Minimal dissipation protocols of an instantaneous equilibrium Brownian particle under time-dependent temperature and potential variations

We consider the instantaneous equilibrium (ieq) transition of an underdamped Brownian particle under arbitrary time-dependent temperature and potential variations derive analytic results for protocols minimal dissipative work. Explicit dissipation associated energetics are obtained cases pure variation, parameter (stiffness) isentropic ieq processes. The condition enforces forward backward to be time-reversal related, with same Remarkably, it is shown that path also transition. in mdieq transitions analyzed detail emphasis on conditions maximizing mean work power can extracted. Furthermore, exact overdamped limit derived, indicating there great freedom choose protocols, thus allowing realization colloidal systems relative ease.