*(English)*Zbl 0928.60092

The authors consider mean-field measures ${\mu}^{\left(N\right)}$ with interacting potential $F$. The probability measure ${\mu}^{\left(N\right)}$ is the law of an interacting system of size $N$. The authors are interested in the asymptotic independence properties of these measures, as $N$ tends to infinity. This asymptotic behavior is called asymptotic chaoticity. Under mild assumptions, one proves that the empirical measures of the system converge in law to a deterministic measure ${\mu}^{*}$, and equivalently for any finite $k$, the convergence of the law of $k$-subsystem under ${\mu}^{\left(N\right)}$ to ${\left({\mu}^{*}\right)}^{\otimes k}$.

The authors estimate the relative entropy distance between ${\mu}^{\left(N\right)}$ and appropriate simpler exchangeable measures ${\nu}^{N}$ which are related to the law ${\mu}^{*}$. They prove that if the minimum of the function $H(\xb7\mid \mu )-F(\xb7)$ is unique, say ${\mu}^{*}$, then ${\nu}^{N}={\left({\mu}^{*}\right)}^{\otimes N}$, and propagation of chaos holds for blocks of size $o\left(N\right)$ if this minimum is nondegenerate. Certain degenerate situations are also studied. Next these results are applied to the Langevin dynamics of interacting particles leading to a McKean-Vlasov limit.