  Given a closed quantum system, the states that can be reached with a cyclic
process are those with the same spectrum as the initial state. Here we prove
that, under a very general assumption on the Hamiltonian, the distribution of
the mean extractable work is very close to a gaussian with respect to the Haar
measure. We derive bounds for both the moments of the distribution of the mean
energy of the state and for its characteristic function, showing that the
discrepancy with the normal distribution is increasingly suppressed for large
dimensions of the system Hilbert space.
