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≫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 . Qualitatively the 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 not generally apply, since such a system —starting from the canonical equilibrium density matrix ρ_T at the bath temperature T— can enhance the work exracted from the microcanonical bath without changing its state ρ_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 microcanonical free-energy and show that in contrast to the canonical case it contains a linear entropy instead of the von Neumann entropy.