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.

中文翻译：

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粘滞摩擦对欠阻尼和过阻尼介质中熵，熵产生和熵提取速率的影响

考虑到随时间变化的粘滞摩擦，熵产生，自由能和熵提取速率的一般表达式可推导为在过阻尼和欠阻尼介质中行走的布朗粒子。通过对欠阻尼和过阻尼介质的众所周知的随机方法，首先在轨迹水平上导出热力学表达式，然后将其推广到整体水平。为了研究在其粘度随时间变化的介质中跳跃的布朗粒子的非平衡热力学特征，考虑了在周期性等温介质（有或无负荷）下行走的布朗粒子。精确的分析结果表明，在没有负载的情况下$F=0$，熵产生率 ${\dot{Ë}}_p$ 接近熵提取率 ${\dot{H}}_d=0$。这是合理的，因为任何与均匀温度接触的系统都应长期遵守细节平衡条件。在有载荷的情况下，当粘滞摩擦在空间或时间上减小时，熵$\mathrm{小号}（Ť）$ 随着时间单调增加并饱和到一个恒定值 $Ť$进一步加强。熵产率${\dot{Ë}}_p$ 在稳定状态下（在有负载的情况下）时间减少 ${\dot{Ë}}_p={\dot{H}}_d>0$。相反，当粘滞摩擦在空间或时间上增加时，熵产生速率和熵提取速率单调增加，表明这种系统固有地是不可逆的。此外，考虑到空间变化的粘度，研究了布朗粒子的非平衡热力学特征，该布朗粒子随负载跃变棘轮电位。在这种情况下，粒子速度的方向由外部载荷的大小决定。$F$。远离失速负载，${\dot{Ë}}_p={\dot{H}}_d>0$ 并在失速状态下 ${\dot{Ë}}_p={\dot{H}}_d=0$揭示在这种特定的参数选择下系统是可逆的。在没有负载的情况下${\dot{Ë}}_p={\dot{H}}_d>0$只要在热浴和冷浴之间保持明显的温差即可。此外，考虑到乘性噪声，我们探索了模型系统的热力学特征。