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We extend certain basic and general concepts of thermodynamics to discreteMarkov systems exchanging work and heat with reservoirs. In this framework weshow that the celebrated Clausius inequality can be generalized and becomes anequality, significantly extending several recent results. We further show thatachieving zero dissipation in a system implies that detailed balance obtains,and as a consequence there is zero power production. We obtain inequalities forpower production under more general circumstances and show that nearequilibrium obtaining maximum power production requires dissipation to be ofthe same order of magnitude.

Generalized Clausius relation and power dissipation in non equilibrium stochastic systems