Optimal work fluctuations for thermally isolated systems in weak processes

The fluctuation-dissipation relation for the classical definition of work is extended to thermally isolated systems, in classical and quantum realms. From this, the optimal work variance is calculated, showing it achieves its minimum possible value, independent of the rate of the process, in a so-called quasistatic variance, related to the difference between the quasistatic work and the difference of Helmholtz's free energy of the system. The result is corroborated by the example of the classical and driven harmonic oscillator, whose probability density function of the work distribution is non-Gaussian and constant for different rates. The optimal variance is calculated for the quantum Ising chain as well, showing its finiteness if the linear response validity criterion is complied with. A stronger definition of the arbitrary constant for the relaxation function of thermally isolated systems is obtained along the work.