Quantum evolution of charge and phase on a capacitor in a dissipative nanoscale circuit

Recently, the kinetic equation for the reduced density matrix that describes a capacitor coupled to a resistor was derived from a fundamental hamiltonian. In this work, a solution of the Markoffian Master equation that describes the capacitor is presented. A preliminary look at a solution that takes into account the nonMarkoffian terms is also included. The total solution reveals for the first time the quantum dynamical features of the RC circuit and the results are dramatic. The solution for the quasi probability distribution shows that even at zero temperature, the expectation value of the square of the phase, varies linearly with time, for times longer than the characteristic time constant of the coupled system. This yields a finite value for the diffusion constant at zero temperature. Using the formula it is now possible to answer a question that has been asked by researchers since 1905: is there a relationship between diffusion and mobility at absolute zero temperature.