Fluctuation relations and strong inequalities for thermally isolated systems

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.