Entropy production and entropy extraction rates for a Brownian particle that walks in underdamped medium

Exact time-dependent analytical solutions for entropy production rate in a system operating in a heat bath in which temperature varies linearly in space

The nonequilibrium thermodynamics feature of a Brownian motor is investigated by obtaining exact time-dependent solutions. This in turn enables us to investigate not only the long time property (steady state) but also the short time the behavior of the system. The general expressions for the free energy, entropy production e[over ̇]_{p}(t) as well as entropy extraction h[over ̇]_{d}(t) rates are derived for a system that is genuinely driven out of equilibrium by time-independent force as well as by spatially varying thermal background. We show that for a system that operates between hot and cold reservoirs, most of the thermodynamics quantities approach a nonequilibrium steady state in the long time limit. The change in free energy becomes minimal at a steady state. However, for a system that operates in a heat bath where its temperature varies linearly in space, the entropy production and extraction rates approach a nonequilibrium steady state while the change in free energy varies linearly in space. This reveals that unlike systems at equilibrium, when systems are driven out of equilibrium, their free energy may not be minimized. The thermodynamic properties of a system that operates between the hot and cold baths are further compared and contrasted with a system that operates in a heat bath where its temperature varies linearly in space along with the reaction coordinate. We show that the entropy, entropy production, and extraction rates are considerably larger for the linearly varying temperature case than a system that operates between the hot and cold baths revealing such systems are inherently irreversible. For both cases, in the presence of load or when a distinct temperature difference is retained, the entropy S(t) monotonously increases with time and saturates to a constant value as t further steps up. The entropy production rate e[over ̇]_{p} decreases in time and at steady state, e[over ̇]_{p}=h[over ̇]_{d}>0, which agrees with the results shown in M. Asfaw's [Phys. Rev. E 89, 012143 (2014)1539-375510.1103/PhysRevE.89.012143; Phys. Rev. E 92, 032126 (2015)10.1103/PhysRevE.92.032126]. Moreover, the velocity, as well as the efficiency of the system that operates between the hot and cold baths, are also collated and contrasted with a system that operates in a heat bath where its temperature varies linearly in space along with the reaction coordinate. A system that operates between the hot and cold baths has significantly lower velocity but a higher efficiency in comparison with a linearly varying temperature case.

Excess and loss of entropy production for different levels of coarse graining

We investigate the effect of coarse graining on the thermodynamic properties of a system, focusing on entropy production. As a case of study, we consider a one-dimensional colloidal particle in contact with a thermal bath, moving in a sinusoidal potential and driven out of equilibrium by a small constant force. Different levels of coarse graining are evaluated: At first, we compare the results in the underdamped dynamics with those in the overdamped one (first coarse graining). For large values of the friction coefficient, the two dynamics have the same thermodynamics properties, while, for smaller friction values, the overdamped approximation produces an excess of entropy production with respect to that of the underdamped dynamics. Moreover, for further smaller values of the drag coefficient, the excess of entropy production turns into a loss. These regimes are explained by evaluating the jump statistics, observing that the inertia is able to induce multiple jumps and affect the average jump rate. The periodic shape of the potential allows us to approximate the continuous dynamics via a Markov chain after the introduction of a suitable time and space discretization (second level of coarse graining). This discretization procedure is implemented starting both from the underdamped and the overdamped evolution and is analyzed for different values of the friction coefficient.

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 e[over ̇]_{p} approaches the entropy extraction rate h[over ̇]_{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 e[over ̇]_{p} decreases in time and at steady state (in the presence of load) e[over ̇]_{p}=h[over ̇]_{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, e[over ̇]_{p}=h[over ̇]_{d}>0 and at stall force e[over ̇]_{p}=h[over ̇]_{d}=0 revealing the system is reversible at this particular choice of parameter. In the absence of load, e[over ̇]_{p}=h[over ̇]_{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.