A.N. Gorban
Thermodynamic Tree: The Space of Admissible Paths
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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.