  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.

Statistical Mechanics