Analytical description of the quantum-mesoscopic-classical transition in systems with quasidiscrete environments

We investigate the dynamic properties of inhomogeneous nanomaterials, which appear in analytical descriptions typically as a series of δ functions with corresponding Gibbs weights. We focus on observables relevant for transport theories of Josephson junction arrays and granular systems near the superconductor-insulator transition. Furthermore, our description applies to the theory of tunnel junctions exchanging energy with a “bath,” the latter having a discrete spectrum. Using the matrix θ-function formalism, we find an analytical expression for the transport characteristics capturing the complete temperature-driven transition from the quantum to the classical regime.