Modified Jarzynski equality in a microcanonical ensemble

We show that the conventional Jarzynski equality does not hold for a system prepared in a microcanonical ensemble. We derive a modified equality that connects microcanonical work fluctuations to entropy production, in an analogous way to the Jarzynski equality, but with reference to an inverse temperature that depends on the path of the work protocol. For close to isothermal processes the modified equality can improve on the bound $\langle W\rangle\ge \Delta F$. Our result is a special case of a general expression for the microcanonical moment-generating function for any extensive quantity, which enables calculation of the breakdown of ensemble equivalence for thermodynamic fluctuations. We demonstrate our microcanonical Jarzynski equality in a system of driven two-level spins.