Analytical Insights and Universal Behavior in Fast Thermal Equilibration Protocols

When a system deviates from equilibrium, it is possible to manipulate and control it to drive it towards equilibrium within a finite time $t_f$, even reducing its natural relaxation time scale $\tau_{relax}$. Although numerous theoretical and experimental studies have explored these shortcut protocols, few have yielded analytical results for the probability distribution of work, heat and entropy production. In this paper, we propose a two-step protocol that captures the essential characteristics of more general protocols and has analytical solution for the relevant thermodynamic probability distributions. Additionally, we demonstrate that for very short protocol duration $t_f\ll \tau_{relax}$, all protocols exhibit universal behavior in their shape and for the ratio of probability distribution functions of positive and negative work, heat and the entropy production.