Fixed-particle finite-temperature density-matrix free energy

Let $n\ge 2$, $r\in \{1,\dots ,n-1\}$, and $\beta >0$. The fixed-particle density-matrix domain is ${\mathcal{D}}_r:=\{D\in {\mathbb{H}}_n:0⪯D⪯I,\mathrm{tr}D=r\},$ where $0⪯D⪯I$ means that both $D$ and $I-D$ are positive semidefinite in the operator order (Non-negative and positive operators). Define the continuous scalar entropy integrand on $[0,1]$ by $\varphi (x):=x\log x+(1-x)\log (1-x),0\log 0:=0.$ For $D\in {\mathcal{D}}_r$, $\varphi (D)$ is defined by the spectral functional calculus (The spectral functional calculus f(T) for a normal endomorphism, For normal endomorphisms, the spectral functional calculus respects sums, products, adjoints, and composition of scalar functions). The logarithm is the natural logarithm on $(0,\mathrm{\infty })$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t). Supply a real $C^1$ energy $\mathcal{E}$ on an open neighborhood of ${\mathcal{D}}_r$ and its Hermitian Frobenius gradient $F_{\mathcal{E}}(D)$, defined by $D\mathcal{E}(D)[H]=⟨F_{\mathcal{E}}(D),H{⟩}_F$ for every Hermitian direction $H$. Define ${\mathcal{F}}_{\beta }(D):=\mathcal{E}(D)+{\beta }^{-1}\mathrm{tr}\varphi (D)\mathrm{.}$ The fixed-Hermitian-field case is $\mathcal{E}(D)=\mathrm{tr}(FD)$ for a fixed Hermitian $F$. For nonlinear self-consistent models, $\mathcal{E}$ and its gradient may depend on $D$; the definition alone asserts no uniqueness of a minimizing density.