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

The expressions for entropy production, free energy, and entropy extraction rates are derived for a Brownian particle that walks in an underdamped medium. Our analysis indicates that as long as the system is driven out of equilibrium, it constantly produces entropy at the same time it extracts entropy out of the system. At steady state, the rate of entropy production ${\dot e}_{p}$ balances the rate of entropy extraction ${\dot h}_{d}$. At equilibrium both entropy production and extraction rates become zero. The entropy production and entropy extraction rates are also sensitive to time. As time progresses, both entropy production and extraction rates increase in time and saturate to constant values. Moreover employing microscopic stochastic approach, several thermodynamic relations for different model systems are explored analytically and via numerical simulations by considering a Brownian particle that moves in overdamped medium. Our analysis indicates that the results obtained for underdamped cases quantitatively agree with overdamped cases at steady state. The fluctuation theorem is also discussed in detailed.