Testing entropy production hypotheses in non-linear steady states

In the last few decades, some hypotheses for entropy production (EP) principles have been forwarded as possible candidates for organizational principles in non-linear nonequilibrium systems. Two important hypotheses will be studied: the maximum entropy production (MaxEP) principle that claims that the selected steady state has the highest EP, and the gradient response principle that claims that the EP of the selected steady state (maximally) increases when the external thermodynamic driving force increases. We will formulate these hypotheses more rigorously and present a simple chemical reaction model to test these hypotheses. With the help of this model, we will clearly demonstrate that there are different MaxEP hypotheses being discussed in the literature and we will look at some parts in the literature where these differences are not always clarified. Furthermore, our chemical model will be a general counter example to all of these MaxEP and gradient response hypotheses. pacs numbers: 05.70.Ln, 65.40.Gr, 82.60.-s