Stability criteria of spatially inhomogeneous solutions to the Vlasov equation for the Hamiltonian mean-field model
by
Shun Ogawa
→
Europe/Rome
281 (Seminar Room)
281
Seminar Room
Description
In this talk the Hamiltonian mean-field (HMF) model is focused on mainly. The HMF model is a simple toy model which shows typical long-range features [1]. For example, non-equilibrium quasi-stationary states (QSSs) can be observed in the HMF model [2].
The QSSs are supposed to be associated with stable stationary solutions to the Vlasov equation. Hence, finding a stability criterion for stationary solutions to the Vlasov equation is the first step to investigate QSSs, since such a criterion makes it possible to decide
whether a stationary solution can be a QSS or not. The stability criterion of spatially homogeneous states has been given in the form of necessary and sufficient condition for the HMF model [2]. Meanwhile, such a criterion has not been given explicitly for spatially inhomogeneous solutions, since it has been necessary to determine an infinite number of Lagrangian multipliers associated with the infinite number of Casimir invariants [3]. In this talk, it is shown that the problem of Lagrangian multipliers can be avoided by using the angle-action coordinates [4]. Then the formal and spectral stability criteria for spatially inhomogeneous solutions are given explicitly for the HMF model.
[1] A. Campa, T. Dauxois and S. Ruffo, Phys. Rep. 480, 57 (2009).
[2] Y. Y. Yamaguchi, J. Barre ́, F. Bouchet, T. Dauxois and S. Ruffo, Physica A 337, 36 (2004).
[3] A. Campa and P. H. Chavanis, J. Stat. Mech. P06001 (2010).
[4] S. Ogawa arXiv:1301.1130.
