12-21 A.N. Gorban
Thermodynamic Tree: The Space of Admissible Paths (594K, PDF) Feb 11, 12
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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 and have a convex Lyapunov function $G$. An admissible path is a continuous curve along which $G$ does not increase. For $x,y \in D$, $x\succcurlyeq y$ ($x$ precedes $y$) if there exists an admissible path from $x$ to $y$ and $x\sim y$ if $x\succcurlyeq y$ and $y \succcurlyeq x$. The tree of $G$ in $D$ is a quotient space $D/\sim$. 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.

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