Derivation of the density operator with quantum analysis for the generalized Gibbs ensemble in quantum statistics

TL;DR

This paper derives the density operator equations for generalized entropy and expectation values in quantum systems, extending the canonical ensemble argument to generalized Gibbs ensembles under certain conditions.

Contribution

It provides a unified derivation of density operators for various entropies and expectation values, extending the canonical ensemble framework to generalized Gibbs ensembles.

Findings

Derived density operators for von Neumann, Tsallis, and Rényi entropies.

Extended the argument of the canonical ensemble to generalized Gibbs ensembles.

Simplified equations under commutation relations and conventional expectation values.

Abstract

We derived the equation of the density operator for generalized entropy and generalized expectation value with quantum analysis when conserved quantities exist. The derived equation is simplified when the conventional expectation value is employed. The derived equation is also simplified when the commutation relations, $[\hat{\rho}, \hat{H}]$ and $[\hat{\rho}, \hat{Q}^{[a]}]$, are the functions of the density operator $\hat{\rho}$, where $\hat{H}$ is the Hamiltonian, and $\hat{Q}^{[a]}$ is the conserved quantity. We derived the density operators for the von Neumann entropy, the Tsallis entropy, and the R\'enyi entropy in the case of the conventional expectation value. We also derived the density operators for the Tsallis entropy and the R\'enyi entropy in the case of the escort average (the normalized $q$-expectation value), when the density operator commutes with the Hamiltonian and the conserved quantities. We found that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in the case of the conventional expectation value, even when conserved quantities do not commute. The simple extension of the argument is also shown in the case of the escort average, when the density operator $\hat{\rho}$ commutes with the Hamiltonian $\hat{H}$ and the conserved quantity $\hat{Q}^{[a]}$: $[\hat{\rho}, \hat{H}] = [\hat{\rho}, \hat{Q}^{[a]}]=0$. These findings imply that the argument of the density operator for the canonical ensemble is simply extended to the argument for the generalized Gibbs ensemble in some systems.