free semicircular family, free from the algebra $ B=W^{*}({X_{1},\dots ,X_{n}}) $. Assume that $ B $ is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor. Then for all $ t>0 $, we have $$\chi (X_{1}+\sqrt{t}S_{1},\dots ,X_{n}+\sqrt{t}S_{n})\geq n\log 2\pi et>-\infty .$$ The same estimate holds for $ \chi ^{\omega } $.

By the change of variables formula and Theorem \[thrm: whenfreefromB\], we have $$\begin{aligned}
\chi (X_{1}+\sqrt{t}S_{1},\dots ,X_{n}+\sqrt{t}S_{n})\geq \chi (X_{1}+\sqrt{t}S_{1},\dots ,X_{n}+\sqrt{t}S_{n}|B)= &  & \\
\chi (\sqrt{t}S_{1},\dots ,\sqrt{t}S_{n})=\frac{n}{2}\log 2\pi et. & \end{aligned}$$

Let $ P_{1},\dots ,P_{n} $ be non-commutative polynomials in $ n $ variables with coefficients from $ B $. Assume that $ \chi ({X_{1},\dots ,X_{n}}|B)>-\infty  $. Then $$\begin{aligned}
\frac{d}{d\epsilon }\chi (X_{1}+\epsilon P_{1}(X_{1},\dots ,X_{n}),\dots ,X_{n}+\epsilon P_{n}({X_{1},\dots ,X_{n}})|B)= &  & \\
\sum _{i}\langle J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n}),P_{i}\rangle , &  & \end{aligned}$$ where $ J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n})) $ is the first-order conjugate variable to $ X_{i} $ with respect to $ B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n}) $ (cf. [@dvv:entropy5]). The same equality holds for $ \chi ^{\omega } $.

For $ \epsilon >0 $ sufficiently small, the transformation $ F^{\epsilon } $ defined by $$F_{i}^{\epsilon }(X_{1},\dots ,X_{n})=X_{i}+\epsilon P_{i}(X_{1},\dots ,X_{n})$$ is a non-commutative power series in $ {X_{1},\dots ,X_{n}}$ with coefficients from 