A Quantum Violation of the Second Law?

An apparent violation of the second law of thermodynamics occurs when an atom coupled to a zero-temperature bath, being necessarily in an excited state, is used to extract work from the bath. Here the fallacy is that it takes work to couple the atom to the bath and this work must exceed that obtained from the atom. For the example of an oscillator coupled to a bath described by the single relaxation time model, the mean oscillator energy and the minimum work required to couple the oscillator to the bath are both calculated explicitly and in closed form. It is shown that the minimum work always exceeds the mean oscillator energy, so there is no violation of the second law.

Figure 1

Three-dimensional plot of $⟨H_O{⟩}_{T=0}/E_0$ as a function of $\gamma /{\omega }_0$ and $\Omega /{\omega }_0$.

Figure 2

Plot of $F_O(0)/E_0$ and $⟨H_O{⟩}_{T=0}/E_0$ as functions of the coupling strength as measured by $\gamma /{\omega }_0$. The relaxation time is chosen so that $\Omega =5{\omega }_0$. The overdamped oscillator corresponds to $\gamma /{\omega }_0>2$.