  The retainability of canonical distributions for a Brownian particle
controlled by a time-dependent harmonic potential is investigated in the
overdamped and underdamped situations, respectively. Because of different time
scales, the overdamped and underdamped Langevin equations (as well as the
corresponding Fokker-Planck equations) lead to distinctive restrictions on
protocols maintaining canonical distributions. Two special cases are analyzed
in details: First, a Brownian particle is controlled by a time-dependent
harmonic potential and embedded in medium with constant temperature; Second, a
Brownian particle is controlled by a time-dependent harmonic potential and
embedded in a medium whose temperature is tuned together with the potential
stiffness to keep a constant effective temperature of the Brownian particle. We
find that the canonical distributions are usually retainable for both the
overdamped and underdamped situations in the former case. However, the
canonical distributions are retainable merely for the overdamped situation in
the latter case. We also investigate general time-dependent potentials beyond
the harmonic form and find that the retainability of canonical distributions
depends sensitively on the specific form of potentials.
