  For processes during which a macroscopic system exchanges no heat with its
surroundings, the second law of thermodynamics places two lower bounds on the
amount of work performed on the system: a weak bound, expressed in terms of a
fixed-temperature free energy difference, $W \ge \Delta F_T$ , and a strong
bound, given by a fixed-entropy internal energy difference, $W \ge \Delta E_S$
. It is known that statistical inequalities related to the weak bound can be
obtained from the nonequilibrium work relation, $\langle\exp (-\beta W)\rangle
= \exp(-\beta\Delta F_T)$ . Here we derive an integral fluctuation relation
$\langle\exp(-\beta X) \rangle = 1 $ that is constructed specifically for
adiabatic processes, and we use this result to obtain inequalities related to
the strong bound, $W \ge \Delta E_S$ . We provide both classical and quantum
derivations of these results.
