  The systematic expansion method of the solution of the Fokker-Planck equation
is developed by generalizing the formulation proposed in [J. Phys. A50, 325001
(2017)]. Using this method, we obtain a new formula to calculate the mean work
perturbatively which is applicable to systems with degeneracy in the
eigenvalues of the Fokker-Planck operator. This method enables us to study how
the geometrical symmetry affects thermodynamic description of a Brownian
particle. To illustrate the application of the derived theory, we consider the
Fokker-Planck equation with a two-dimensional harmonic potential. To
investigate the effect of symmetry of the potential, we study thermodynamic
properties in symmetric and asymmetric deformation processes of the potential:
the rotational symmetry of the harmonic potential is held in the former, but it
is broken in the latter. Optimized deformations in these processes are defined
by minimizing mean works. Comparing these optimized processes, we find that the
difference between the symmetric and asymmetric processes is maximized when the
deformation time of the potential is given by a critical time which is
characterized by the relaxation time of the Fokker-Planck equation. This
critical time in the mean work is smaller than that of the change of the mean
energy because of the hysteresis effect in the irreversible processes.
