Exact analytical expressions for entropy production and free energy

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.