第3回 理論グループ共通セミナー （講師：Prof. Tomio Yamakoshi Petrosky）
|場 所：||63講義室（工6号館 269号室）|
|講演者：||Prof. Tomio Yamakoshi Petrosky|
|(Center for Complex Quantum Systems, The University of Texas)|
|題 目：||Level Repulsion of Eigenstates of the Liouvillian in Asteroid belt in the Solar System, and its analogy to the Band-Spectrum of an Electron in Solid State Physics.|
In spite of the fact that the Liouville equation is the fundamental equation in classical dynamics, this equation is not yet well investigated except for non-equilibrium statistical mechanics. In this lecture, we discuss an interesting application of the Liouvillian formulation to a classical dynamics on the solar system. We estimate the size of the Kirkwood gaps in the distribution of main belt asteroids in terms of the eigenvalue problem of the Liouvillian. This is a long standing problem in a non-integrable system, because of the small denominator difficulty of the nonlinear system due to the resonance singularity. We show that this problem is solved on the level of distribution function instead of a trajectory level. The problem is treated as an example of a restricted three-body problem that consists of an asteroid, Sun, and Jupiter. Jupiter is treated as a perturbation on the asteroid-Sun two-body problem. We found that the eigenstate of the Liouvillian of this two-body problem has a threefold degeneracy at the resonance point inside the Kirkwood gaps. Using the degenerate perturbation theory which has been extensively developed in quantum mechanics, we can analyze the resonance effect on the resonance point without divergence. The perturbation due to Jupiter removes the degeneracy and results a level repulsion in the eigenvalues, just as the same mechanism of the level repulsion in the eigenvalues of the Hamiltonian for an electron in quantum solid-state physics. As a result, the spectrum of the Liouvillian for the three-body problem has a band structure in the frequency space. Since the degeneracy is threefold, there is a stable eigenstate inside the resonance region. The stability is independent of the intensity of the perturbation determined by the eccentricity of the asteroid. This explains the stability of some asteroids with moderately large eccentricity that are located on the resonance orbit inside the Kirkwood gaps.
参考文献：T. Y. Petrosky, Prog. Theor. Phys. 125 (2011) 411.