Number of Classes of Invariant Equilibrium States in Complex Thermodynamic Systems

The values of the Gibbs function of a system with C components create a 2？dimensional topological manifold that is piecewise smooth and continuous. Each of the C+2 smooth elements of such a manifold represents the states of a phase within the system. The elements are glued together along the C types of phase transformation lines, which converge to a single point that represents the invariant state of the system (i.e. a state with zero degrees of freedom). Transformation lines, treated as edges, and the smooth elements of the manifold, i.e. faces, constitute a zero-vertex graph that represents the invariant state. This graph is referred to here as the graph-map of the invariant state. The distribution of each component in an invariant state depends on the configuration (distribution) of the phase transformation lines. Because the smoothness and continuity of the manifold makes certain configurations of the lines forbidden, some forms of invariant states are also forbidden, even though they satisfy the Gibbs phase rule. Some academic handbooks do not take this fact into account, and provide forbidden configurations as examples of invariant states. States that only differ in terms of the permutation of two or more of their components will belong to the same class. This study shows that all real graph-maps can be represented by C-vertex graphs with C+2 edges that have an even value of the vertex valence. The number of such graphs, i.e. the number of classes of invariant states, ho(C), is shown to meet the recurrence relation ho(2k+1) = 2*ho(2k) - ho(2k-1), where k = 1, 2, 3, 4. Knowing the number ho(C) for several small values of C allows us to determine the number of invariant states in a thermodynamic system using the above equation, regardless of the complexity of the system