Behavior of entropy under convolutions of random walk operator on a finitely generated group?

Question

Let $G$ be a finitely generated group with generating set $S=\{s_1,s_2,....,s_n\}$. Let $\mu$ be a uniform probability distribution on $S$. We can define a random walk $X_n$ on $G$ by declaring that $\mu$ is the law/step distribution for the random walk. Then the n'th convolution $\mu^{*n}$ is the distribution for $X_n$ (this is the simple random walk on $G$ for $\mu$ uniform). For any distribution $\mu$ on $G$, we can define its entropy as: $H(\mu)=\sum_{x\in G}\mu(x)log(\mu(x))$. I'm wondering if anyone knows anything about how $H(\mu^{*n})$ behaves? I'm most interested in equalities/inequalities relating $\mu^{*n}$ to $\mu$ or $\mu^{*k}$, $k\neq n$.