Chapitre III. Les projets d’établissement : peupler l’Empire (c.1760-c.1765)

Out-of-equilibrium phase transitions in the Hamiltonian mean-field model: A closer look

“…Therefore, the Lynden-Bell theory remains a valuable tool even if it is difficult to specify its general domain of validity. In fact, Mouhot & Villani [87] do not totally reject this statistical approach and point out limitations in the application of their results. In particular, their theory is based on smooth functions (which is not the norm in statistical theories) and Landau damping is a thin effect which might be neglected when it comes to predict the "final" state in a "turbulent" situation (which is precisely the aim of Lynden-Bell's statistical theory).…”

“…Very recently, a mathematical "tour de force" has been accomplished by Mouhot & Villani [87] who rigorously proved that systems with long-range interactions described by the Vlasov equation possess some asymptotic "stabilization" property in large time, although the Vlasov equation is time-reversible. More precisely, they show that if a stable steady state of the Vlasov equation is slightly perturbed, the perturbation converges in a weak sense towards a steady distribution through phase mixing without the help of any extra diffusion or ensemble averaging.…”