Hard-disk Lorentz gas

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

The periodic hard-disk Lorentz gas is a two-dimensional billiard in which a point particle undergoes elastic collisions on hard disks which are fixed in the plane in the form of a spatially periodic lattice. Bunimovich and Sinai have proved that... 

Figure 9. Two trajectories of the periodic hard-disk Lorentz gas. They start from the same position but have velocities that differ by one part in a million, (a) Both trajectories depicted on large spatial scales, (b) Initial segments of both trajectories showing the sensitivity to initial conditions.

Figure 10. The diffusive inodes of the periodic hard-disk Lorentz gas represented by their cumulative function depicted in the complex plane (ReF, ImF ) versus the wavenumber A .

This other Lorentz gas is similar to the previous one except that the hard disks are replaced by Yukawa potentials centered here at the vertices of a square lattice. The Hamiltonian of this system is given by... 

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