Topological term in the First Law of Thermodynamics*

We consider entropy and persistent currents induced by the Aharonov-Bohm effect in multiply-connected open quantum systems threaded by a magnetic flux at finite temperature. We prove a strong form of the Nernst theorem (third law of thermodynamics) for ``fully open'' quantum systems: the entropy goes strictly to zero as temperature approaches absolute zero. The conventional formula for the heat current is shown to be problematic for persistent currents, implying a divergent entropy current as temperature goes to zero, in contradiction to the third law. The apparent paradox is resolved through the inclusion of a topological work term in the first law corresponding to the ``persistent electrical work'' done in establishing the Aharonov-Bohm flux.