Thermodynamic paths and stochastic order in open systems

The theorem of extremum entropy generation is related to the stochastic order of the paths inside the phase space; indeed, the system evolves, from an indistinguishable configuration to another one, on the most probable path in relation to the paths stochastic order. The result is that, at the stationary state, the entropy generation is maximal and, this maximum value is a consequence of the stochastic order of the paths in the phase space. Conversely, the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation for the open systems at the stationary states.