2.2 Work Done by Voltage Sources

  To evaluate the available energy for a given tunnel event, the work done on the system by the power supplies has to be included, since thermodynamically the interacting islands represent an  open system. The work done by the voltage sources may be written as the time integral over the power delivered to the system.
 $$\begin{gather} W = \sum_{\text{sources}}\int V(t)I(t)\,dt \end{gather}$$
Following any tunnel event, charges flow to and from the contacts until equilibrium is established. The transferred charge includes the electron which tunnels into or out of an island as well as the continuous   polarization charge that builds up in response to the change of electrostatic potential on islands. It is assumed that the duration of this  charge relaxation caused by tunneling or changing voltage sources is much shorter than the time between two tunnel events. Voltage sources are considered to be ideal, that is their internal resistance is zero. Then, for constant voltage sources, the change in work may be written as
$$\begin{gather}\Delta W=\pm eV+\sum_i V_i\Delta q_i, \end{gather}$$
where the first term is the contribution from the tunneled electron and the second term is the sum of the polarization work.