Fluctuation theorem for time-averaged work

There is evidence that taking the time average of the work performed by a thermally isolated system effectively "transforms" the adiabatic process into an isothermal one. This approach allows inherent quantities of adiabatic processes to be accessed through the definitions of isothermal processes. A fluctuation theorem is then established, linking the time-averaged work to the quasistatic work. Numerical evidence supporting this equality is provided for a classical harmonic oscillator with a driven linear equilibrium position parameter. Furthermore, the strong inequality for the averaged work is derived from the deduced fluctuation theorem using optimality arguments.