  The nonequilibrium thermodynamics feature of a Brownian motor operating
between two different heat baths is explored as a function of time $t$. Using
the Gibbs entropy and Schnakenberg microscopic stochastic approach, we find
exact closed form expressions for the free energy, the rate of entropy
production and the rate of entropy flow from the system to the outside. We show
that when the system is out of equilibrium, it constantly produces entropy and
at the same time extract entropy out of the system. Its entropy production and
extraction rates decrease in time and saturate to constant value. In long time
limit, the rate of entropy production balances the rate of entropy extraction
and at equilibrium both entropy production and extraction rates become zero.
Furthermore, via the present model, not only many thermodynamic theories can be
checked but also the wrong conclusions that are given due to lack of exact
analytic expressions will be corrected.
