Optimal work fluctuations for thermally isolated systems in weakly driven processes

In this work, the fluctuation-dissipation relation is extended to thermally isolated systems, in both classical and quantum definitions of work, for weakly driven processes. In the classical case, the optimal work variance is calculated, showing that it attains its minimal possible value in a so-called quasistatic variance, related to the difference between the quasistatic work and the system's Helmholtz free-energy difference. In the quantum case, the corresponding optimality condition involves higher cumulants and is expressed as a constraint on a weighted cumulant sum. The result is illustrated by the examples of the classical driven harmonic oscillator and the transverse-field quantum Ising chain. Along the way, a stronger, physically motivated definition of the arbitrary constant in the relaxation function of thermally isolated systems is derived for both the classical and quantum cases.