Finite unitary dynamics and stationary Gibbs states

Let $\mathcal{H}$ be a complex Hilbert space of finite dimension $d\ge 1$, let $H=H^{\ast }$ be a Hamiltonian with energy unit J, and let $\mathrm{\hslash }>0$ have unit J s. A finite density matrix is a positive semidefinite matrix $\rho$ with $\mathrm{Tr}\rho =1$. For time $t\in \mathbb{R}$ measured in s, set $U(t)=e^{-itH\mathrm{/}\mathrm{\hslash }}$ and ${\alpha }_t(A)=U(t{)}^{\ast }AU(t)$. Then $U(t)$ is a unitary group, ${\alpha }_t$ is a group of matrix $\ast$-automorphisms, and for every finite density matrix $\rho$ and matrix $A$,

$\mathrm{Tr}(U(t)\rho U(t{)}^{\ast }A)=\mathrm{Tr}(\rho {\alpha }_t(A))\mathrm{.}$

The von Neumann entropy $S(\rho )=-k_B\mathrm{Tr}(\rho \log \rho )$ is defined using $\log \rho$ on the positive spectral support, with zero eigenvalues contributing $0$ (the convention $0\log 0:=0$), and is finite and unchanged by $U(t)\rho U(t{)}^{\ast }$. For $\beta >0$ with unit J${}^{-1}$, let $Z_{\beta }=\mathrm{Tr}(e^{-\beta H})$ and ${\sigma }_{\beta }=Z_{\beta }^{-1}{e}^{-\beta H}$. Then $Z_{\beta }>0$, ${\sigma }_{\beta }$ is stationary, and the finite Gibbs correlation has the KMS strip identity

$F_{A,B}(t+i\beta \mathrm{\hslash })=\mathrm{Tr}({\sigma }_{\beta }{\alpha }_t(B)A),F_{A,B}(z)=\mathrm{Tr}({\sigma }_{\beta }A{\alpha }_z(B)),$

where ${\alpha }_z(B)=e^{izH\mathrm{/}\mathrm{\hslash }}B{e}^{-izH\mathrm{/}\mathrm{\hslash }}$. The function $F_{A,B}$ is entire and bounded on $0\le \mathrm{Im}z\le \beta \mathrm{\hslash }$. This finite-dimensional result asserts no approach to equilibrium.