Let $X_0$ be some vector on the unit sphere of $\mathbb{R}^n$ and let $\rho_0$ be the distribution of its entries, $$ \rho_0(z) = \frac{1}{n} \sum_i \delta\left(z-X_0^{i}\right) $$ Assume that $n$ is large enough and that $X_0$ is not localized enugh, so that we can approximate $\rho_0$ as a continuous function.
Now define a random walk on the unit sphere starting at $X_0$ and such that $$ \mathrm{d}X_t = \frac{1}{\sqrt{n}}\mathrm{d}A_t X_t, $$ where $A_t$ is a Brownian motion on the space of skew-symmetric matrices. It is known that the law of $X_t$ converges to the uniform measure on the sphere, which implies that $\rho_t$ goes to a normal distribution $\mathcal{N}(0,1/\sqrt{n})$ (a precise quantification would require dealing with the $n\rightarrow \infty$ but I'm not bothered by that). I'm interested in measuring how close $\rho_t$ is to $\mathcal{N}(0,1/\sqrt{n})$ using differential entropy, $$ \mathcal{H}[\rho_t] = - \int \mathrm{d}x \rho_t(x) \ln \rho_t(x). $$ I expect $\mathcal{H}$ to increase until it saturates around the value of the normal distribution.
Is there a way to quantify the previous statement in the form of an evolution equation for $\mathcal{H}$ (possibly depending on $\rho_t$) ?