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The nonequilibrium thermodynamics feature of a Brownian motor is investigatedby obtaining exact time-dependent solutions. This in turn enables us toinvestigate not only the long time property (steady-state) but also the shorttime the behavior of the system. The general expressions for the free energy,entropy production ${\dot e}_{p}(t)$ as well as entropy extraction ${\doth}_{d}(t)$ rates are derived for a system that is genuinely driven out ofequilibrium by time-independent force as well as by spatially varying thermalbackground. We show that for a system that operates between hot and coldreservoirs, most of the thermodynamics quantities approach a non-equilibriumsteady state in the long time limit. The change in free energy becomes minimalat a steady state. However for a system that operates in a heat bath where itstemperature varies linearly in space, the entropy production and extractionrates approach a non-equilibrium steady state while the change in free energyvaries linearly in space. This reveals that unlike systems at equilibrium, whensystems are driven out of equilibrium, their free energy may not be minimized.The thermodynamic properties of a system that operates between the hot and coldbaths are further compared and contrasted with a system that operates in a heatbath where its temperature varies linearly in space along with the reactioncoordinate. We show that the entropy, entropy production, and extraction ratesare considerably larger for linearly varying temperature case than a systemthat operates between the hot and cold baths revealing such systems areinherently irreversible. For both cases, in the presence of load or when adistinct temperature difference is retained, the entropy $S(t)$ monotonouslyincreases with time and saturates to a constant value as $t$ further steps up.

physical review. e

Exact time-dependent analytical solutions for entropy production rate  for a system that operates in a heat bath where its temperature varies  linearly in space