  We discuss the systematic expansion of the solution of the Fokker-Planck
equation with the help of the eigenfunctions of the time-dependent
Fokker-Planck operator. The expansion parameter is the time derivative of the
external parameter which controls the form of an external potential. Our
expansion corresponds to the perturbative calculation of the adiabatic motion
in quantum mechanics. With this method, we derive a new formula to calculate
the irreversible work order by order, which is expressed as the expectation
value with a pseudo density matrix. Applying this method to the case of the
harmonic potential, we show that the first order term of the expansion gives
the exact result. Because we do not need to solve the coupled differential
equations of moments, our method simplifies the calculations of various
functions such as the fluctuation of the irreversible work per unit time. We
further investigate the exact optimized protocol to minimize the irreversible
work by calculating its variation with respect to the control parameter itself.
