Work extraction from microcanonical bath

We determine the maximal work extractable via a cyclic Hamiltonian process from a positive-temperature ($T>0$) microcanonical state of a $N\gg 1$ spin bath. The work is much smaller than the total energy of the bath, but can be still much larger than the energy of a single bath spin, e.g. it can scale as ${\cal O}(\sqrt{N\ln N})$. Qualitatively same results are obtained for those cases, where the canonical state is unstable (e.g., due to a negative specific heat) and the microcanonical state is the only description of equilibrium. For a system coupled to a microcanonical bath the concept of free energy does {\it not generally} apply, since such a system|starting from the canonical equilibrium density matrix $\rho_T$ at the bath temperature $T$|can enhance the work extracted from the microcanonical bath without changing its state $\rho_T$. This is impossible for any system coupled to a canonical thermal bath due to the relation between the maximal work and free energy. But the concept of free energy still applies for a sufficiently large $T$. Here we find a compact expression for the {\it microcanonical free-energy} and show that in contrast to the canonical case it contains a {\it linear entropy} instead of the von Neumann entropy.