Quantum Analysis and Nonequilibrium Response

TL;DR

This paper develops quantum derivatives for key operators in quantum statistical physics, extends Zubarev's nonequilibrium theory to infinite order, and rederives Kubo's linear response using quantum analysis.

Contribution

It introduces quantum derivatives for unbounded operators, extends nonequilibrium entropy operator theory, and provides a new derivation of linear response in quantum systems.

Findings

Quantum derivatives are proven to converge for unbounded positive operators.

The entropy operator's basic equation is derived for nonequilibrium quantum systems.

Kubo's linear response is rederived using quantum analysis and inner derivation.

Abstract

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.