Quantum Analysis and Nonequilibrium Response

The quantum derivatives of $e^{-A}, A^{-1}$ and $\log A$, which play a basic role in quantum statistical physics, are derived and their convergence is proven for an unbounded positive operator $A$ in a Hilbert space. Using the quantum analysis based on these quantum derivatives, a basic equation for the entropy operator in nonequilibrium systems is derived, and Zubarev's theory is extended to infinite order with respect to a perturbation. Using the first-order term of this general perturbational expansion of the entropy operator, Kubo's linear response is rederived and expressed in terms of the inner derivation $\delta_{{\cal H}}$ for the relevant Hamiltonian ${\cal H}$. Some remarks on the conductivity $\sigma (\omega)$ are given.