Fluctuation relations and strong inequalities for thermally isolated systems

Abstract

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, $W \ge ΔF_T$ , and a strong bound, given by a fixed-entropy internal energy difference, $W \ge ΔE_S$ . It is known that statistical inequalities related to the weak bound can be obtained from the nonequilibrium work relation, $\langle\exp (-βW)\rangle = \exp(-βΔF_T)$ . Here we derive an integral fluctuation relation $\langle\exp(-βX) \rangle = 1 $ that is constructed specifically for adiabatic processes, and we use this result to obtain inequalities related to the strong bound, $W \ge ΔE_S$ . We provide both classical and quantum derivations of these results.

Dedicated to the memory of Christian Van den Broeck