  We present an alternative approach to the theory of free Gibbs states with
convex potentials. Instead of solving SDE's, we combine PDE techniques with a
notion of asymptotic approximability by trace polynomials for a sequence of
functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$
is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly
convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$
is asymptotically approximable by trace polynomials. Then the moments of
$\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free
entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$
agree and equal the limit of the normalized classical entropies of $\mu_N$.
