Dispersion of the conductance of quantum nanowires and Joule heating

The high-frequency ballistic conductance $G(\omega)$ of a quantum wire connecting two classical reservoirs is discussed. It is supposed that the transverse size of the wire is on the order of the de Broglie wavelength of the conduction electrons. An expression for $G(\omega)$ in a wide range of frequencies $\omega$ is given. The behavior of both active $\operatorname{Re}G(\omega)$ and reactive $\operatorname{Im}G(\omega)$ parts of the conductance is investigated. The frequency range is determined where the so-called kinetic inductance is dominant, i.e., $\operatorname{Im}G(\omega)$ is positive and larger than $\operatorname{Re}G(\omega)$. This range is defined by the condition that the time of flight of the conduction electrons along the wire length $L$ exceeds the period of oscillation $2\pi/\omega$ of the electric potential. The Joule heat generation that accompanies the current flow through the quantum wire takes place in the reservoirs over a distance on the order of the mean free path of conduction electrons. The total rates of Joule heat generation are the same in both reservoirs.