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arXiv · 1010.4647

Stability of inhomogeneous states in mean-field models with a local potential

Romain Bachelard·F. Staniscia·Thierry Dauxois·G. De Ninno·S. Ruffo
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Abstract

The Vlasov equation is well known to provide a good description of the dynamics of mean-field systems in the $N \to \infty$ limit. This equation has an infinity of stationary states and the case of {\it homogeneous} states, for which the single-particle distribution function is independent of the spatial variable, is well characterized analytically. On the other hand, the inhomogeneous case often requires some approximations for an analytical treatment: the dynamics is then best treated in action-angle variables, and the potential generating inhomogeneity is generally very complex in these new variables. We here treat analytically the linear stability of toy-models where the inhomogeneity is created by an external field. Transforming the Vlasov equation into action-angle variables, we derive a dispersion relation that we accomplish to solve for both the growth rate of the instability and the stability threshold for two specific models: the Hamiltonian Mean-Field model with additional asymmetry and the mean-field $\phi^4$ model. The results are compared with numerical simulations of the $N$-body dynamics. When the {\it inhomogeneous} state is stationary, we expect to observe in the $N$-body dynamics Quasi-Stationary-States (QSS), whose lifetime diverge algebraically with $N$.

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Romain Bachelard, F. Staniscia, Thierry Dauxois, G. De Ninno, S. Ruffo. 2010-10-22. Stability of inhomogeneous states in mean-field models with a local potential. https://doi.org/10.1088/1742-5468%2F2011%2F03%2Fp03022

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