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

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 overdamped and underdamped media. Via the well known stochastic approaches to underdamped and overdamped media, the thermodynamic expressions are first derived at a trajectory level then generalized to an ensemble level. To study the nonequilibrium 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 $S\left(t\right)$ 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. Furthermore, considering a spatially varying viscosity, the nonequilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. In this case, the direction of the particle velocity is dictated by the magnitude of the external load of $f$. Far from the stall load, ${\dot{e}}_p={\dot{h}}_d>0$ and at stall force ${\dot{e}}_p={\dot{h}}_d=0$ revealing the system is reversible at this particular choice of parameter. In the absence of load, ${\dot{e}}_p={\dot{h}}_d>0$ as long as a distinct temperature difference is retained between the hot and cold baths. Moreover, considering a multiplicative noise, we explore the thermodynamic features of the model system.

Figure 1

(a) The entropy extraction rate ${\dot{h}}_d\left(t\right)$ as a function of $t$ evaluated analytically by substituting Eqs. (52) and (54) into Eq. (57). (b) The plot of entropy production rate ${\dot{e}}_p\left(t\right)$ as a function of $t$. ${\dot{e}}_p\left(t\right)$ is analyzed analytically by substituting Eqs. (52) and (54) into Eq. (56). The two figures exhibit that ${\dot{e}}_p\left(t\right)$ and ${\dot{h}}_d\left(t\right)$ decrease in time and as time further steps up, ${\dot{e}}_p\left(t\right)$ and ${\dot{h}}_d\left(t\right)$ increase. In both figures, the parameters are fixed as $f=1.0$, $\tau =1.0$, and $z=-0.5$.

Figure 2

(a) ${\dot{h}}_d\left(t\right)$ as a function of $t$ evaluated analytically by substituting Eqs. (52) and (54) into Eq. (57). (b) ${\dot{e}}_p\left(t\right)$ as a function of $t$. ${\dot{e}}_p\left(t\right)$ is analyzed analytically by substituting Eqs. (52) and (54) into Eq. (56). The figure exhibits that ${\dot{e}}_p\left(t\right)$ and ${\dot{h}}_d\left(t\right)$ decrease in time and as time further steps up, ${\dot{e}}_p\left(t\right)$ and ${\dot{h}}_d\left(t\right)$ approach a constant value. In both figures, the parameters are fixed as $f=1.0$, $\tau =1.0$, and $z=1.0$.

Figure 3

Schematic diagram for a particle that walks in a piecewise linear potential in the absence of an external load.

Figure 4

(a) The dependence of $J$ on $U_0$ for fixed $\tau =2.0$, ${\gamma }^{\prime }=1$, and $f=0.5$. The parameter $C$ is also fixed as 0.4 (solid line), 0.2 (dashed line), and 0.04 (dotted line). (b) The plot $J$ as a function of $f$ for parameter choice $U_0=2.0$ and $\tau =2.0$. The parameter $C$ is fixed as 0.4 (solid line), 0.2 (dashed line), and 0.04 (dotted line).

Figure 5

(a) The plot for ${\dot{e}}_p\left(t\right)$ and ${\dot{h}}_d\left(t\right)$ as a function of $U_0$ for parameter choice of $\tau =2.0$, $C=0.04$, and $f=0.5$. (b) The plot ${\dot{e}}_p\left(t\right)$ and ${\dot{h}}_d\left(t\right)$ as a function of $f$ for parameter choice of $U_0=2.0$, $C=0.04$, and $\tau =2.0$.

Figure 6

The plot ${\dot{h}}_d\left(t\right)$ and ${\dot{e}}_p\left(t\right)$ as a function of $t$ for parameter choice $\tau =1$, $D=1.0$, and $z=-4.0$ is depicted in (a) and (b), respectively. The figures depict that ${\dot{h}}_d\left(t\right)$ and ${\dot{e}}_p\left(t\right)$ decrease as time increases and in long time limit it approaches its stationary value ${\dot{h}}_d\left(t\right)={\dot{e}}_p\left(t\right)=0$.