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The expressions for entropy production, free energy, and entropy extractionrates 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 thesystem. At steady state, the rate of entropy production ${\dot e}_{p}$ balancesthe rate of entropy extraction ${\dot h}_{d}$. At equilibrium both entropyproduction and extraction rates become zero. The entropy production and entropyextraction rates are also sensitive to time. As time progresses, both entropyproduction and extraction rates increase in time and saturate to constantvalues. Moreover employing microscopic stochastic approach, severalthermodynamic relations for different model systems are explored analyticallyand via numerical simulations by considering a Brownian particle that moves inoverdamped medium. Our analysis indicates that the results obtained forunderdamped cases quantitatively agree with overdamped cases at steady state.The fluctuation theorem is also discussed in detailed.

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