Microscopic effects in the energy currents of interacting systems

We derive an expression for the rate of change of the energy of an interacting many-body system connected to macroscopic reservoirs. We show that the energy variation is the sum of different contributions each coming from the single terminals. We show that unlike in the case of particle currents, where the rate of change of particles in the system is the opposite of the one in the reservoirs, each of the terms that appears in our expression can differ from the rate of change of the energy in the terminal. The difference between these two currents is due to the direct exchange of energy among two of the terminals and the microscopic mechanism behind it is the coupling of the latter through virtual processes involving the central interacting region.