  Considering viscous friction that varies spatially and temporally, the
general expressions for entropy production, free energy, and entropy extraction
rates are derived to a Brownian particle that walks in an overdamped and
underdamped media. Via the well known stochastic approaches to underdamped and
overdamped media, the thermodynamic expressions first derived at trajectory
level then generalized to an ensemble level. To study the non-equilibrium
thermodynamic features of a Brownian particle that hops in a medium where its
viscosity varies on time, a Brownian particle that walks on a periodic
isothermal medium (in the presence or absence of load) is considered. The exact
analytical results depict that in the absence of load $f=0$, the entropy
production rate ${\dot e}_{p}$ approaches the entropy extraction rate ${\dot
h}_{d}=0$. This is reasonable since any system which is in contact with a
uniform temperature should obey the detail balance condition in a long time
limit. In the presence of load and when the viscous friction decreases either
spatially or temporally, the entropy $S(t)$ monotonously increases with time
and saturates to a constant value as $t$ further steps up. The entropy
production rate ${\dot e}_{p}$ decreases in time and at steady state (in the
presence of load), ${\dot e}_{p}={\dot h}_{d}>0$. On the contrary, when the
viscous friction increases either spatially or temporally, the rate of entropy
production as well as the rate of entropy extraction monotonously steps up
showing that such systems are inherently irreversible.
