Stability of inhomogeneous states in mean-field models with an external potential

Abstract

The Vlasov equation is well known to provide a good description of the dynamics ofmean-field systems in the limit. This equation has an infinity of stationary states and the case of homogeneousstates, for which the single-particle distribution function is independent of the spatialvariable, is well characterized analytically. On the other hand, the inhomogeneous caseoften requires some approximations for an analytical treatment: the dynamicsis then best treated in action–angle variables, and the potential generatinginhomogeneity is generally very complex in these new variables. We here treatanalytically the linear stability of toy models where the inhomogeneity is created byan external field. Transforming the Vlasov equation into action–angle variables,we derive a dispersion relation that we accomplish solving for both the growthrate of the instability and the stability threshold for two specific models: theHamiltonian mean-field model with additional asymmetry and the mean-fieldϕ4 model. The results are compared with numerical simulations of theN-bodydynamics. When the inhomogeneous state is a stationary stable one, we expect to observe in theN-body dynamics quasi-stationary states, whose lifetime diverges algebraically withN.