Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation

1. (Conventions)¶ The conventions of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation are in force. Rays such as $(0,\infty)$, integrability of a real function on a Borel subset $B$ of $\mathbb{R}$ and the integral $\int_{B}f(t)\,dt$ are those of Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §rays and Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §integral, and $(a,b)$ is an open interval; $\log$ is the natural logarithm, $e=\exp(1)$ for the exponential function, and $\pi$ is the number $\pi$.

2. (Free products)¶ For $m,n\in\mathbb{N}$, freeness of a law in $\Sigma_{m+n}$ is that of Freeness of Two Groups of Variables under a Noncommutative Law §free, and for $\alpha\in\Sigma_{m}$ and $\beta\in\Sigma_{n}$, $\alpha\star\beta$ is their free product. For $\alpha,\beta\in\Sigma_{d}$, $\alpha\boxplus\beta$ is their free additive convolution.

3. (Semicircular laws)¶ For $n\in\mathbb{N}$ and real $t\ge0$, $\mathrm{sc}_{n}$ and $\mathrm{sc}_{n,t}$ are the semicircular law and the semicircular law of variance $t$; the Schwinger-Dyson equations are those of The Schwinger-Dyson Equation for a Noncommutative Law.

4. (Free heat flow)¶ For real $t\ge0$, $\Gamma_{t}$ is the affine datum of the free heat flow $\lambda\mapsto\lambda\boxplus\mathrm{sc}_{d,t}$.

5. (Free Fisher information)¶ $\Phi^{*}$ is the free Fisher information; for $\lambda\in\Sigma_{d}$, $f_{\lambda}$ and $g_{\lambda}$ are the functions on $(0,\infty)$ of Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law §integrand.

6. (Free entropy)¶ $\mathcal{D}_{\chi}$ is the domain and $\chi^{*}$ the non-microstates free entropy.

7. (Background)¶ The following results are in force and may be used without restating them: The Chain Rule for Free Difference Quotients under an Affine Substitution, The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity, Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations, The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity, Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information, Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits, Logarithmic Integrals of Reciprocal Linear Functions over Bounded and Unbounded Intervals and The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability.