Thermodynamic Tree: The Space of Admissible Paths

Is a spontaneous transition from a state x to a state y allowed by thermodynamics? Such a question arises often in chemical thermodynamics and kinetics. We ask the more formal question: is there a continuous path between these states, along which the conservation laws hold, the concentrations remain non-negative and the relevant thermodynamic potential G (Gibbs energy, for example) monotonically decreases? The obvious necessary condition, G(x) ≥ G(y), is not sufficient, and we construct the necessary and sufficient conditions. For example, it is impossible to overstep the equilibrium in 1-dimensional (1D) systems (with n components and n−1 conservation laws). The system cannot come from a state x to a state y if they are on the opposite sides of the equilibrium even if G(x) > G(y). We find the general multidimensional analogue of this 1D rule and constructively solve the problem of the thermodynamically admissible transitions. We study dynamical systems, which are given in a positively invariant convex polyhedron and have a convex Lyapunov function G. An admissible path is a continuous curve along which G does not increase. For x, y ∈ D, x < y (x precedes y) if there exists an admissible path from x to y and x ∼ y if x < y and y < x. The tree of G in D is a quotient space D/ ∼. We provide an algorithm for the construction of this tree. In this algorithm, the restriction of G onto the 1-skeleton of D (the union of edges) is used. The problem of existence of admissible paths between states is solved constructively. The regions attainable by the admissible paths are described.