Thermodynamics of discrete-charge quantum circuits

We study a dilute gas made of quantum circuits with discrete charge, interacting with a thermal reservoir at absolute temperature T, using Boltzmann statistics. This system is described by a quantum Hamiltonian that explicitly includes the effect of the discrete nature of the electric charge within a circuit approach. The eigenfunctions can be expressed as solutions of the Mathieu equation, and the energy spectrum is related to its characteristic values. We also make a comparative numerical study of the thermal properties, between the quantum case with discrete charge, and the so-called semiclassical approximation.