A.N. Gorban
Thermodynamic Tree: The Space of Admissible Paths
(594K, PDF)

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.