Rings and Coulomb boxes in dissipative environments

We study a particle on a ring in the presence of a dissipative Caldeira-Leggett environment and derive its response to a dc field. We show how this non-equilibrium response is related to a flux averaged equilibrium response. We find, through a two-loop renormalization group analysis, that a large dissipation parameter η flows to a fixed point ηR=/(2π). 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 η. For finite η>ηR we find that a certain average of the relaxation resistance is quantized. We propose a Coulomb-box experiment to measure a quantized noise.