limit  Q(t) V U

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Result 1. For any adequately little motions of the boundary with transversally converge segments scattering billiard with has the exponential difference of directions.

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Result 2. Consider a period subordinate billiard comprising of centering (with steady ebb and flow) and impartial segments (for instance, stadium). Assume that in this billiard Then for sufficiently little limit irritations this billiard is turbulent.

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n - th and ( n +1)- th impressions of the slender light emission for the billiard on a circle where

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Thus, where Result 3 . Scattering billiard with transversally meet parts for which is turbulent.

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