Gibbs statement concerning equilibrium

Gibbs based his entire treatment of heterogenous equilibrium on the following statement [3]. 

The criterion of equilibrium of a material system which is isolated from all external influences may be expressed in either of the following entirely equivalent forms  

For equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the variations of its entropy shall either vanish or be negative. 

We find in Section 5.3 that the equality sign is applicable when all variations of the system can be carried out in either direction. When any variation of the system can be carried out in only one direction, the inequality sign is applicable. 

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The aim of thermodynamics is to express functions of state such as Gibbs energy (free enthalpy) in terms of state variables and to obtain relevant information on the equilibrium state. Both the first and second laws make statements concerning the variation of a particular extensive state function of a given system with regard to... 

This is an alternative but more convenient form of the more familiar statement that at equilibrium the total Gibbs free energy of a system will be a minimum. (If you are unfamiliar with the former approach Box 1.2 may convince you of its equivalence to the latter.) We can relate chemical potentials to the partial pressures, P and Pq, of the gases concerned if we assume them to be ideal ... 

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