Scaling Behaviors of Work Cumulants in Slow Isothermal Processes

We study the cumulants of work in a slow isothermal process for gapped systems. Using the Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) formalism and the properties of connected correlation functions, we show that in this process, the $n$-th cumulant of work scales as $1\mathrm{/}{T}^{n-1}$ , where $T$ is the time duration. This result holds generally for arbitrary smooth protocols. Furthermore, we derive the coefficients of the cumulants from equilibrium properties. These coefficients are found to be relevant to thermodynamic geometric tensors.