Complexity Thermodynamics, Equiprobability Principle, Percolation, and Goldstein's Conjectures

The configurational states as introduced by Goldstein represent the system's basins and are characterized by their free energies $\varphi(T,V)$ as we show here. We find that the energies of some of the special points (termed basin identifiers here) like the basin minima, maxima, lowest energy barriers, etc. cannot be used to characterize the configurational states of the system in all cases due to their possible non-monotonic behavior as we explain. The complexity $\mathcal{S(}\varphi,T,V\mathcal{)},$ represents the configurational state entropy. We prove that $S(T,V) \equiv S(\varphi_b,T,V) + S_b(T,V)$, where $S_{\text{b}},$ and $\varphi_{\text{b}}$ are the basin entropy and free energy, respectively. We further prove that all basins at equilibrium have the same equilibrium basin energy $E(T,V)$ and entropy $S_{\text{b}% }(T,V).$ Here, $\varphi$\ and $E$ are measured with respect to the zero of the potential energy. The Boltzmann equiprobability principle is shown to apply to the basins in that each equilibrium basin has an equal probability $\mathcal{P=}\exp(-\mathcal{S})$ to be explored. This principle allow us to draw some useful conclusions about the time-dependence in the system. We discuss the percolation due to basin connectivity and its relevance for the dynamic transition. Our analysis validates modified Goldstein's conjectures. All the above results are shown to be valid at all temperatures, and not just low temperatures as originally propsed by Goldstein.