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Considering viscous friction that varies spatially and temporally, thegeneral expressions for entropy production, free energy, and entropy extractionrates are derived to a Brownian particle that walks in an overdamped andunderdamped media. Via the well known stochastic approaches to underdamped andoverdamped media, the thermodynamic expressions first derived at trajectorylevel then generalized to an ensemble level. To study the non-equilibriumthermodynamic features of a Brownian particle that hops in a medium where itsviscosity varies on time, a Brownian particle that walks on a periodicisothermal medium (in the presence or absence of load) is considered. The exactanalytical results depict that in the absence of load $f=0$, the entropyproduction rate ${\dot e}_{p}$ approaches the entropy extraction rate ${\doth}_{d}=0$. This is reasonable since any system which is in contact with auniform temperature should obey the detail balance condition in a long timelimit. In the presence of load and when the viscous friction decreases eitherspatially or temporally, the entropy $S(t)$ monotonously increases with timeand saturates to a constant value as $t$ further steps up. The entropyproduction rate ${\dot e}_{p}$ decreases in time and at steady state (in thepresence of load), ${\dot e}_{p}={\dot h}_{d}>0$. On the contrary, when theviscous friction increases either spatially or temporally, the rate of entropyproduction as well as the rate of entropy extraction monotonously steps upshowing that such systems are inherently irreversible.

Effect of viscous friction on entropy, entropy production and entropy extraction rates in underdamped and overdamped media