Lszl Oroszlny, Zoltn Csaba Nagy : Classical Magnetic Sinai Billiard Supervisors: Jzsef Cserti Tams T

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László Oroszlány, Zoltán Csaba Nagy Classical
Magnetic Sinai Billiard SupervisorsJózsef
Cserti Tamás Tasnádi Péter Pollner
Eötvös Loránd University Department of Physics
of Complex Systems

• We would like to thank the stuff of the
  Department of Physics of Complex Systems for
  their useful help, and their friendly support.
• Special thanks to our classmates, and our
  HAPS-mates for their infinite calmness

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Motivations
BBn

• Investigation of classical and quantum billiards
  (80s)
• Sinai billiard Bunimovich
  billiard Lorentz gas
• Nanotechnology 2DEG realiseable
• B 2DEG superconductor

B0
Classical investigation
Magnetic field B
Cyclotron radius Rc B1/ Rc
-Square lattice -Hexagonal lattice

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Poincaré sections
Chaotic behaviour by stong magnetic filed
B ??
appearing and disappearing of non-chaotic
islands in the phase space
Analyticaly provable!
Hexagonal lattice Square lattice
Rc0.4
Rc0.5
sin(m)
sin(m)
j
Rc0.9
Rc1.6
B
sin(m)
Rc
Rc2
Rc1.2
sin(m)
j
j
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Poincaré sections
Chaotic behaviour by stong magnetic filed
B ??
appearing and disappearing of non-chaotic
islands in the phase space
Analyticaly provable!
Hexagonal lattice Square lattice
Rc0.4
Rc0.5
sin(m)
sin(m)
j
Rc0.9
Rc1.6
B
sin(m)
Rc
Rc2
Rc1.2
sin(m)
j
j
5

Sin(m)
j
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Non-Chaotic/Chaotic phase space volume ratio
rVchao/Vfull
square
hexagon
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The Lyapunov exponent
dx(t)eltdx(0)
a)Rc0.9 b)Rc0.98 c)Rc0.99 d)Rc1.02
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The Lyapunov exponent
dx(t)eltdx(0)
a)Rc0.9 b)Rc0.98 c)Rc0.99 d)Rc1.02
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Stability analysis I
Stability matrices
The stability condition
MP(PiTiEiTi)
0?Tr(M)- 2
Free motion
Boundary transition
Motion in b magnetic field
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Stability analysis II
stability
Rc
Rc1.26
Rc1.273
sin(m)
sin(m)
j
j
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From diffusion to motion in an avarage flux
Investigating the whole plane
Diffusion
B ? ?
10R ?Rc
Diffusion of small circles
Hexagonal
Square
y
y
Motion in an avarage flux
x
x
Initial position
B ? 0
b0.5098
b0.4858
log(r)
log(r)
log(t)
log(t)
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Summary

• Magnetically tunable Sinai billiard
• Lyapunov exponent l(B) ? Phase space volume ratio
  r(B)
• Fast bifurcation of a large non-chaotic island
• ? l(B) , r(B) peak
• Diffusion B ? ?
• Average flux B ? 0
• Continuing our work
• Conductivity sij ltvi(0)vj(t)gt (Kubo formula)
• Quantum Mechanics

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