Rings and boxes in dissipative environments

We study a particle on a ring in presence of a dissipative Caldeira-Leggett environment and derive its response to a DC field. We find, through a 2-loop renormalization group analysis, that a large dissipation parameter $\eta$ flows to a fixed point ${\eta }_R={\eta }_c=\mathrm{\hslash }\mathrm{/}2\pi$. We also reexamine the mapping of this problem to that of the Coulomb box and show that the relaxation resistance, of recent interest, is quantized for large $\eta$. For finite $\eta >{\eta }_c$ we find that a certain average of the relaxation resistance is quantized. We propose a box experiment to measure a quantized noise.