  We discuss the thermodynamics of closed quantum systems driven out of
equilibrium by a change in a control parameter and undergoing a unitary
process. We compare the work actually done on the system with the one that
would be performed along ideal adiabatic and isothermal transformations. The
comparison with the latter leads to the introduction of irreversible work,
while that with the former leads to the introduction of inner friction. We show
that these two quantities can be treated on equal footing, as both can be
linked with the heat exchanged in thermalization processes and both can be
expressed as relative entropies. Furthermore, we show that a specific
fluctuation relation for the entropy production associated with the inner
friction exists, which allows the inner friction to be written in terms of its
cumulants.
