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.