we study tuples $(x_1,\dots,x_d)$ of self-adjoint operators in a tracial $w^*$-algebra whose non-commutative distribution is the free gibbs law for a (sufficiently regular) convex potential $v$. such tuples model the large $n$ behavior of random matrices $(x_1^{(n)}, \dots, x_d^{(n)})$ chosen according to the measure $e^{-n^2 v(x)}\,dx$ on $m_n(\mathbb{c})_{sa}^d$. previous work showed that $w^*(x_1,\dots,x_d)$ is isomorphic to the free group factor $l(\mathbb{f}_d)$. in a recent preprint, we showed that an isomorphism $\phi: w^*(x_1,\dots,x_d)$ can be chosen so that $w^*(x_1,\dots,x_k)$ is mapped to the canonical copy of $l(\mathbb{f}_k)$ inside $l(\mathbb{f}_d)$ for each $k$. the idea behind the proof is to apply pde methods for constructing transport to gaussian to the conditional density of $x_j^{(n)}$ given $x_1^{(n)}, \dots, x_{j-1}^{(n)}$. then we analyze the asymptotic behavior of these transport maps as $n \to \infty$ using a new type of functional calculus, which applies certain $\|\cdot\|_2$-continuous functions to tuples of self-adjoint operators to self-adjoint tuples in (connes-embeddable) tracial $w^*$-algebras.