Thermodynamic Tree: The Space of Admissible Paths

TL;DR

This paper introduces a comprehensive framework for determining whether a thermodynamic transition between two states is permissible, based on the construction of a thermodynamic tree that encodes all admissible paths respecting conservation laws and monotonic decrease of thermodynamic potential.

Contribution

It develops necessary and sufficient conditions for thermodynamically admissible state transitions and provides an algorithm to construct the thermodynamic tree representing these paths.

Findings

The thermodynamic tree encodes all admissible state transitions.

The algorithm uses the restriction of G on the 1-skeleton of D.

Regions attainable by admissible paths are explicitly described.

Abstract

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)\geq 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 D and have a convex Lyapunov function G. An admissible path is a continuous curve along which $G$ does not increase. For x,y from D, x\geq y (x precedes y) if there exists an admissible path from x to y and x \sim y if x\geq y and y\geq 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.