Quantum Statistical Thermodynamics of Two-Level Systems

TL;DR

This paper analyzes Gibbs distributions for two-level systems across different mathematical frameworks, revealing relationships between average polarization and temperature, and proposing an alternative to the classical Brillouin function.

Contribution

It introduces a unified analysis of Gibbs distributions for various two-level systems and proposes an alternative polarization function satisfying specific probabilistic criteria.

Findings

Derived relationships between average polarization and temperature parameters.

Identified a new alternative to the Brillouin function for the standard complex case.

Established structure functions for different two-level systems.

Abstract

We study four distinct families of Gibbs canonical distributions defined on the standard complex, quaternionic, real and classical (nonquantum) two-level systems. The structure function or density of states for any two-level system is a simple power (1, 3, 0 or -1) of the length of its polarization vector, while the magnitude of the energy of the system, in all four cases, is the negative of the logarithm of the determinant of the corresponding two-dimensional density matrix. Functional relationships (proportional to ratios of gamma functions) are found between the average polarizations with respect to the Gibbs distributions and the effective polarization temperature parameters. In the standard complex case, this yields an interesting alternative, meeting certain probabilistic requirements recently set forth by Lavenda, to the more conventional (hyperbolic tangent) Brillouin function of paramagnetism (which, Lavenda argues, fails to meet such specifications).